Recent zbMATH articles in MSC 05https://zbmath.org/atom/cc/052021-11-25T18:46:10.358925ZWerkzeugBook review of: E. Brown and R. K. Guy, The unity of combinatoricshttps://zbmath.org/1472.000102021-11-25T18:46:10.358925Z"Calkin, Neil"https://zbmath.org/authors/?q=ai:calkin.neil-j"Mulcahy, Colm"https://zbmath.org/authors/?q=ai:mulcahy.colmReview of [Zbl 1458.05001].Book review of: S. Wagner and H. Wang, Introduction to chemical graph theoryhttps://zbmath.org/1472.000292021-11-25T18:46:10.358925Z"Rada, Juan"https://zbmath.org/authors/?q=ai:rada.juanReview of [Zbl 1409.92002].Prospects for a theory of decyclinghttps://zbmath.org/1472.030082021-11-25T18:46:10.358925Z"Litland, Jon Erling"https://zbmath.org/authors/?q=ai:litland.jon-erlingThe solutions presented in the author's [``Grounding, explanation, and the limit of internality'', Philos. Review 124, No. 4, 481--532 (2015; \url{doi:10.1215/00318108-3147011})] and \textit{K. Fine} [Notre Dame J. Formal Logic 51, 97--118 (2010; Zbl 1256.03034)] to a range of puzzles about what grounds what belong to the family of what the author refers to as ``decycling'' solutions. Puzzles of a self-referential nature, one of which is called (S1) in this paper and is used as a test case, ``present more serious challenges to such decycling solutions than has previously been realized. The goal of this paper is to examine to what extent these challenges can be met.'' First, a number of general constraints that any reasonable decycling solution has to meet are identified. Any solution that meets all the constraints agrees on the treatment of (S1). However, the existing decycling solutions in [loc. cit.] fail to meet some of those constraints. On the other hand, the author constructs a solution that meets all those constraints and shows that that particular solution is not only not unique, but that there are uncountably many distinct solutions that meet the constraints. This ``can be interpreted to mean that the notion of ground is indeterminate.''First-order definitions of subgraph isomorphism through the adjacency and order relationshttps://zbmath.org/1472.030262021-11-25T18:46:10.358925Z"Grigoryan, Oleg"https://zbmath.org/authors/?q=ai:grigoryan.oleg"Makarov, Mikhail"https://zbmath.org/authors/?q=ai:makarov.mikhail-v"Zhukovskii, Maksim"https://zbmath.org/authors/?q=ai:zhukovskii.maximGiven a graph \(F\), let \(\mathcal S(F)\) be the graphical property of having \(F\) as a (not necessarily induced) subgraph. If \(\sim\) is the vertex adjacency relation symbol, let \(\Sigma = \{ \sim, < \}\) be the signature for linearly ordered finite graphs. A first-order sentence \(\varphi\) of signature \(\Sigma\) defines \(\mathcal S(F)\) if, for any graph \(G\), and any linear ordering of \(G\)'s vertices, if \(G'\) is \(G\) expanded by that linear ordering, then \(G' \models \varphi\) iff \(G\) has \(F\) as a (not necessarily induced) subgraph. Let \(D_< (F)\) be the minimum quantifier depth of any \(\Sigma\)-sentence defining \(\mathcal S(F)\) and let \(W_< (F)\) be the minimum number of variables of any \(\Sigma\)-sentence defining \(\mathcal S(F)\).
The primary result of this paper is that if \(F\) is a tree of \(\ell\) vertices, then \(D_< (F) \leq \ell/2 + \lceil \log_2 (\ell + 2) \rceil - 1\); they also observe that \(W_< (F) \leq \ell/2 + 2\). In addition, using Fraïsse-Ehrenfeucht games, \(D_<(F)\) is computed for each graph \(F\) of order less than \(5\).
There are some typos, so readers should take care.Diagonalising an ultrafilter and preserving a $P$-pointhttps://zbmath.org/1472.030472021-11-25T18:46:10.358925Z"Mildenberger, Heike"https://zbmath.org/authors/?q=ai:mildenberger.heikeSummary: With Ramsey-theoretic methods we show: It is consistent that there is a forcing that diagonalises one ultrafilter over $\omega $ and preserves another ultrafilter.Representability of Lyndon-Maddux relation algebrashttps://zbmath.org/1472.030702021-11-25T18:46:10.358925Z"Alm, Jeremy F."https://zbmath.org/authors/?q=ai:alm.jeremy-fSummary: In 2016, the author et al. [Rev. Symb. Log. 9, No. 3, 511--521 (2016; Zbl 1392.03060)] defined relation algebras \(\mathfrak {L}(q,n)\) that generalize Roger Lyndon's relation algebras from projective lines, so that \(\mathfrak {L}(q,0)\) is a Lyndon algebra. In that paper, it was shown that if \(q>2304n^2+1\), then \(\mathfrak {L}(q,n)\) is representable, and if \(q<2n\), then \(\mathfrak {L}(q,n)\) is not representable. In the present paper, we reduced this gap by proving that if \(q\geq n(\log n)^{1+\varepsilon }\), then \(\mathfrak {L}(q,n)\) is representable.Irregularity in graphshttps://zbmath.org/1472.050012021-11-25T18:46:10.358925Z"Ali, Akbar"https://zbmath.org/authors/?q=ai:ali.akbar|ali.akbar.1"Chartrand, Gary"https://zbmath.org/authors/?q=ai:chartrand.gary"Zhang, Ping"https://zbmath.org/authors/?q=ai:zhang.ping.6|zhang.ping.1The book is relatively short (109 pages) but rich in topics. It deals with the less popular extreme of the regularity spectrum, as the title suggests. It has eight chapters: Introduction, Locally irregular graphs, \(F\)-irregular graphs, Irregularity strength, Rainbow mean index, Royal colorings, Traversable irregularity, Ascending subgraph, Decomposition.
Chapters 1--4 deal with vertex degree related types of irregularity and their generalizations (e.g., \(F\)-degree). Chapters 5--6 focus on irregular colorings, Chapter 7 describes irregular Eulerian and Hamiltonian-type walks, and the last chapter title is self-describing.
The book can serve as a reference source, especially for the less-known types of irregularity described in Chapters 5--7, or as a textbook for an undergraduate exploratory graph theory course or seminar. It contains a wealth of easy to understand open problems that can inspire enthusiastic students to turn their focus to further study of graph theory.Algebraic combinatorics. Translated from the Japanesehttps://zbmath.org/1472.050022021-11-25T18:46:10.358925Z"Bannai, Eiichi"https://zbmath.org/authors/?q=ai:bannai.eiichi"Bannai, Etsuko"https://zbmath.org/authors/?q=ai:bannai.etsuko"Ito, Tatsuro"https://zbmath.org/authors/?q=ai:ito.tatsuro"Tanaka, Rie"https://zbmath.org/authors/?q=ai:tanaka.rieThis book is the English translation of the book Introduction to Algebraic Combinatorics, which was published by Kyoritsu Shuppan in 2016 and written by the first three authors in Japanese. Later, the fourth author joined the team and translated the book in its current form in English. Bannai and Ito describe algebraic combinatorics as ``a group theory without groups'' or ``a character theoretical study of combinatorial objects''. The aim of this book is to ``pursue the study of combinatorics as an extension or a generalization of the study of finite permutation groups''. The early chapters of the book provides an accessible introduction to the subject for undergraduates and interested readers. The later chapters are suited for researchers in combinatorics and broader areas.
Chapter 1 is an introduction to classical combinatorics with selected subjects of study, such as graph theory and coding theory. Chapter 2 is an introduction to association schemes. Chapter 3 and 4 present codes and designs in connection to association schemes. Chapter 5 is on algebraic combinatorics on spheres. The book ends with Chapter 6 covering \(P\) and \(Q\)-polynomial schemes.Discrete encountershttps://zbmath.org/1472.050032021-11-25T18:46:10.358925Z"Bauer, Craig P."https://zbmath.org/authors/?q=ai:bauer.craig-p``This book provides a refreshing approach to discrete mathematics. The author blends traditional course topics and applications with historical context, pop culture reference, and open problems. It focuses on the historical development of the subject and provides fascinating details of the people behind the mathematics, along with their motivations, deepening readers' appreciation of mathematics. This unique book covers many of the same topics found in traditional textbooks, but does so in an alternative, entertaining style that better captures readers' attention. In addition to standard discrete mathematics material, the author shows the interplay between the discrete and the continuous and includes high-interest topics such as fractals, chaos theory, cellular automata, money-saving financial mathematics, and much more. Not only will readers gain a greater understanding of mathematics and its culture, they will also be encouraged to further explore the subject. Long lists of references at the end of each chapter make this easy.'' (from the presentation of the book).
\par Its chapters are the following: Continuous vs. discrete; Logic; Proof techniques; Practice with proofs; Set theory; Venn diagrams; The functional view of mathematics; The multiplication principle; Permutations; Combinations; Pascal and the arithmetic triangle; Stirling and Bell numbers; The basics of probability; The Fibonacci sequence; The tower of Hanoi; Population models; Financial mathematics (and more); More difference equations; Chaos theory and fractals; Cellular automata; Graph theory; Trees; Relations, partial orderings, and partitions; Index. Each of them ends with a set of exercises and references/further reading. The book also includes many illustrations and portraits from the history of mathematics and ends with a series of 39 colored illustrations. The text's narrative style is that of a popular book, not a dry textbook. Its multidisciplinary approach makes this nice book ideal for liberal arts mathematics classes, leisure reading, or as a reference for professors looking to supplement traditional courses.Restricted stacks as functionshttps://zbmath.org/1472.050042021-11-25T18:46:10.358925Z"Berlow, Katalin"https://zbmath.org/authors/?q=ai:berlow.katalinSummary: The stack sort algorithm has been the subject of extensive study over the years. In this paper we explore a generalized version of this algorithm where instead of avoiding a single decrease, the stack avoids a set \(T\) of permutations. We let \(s_T\) denote this map. We classify for which sets \(T\) the map \(s_T\) is bijective. A corollary to this answers a question of \textit{J.-L. Baril} et al. [Inf. Process. Lett. 171, Article ID 106138, 9 p. (2021; Zbl 07360091)] about stack sort composed with \(s_{\{ \sigma , \tau \}}\), known as the \((\sigma, \tau)\)-machine. This fully classifies for which \(\sigma\) and \(\tau\) the preimage of the identity under the \((\sigma, \tau)\)-machine is counted by the Catalan numbers. We also prove that the number of preimages of a permutation under the map \(s_T\) is bounded by the Catalan numbers, with a shift of indices. For \(T\) of size 1, we classify exactly when this bound is sharp. We also explore the periodic points and maximum number of preimages of various \(s_T\) for \(T\) containing two length 3 permutations.Preimages under the Queuesort algorithmhttps://zbmath.org/1472.050052021-11-25T18:46:10.358925Z"Cioni, Lapo"https://zbmath.org/authors/?q=ai:cioni.lapo"Ferrari, Luca"https://zbmath.org/authors/?q=ai:ferrari.lucaSummary: Following the footprints of what has been done with the algorithm \texttt{Stacksort}, we investigate the preimages of the map associated with a slightly less well known algorithm, called \texttt{Queuesort}. After having described an equivalent version of \texttt{Queuesort}, we provide a recursive description of the set of all preimages of a given permutation, which can be also translated into a recursive procedure to effectively find such preimages. We then deal with some enumerative issues. More specifically, we investigate the cardinality of the set of preimages of a given permutation, showing that all cardinalities are possible, except for 3. We also give exact enumeration results for the number of permutations having 0, 1 and 2 preimages. Finally, we consider the special case of those permutations \(\pi\) whose set of left-to-right maxima is the disjoint union of a prefix and a suffix of \(\pi \): we determine a closed formula for the number of preimages of such permutations, which involves two different incarnations of ballot numbers, and we show that our formula can be expressed as a linear combination of Catalan numbers.On the nature of four models of symmetric walks avoiding a quadranthttps://zbmath.org/1472.050062021-11-25T18:46:10.358925Z"Dreyfus, Thomas"https://zbmath.org/authors/?q=ai:dreyfus.thomas"Trotignon, Amélie"https://zbmath.org/authors/?q=ai:trotignon.amelieSummary: We study the nature of the generating series of some models of walks with small steps in the three quarter plane. More precisely, we restrict ourselves to the situation where the group is infinite, the kernel has genus one, and the step set is diagonally symmetric (i.e., with no steps in anti-diagonal directions). In that situation, after a transformation of the plane, we derive a quadrant-like functional equation. Among the four models of walks, we obtain, using difference Galois theory, that three of them have a differentially transcendental generating series, and one has a differentially algebraic generating series.On framed simple purely real Hurwitz numbershttps://zbmath.org/1472.050072021-11-25T18:46:10.358925Z"Kazarian, M. E."https://zbmath.org/authors/?q=ai:kazaryan.maxim-e"Lando, S. K."https://zbmath.org/authors/?q=ai:lando.sergei-k"Natanzon, S. M."https://zbmath.org/authors/?q=ai:natanzon.sergei-mOn some problems about ternary paths: a linear algebra approachhttps://zbmath.org/1472.050082021-11-25T18:46:10.358925Z"Prodinger, Helmut"https://zbmath.org/authors/?q=ai:prodinger.helmutSummary: Ternary paths consist of an upstep of one unit, a downstep of two units, never go below the \(x\)-axis, and return to the \(x\)-axis. This paper addresses the enumeration of partial ternary paths, ending at a given level \(i\), reading the path either from left-to-right or from right-to-left. Since the paths are not symmetric with respect to left vs. right, as classical Dyck paths, this leads to different results. The right-to-left enumeration is quite challenging, but leads at the end to very satisfying results. The methods are elementary (solving systems of linear equations). In this way, several conjectures left open in [\textit{N. T. Cameron}, Random walks, trees and extensions of Riordan group techniques. Washington, D.C.: Howard University (PhD Thesis) (2002)] could be successfully settled.Crossings and nestings over some Motzkin objects and \(q\)-Motzkin numbershttps://zbmath.org/1472.050092021-11-25T18:46:10.358925Z"Rakoyomamonjy, Paul M."https://zbmath.org/authors/?q=ai:rakoyomamonjy.paul-m"Andriantsoa, Sandrataniaina R."https://zbmath.org/authors/?q=ai:andriantsoa.sandrataniaina-rSummary: We examine the enumeration of certain Motzkin objects according to the numbers of crossings and nestings. With respect to continued fractions, we compute and express the distributions of the statistics of the numbers of crossings and nestings over three sets, namely the set of \(4321\)-avoiding involutions, the set of \(3412\)-avoiding involutions, and the set of \((321,3\bar{1}42)\)-avoiding permutations. To get our results, we exploit the bijection of \textit{P. Biane} [Eur. J. Comb. 14, No. 4, 277--284 (1993; Zbl 0784.05005)] restricted to the sets of \(4321\)- and \(3412\)-avoiding involutions which was characterized by \textit{M. Barnabei} et al. [Adv. Appl. Math. 47, No. 1, 102--115 (2011; Zbl 1225.05242)] and the bijection between \((321,3\bar{1}42)\)-avoiding permutations and Motzkin paths, presented by \textit{W. Y. C. Chen} et al. [J. Comb. 9, No. 2, Research paper R15, 13 p. (2003; Zbl 1023.05002)]. Furthermore, we manipulate the obtained continued fractions to get the recursion formulas for the polynomial distributions of crossings and nestings, and it follows that the results involve two new \(q\)-Motzkin numbers.Applications of constructed new families of generating-type functions interpolating new and known classes of polynomials and numbershttps://zbmath.org/1472.050102021-11-25T18:46:10.358925Z"Simsek, Yilmaz"https://zbmath.org/authors/?q=ai:simsek.yilmazSummary: The aim of this article is to construct some new families of generating-type functions interpolating a certain class of higher order Bernoulli-type, Euler-type, Apostol-type numbers, and polynomials. Applying the umbral calculus convention method and the shift operator to these functions, these generating functions are investigated in many different aspects such as applications related to the finite calculus, combinatorial analysis, the chordal graph, number theory, and complex analysis especially partial fraction decomposition of rational functions associated with Laurent expansion. By using the falling factorial function and the Stirling numbers of the first kind, we also construct new families of generating functions for certain classes of higher order Apostol-type numbers and polynomials, the Bernoulli numbers and polynomials, the Fubini numbers, and others. Many different relations among these generating functions, difference equation including the Eulerian numbers, the shift operator, minimal polynomial, polynomial of the chordal graph, and other applications are given. Moreover, further remarks and comments on the results of this paper are presented.On the number of derangements and derangements of prime power order of the affine general linear groupshttps://zbmath.org/1472.050112021-11-25T18:46:10.358925Z"Spiga, Pablo"https://zbmath.org/authors/?q=ai:spiga.pabloSummary: A derangement is a permutation that has no fixed points. In this paper, we are interested in the proportion of derangements of the finite affine general linear groups. We prove a remarkably simple and explicit formula for this proportion. We also give a formula for the proportion of derangements of prime power order. Both formulae rely on a result of independent interest on partitions: we determine the generating function for the partitions with \(m\) parts and with the \(k\)th largest part not \(k\), for every \(k\in \mathbb {N}\).Binomial series identities involving generalized harmonic numbershttps://zbmath.org/1472.050122021-11-25T18:46:10.358925Z"Wang, Xiaoyuan"https://zbmath.org/authors/?q=ai:wang.xiaoyuan"Chu, Wenchang"https://zbmath.org/authors/?q=ai:chu.wenchangLet \(R(k)\) be a rational function, \(\binom{2k}{k}\) the central binomial coefficient, and \(H_{k}(\alpha)\) the generalised harmonic number, defined for real \(\alpha\) such that \[ H_{n}(\alpha)=\sum_{k=1}^{n}\frac{1}{k\alpha^{k}}, \] with \(H_{0}(\alpha)=0.\)
In this paper, the authors consider sums of the form
\[
\sum_{k\geq 1}H_{k}(\alpha)\frac{(4\beta)}{(_{k}^{2k})}R(k)
\]
with \(|\alpha|\geq 1\) and \(|\beta|<1,\) which they evaluate by applying Abel's lemma on summation by parts. Their results are motivated by recent progress in this area and span nine classes of infinite series, such as those given in Theorem~1, stated below.
Theorem 1. Let \(\alpha\) and \(\beta\) be two real numbers subject to \(|\alpha|\geq 1\) and \(|\beta|<1\). Then for \( y=\sqrt{\beta/\alpha}\), the following summation formula holds: \[ \sum_{k\geq 1}H_{k}(\alpha)\frac{{(4\beta)^{k}}(1+2k-2\beta k-2\beta)}{(2k+1)\binom{2k}{k}}=\frac{2y\arcsin y}{\sqrt{1-y^{2}}}. \]Looking for a new version of Gordon's identitieshttps://zbmath.org/1472.050132021-11-25T18:46:10.358925Z"Afsharijoo, Pooneh"https://zbmath.org/authors/?q=ai:afsharijoo.poonehSummary: We give a commutative algebra viewpoint on Andrews recursive formula for the partitions appearing in Gordon's identities, which are a generalization of Rogers-Ramanujan identities. Using this approach and differential ideals, we conjecture a family of partition identities which extend Gordon's identities. This family is indexed by \(r\geq 2\). We prove the conjecture for \(r=2\) and \(r=3\).On conjectures concerning the smallest part and missing parts of integer partitionshttps://zbmath.org/1472.050142021-11-25T18:46:10.358925Z"Binner, Damanvir Singh"https://zbmath.org/authors/?q=ai:binner.damanvir-singh"Rattan, Amarpreet"https://zbmath.org/authors/?q=ai:rattan.amarpreetSummary: For positive integers \(L\geq 3\) and \(s\), \textit{A. Berkovich} and \textit{A. K. Uncu} [ibid. 23, No. 2, 263--284 (2019; Zbl 1415.05009)] conjectured an inequality between the sizes of two closely related sets of partitions whose parts lie in the interval \(\{s,\dots,L+s\}\). Further restrictions are placed on the sets by specifying impermissible parts as well as a minimum part. The authors proved their conjecture for the cases \(s=1\) and \(s=2\). In the present article, we prove their conjecture for general \(s\) by proving a stronger theorem. We also prove other related conjectures found in the same paper.Distribution properties for \(t\)-hooks in partitionshttps://zbmath.org/1472.050152021-11-25T18:46:10.358925Z"Craig, William"https://zbmath.org/authors/?q=ai:craig.william"Pun, Anna"https://zbmath.org/authors/?q=ai:pun.anna-yingSummary: Partitions, the partition function \(p(n)\), and the hook lengths of their Ferrers-Young diagrams are important objects in combinatorics, number theory, and representation theory. For positive integers \(n\) and \(t\), we study \(p_t^{\mathrm{e}}(n)\) (resp. \(p_t^{\mathrm{o}}(n))\), the number of partitions of \(n\) with an even (resp. odd) number of \(t\)-hooks. We study the limiting behavior of the ratio \(p_t^{\mathrm{e}}(n)/p(n)\), which also gives \(p_t^{\mathrm{o}}(n)/p(n)\), since \(p_t^{\mathrm{e}}(n)+p_t^{\mathrm{o}}(n)=p(n)\). For even \(t\), we show that
\[
\lim\limits_{n\rightarrow\infty}\frac{p_t^{\mathrm{e}}(n)}{p(n)}=\frac{1}{2},
\]
and for odd \(t\), we establish the non-uniform distribution
\[
\lim\limits_{n\rightarrow\infty}\frac{p^{\mathrm{e}}_t(n)}{p(n)}=
\begin{cases}
\frac{1}{2}+\frac{1}{2^{(t+1)/2}} & \text{ if }2\mid n,\\
\frac{1}{2}-\frac{1}{2^{(t+1)/2}} & \text{ otherwise.}
\end{cases}
\]
Using the Rademacher circle method, we find an exact formula for \(p_t^{\mathrm{e}}(n)\) and \(p_t^{\mathrm{o}}(n)\), and this exact formula yields these distribution properties for large \(n\). We also show that for sufficiently large \(n\), the sign of \(p_t^{\mathrm{e}}(n)-p_t^{\mathrm{o}}(n)\) is periodic.Statistical structure of concave compositionshttps://zbmath.org/1472.050162021-11-25T18:46:10.358925Z"Dalal, Avinash J."https://zbmath.org/authors/?q=ai:dalal.avinash-j"Lohss, Amanda"https://zbmath.org/authors/?q=ai:lohss.amanda"Parry, Daniel"https://zbmath.org/authors/?q=ai:parry.daniel-tSummary: In this paper, we study concave compositions, an extension of partitions that were considered by \textit{G. E. Andrews} et al. [Algebra Number Theory 7, No. 9, 2103--2139 (2013; Zbl 1282.05016)]. They presented several open problems regarding the statistical structure of concave compositions including the distribution of the perimeter and tilt, the number of summands, and the shape of the graph of a typical concave composition. We present solutions to these problems by applying Fristedt's conditioning device on the uniform measure.Enumerating coloured partitions in 2 and 3 dimensionshttps://zbmath.org/1472.050172021-11-25T18:46:10.358925Z"Davison, Ben"https://zbmath.org/authors/?q=ai:davison.ben"Ongaro, Jared"https://zbmath.org/authors/?q=ai:ongaro.jared"Szendrői, Balázs"https://zbmath.org/authors/?q=ai:szendroi.balazsSummary: We study generating functions of ordinary and plane partitions coloured by the action of a finite subgroup of the corresponding special linear group. After reviewing known results for the case of ordinary partitions, we formulate a conjecture concerning a basic factorisation property of the generating function of coloured plane partitions that can be thought of as an orbifold analogue of a conjecture of \textit{D. Maulik} et al. [Compos. Math. 142, No. 5, 1263--1285 (2006; Zbl 1108.14046)], now a theorem, in three-dimensional Donaldson-Thomas theory. We study natural quantisations of the generating functions arising from geometry, discuss a quantised version of our conjecture, and prove a positivity result for the quantised coloured plane partition function under a geometric assumption.Biases in integer partitionshttps://zbmath.org/1472.050182021-11-25T18:46:10.358925Z"Kim, Byungchan"https://zbmath.org/authors/?q=ai:kim.byungchan"Kim, Eunmi"https://zbmath.org/authors/?q=ai:kim.eunmiSummary: We show that there are biases in the number of appearances of the parts in two residue classes in the set of ordinary partitions. More precisely, let \(p_{j,k,m} (n)\) be the number of partitions of \(n\) such that there are more parts congruent to \(j \pmod m\) than parts congruent to \(k \pmod m\) for \(m \geq 2\). We prove that \(p_{1,0,m} (n)\) is in general larger than \(p_{0,1,m} (n)\). We also obtain asymptotic formulas for \(p_{1,0,m}(n)\) and \(p_{0,1,m}(n)\) for \(m \geq 2\).Asking questions to determine the product of circularly arranged numbershttps://zbmath.org/1472.050192021-11-25T18:46:10.358925Z"Sane, Sharad S."https://zbmath.org/authors/?q=ai:sane.sharad-sSummary: Fix positive integers \(k\) and \(n\) with \(k \leq n\). Numbers \(x_0, x_1, x_2, \ldots , x_{n - 1}\), each equal to \(\pm{1} \), are cyclically arranged (so that \(x_0\) follows \(x_{n - 1} \)) in that order. The problem is to find the product \(P = x_0x_1 \cdots x_{n - 1}\) of all \(n\) numbers by asking the smallest number of questions of the type \(Q_i\): what is \(x_ix_{i + 1}x_{i + 2} \cdots x_{i+ k -1}\)? (where all the subscripts are read modulo \(n\)). This paper studies the problem and some of its generalisations.On double sum generating functions in connection with some classical partition theoremshttps://zbmath.org/1472.050202021-11-25T18:46:10.358925Z"Uncu, Ali Kemal"https://zbmath.org/authors/?q=ai:uncu.ali-kemalSummary: We focus on writing double sum representations of the generating functions for the number of partitions satisfying some gap conditions. Some example sets of partitions to be considered are partitions into distinct parts and partitions that satisfy the gap conditions of the Rogers-Ramanujan, Göllnitz-Gordon, and little Göllnitz theorems. We refine our representations by imposing a bound on the largest part and find finite analogues of these new representations. These refinements lead to many \(q\)-series and polynomial identities. Additionally, we present a different construction and a double sum representation for the products similar to the ones that appear in the Rogers-Ramanujan identities.Log-behavior for some sequences related to Cauchy numbers of two kindshttps://zbmath.org/1472.050212021-11-25T18:46:10.358925Z"Zhao, Feng-Zhen"https://zbmath.org/authors/?q=ai:zhao.fengzhenSummary: In this paper, we discuss log-behavior for some sequences related to Cauchy numbers of two kinds. For Cauchy numbers of two kinds \(\{a_n\}_{n\ge 0}\) and \(\{b_n\}_{n\ge 0}\), we prove that two sequences \(\{|a_n|\}_{n\ge 1}\) and \(\{b_n\}_{n\ge 0}\) are 2-log-convex. In addition, we show that sequences \(\{|a_{n+1}|-|a_n|\}_{n\ge 2}\), \(\{b_{n+1}-b_n\}_{n\ge 1}\) are log-convex, and \(\{|a_n|\}_{n\ge 1}\) and \(\{b_n\}_{n\ge 0}\) are ratio log-concave.On some \(q\)-series identities related to a generalized divisor function and their implicationshttps://zbmath.org/1472.050222021-11-25T18:46:10.358925Z"Gupta, Rajat"https://zbmath.org/authors/?q=ai:gupta.rajat"Kumar, Rahul"https://zbmath.org/authors/?q=ai:kumar.rahulSummary: In this article, a \(q\)-series examined by \textit{J. C. Kluyver} [Wiskundige Opgaven 1919, 92--93 (1919; JFM 47.0212.07)] and \textit{K. Uchimura} [J. Comb. Theory, Ser. A 31, 131--135 (1981; Zbl 0473.05006)] is generalized. This allows us to find generalization of the identities in the random acyclic digraph studied by \textit{K. Simon} et al. [``On the distribution of the transitive closure in a random acyclic digraph'', Lect. Notes Comput. Sci. 726, 345--356 (1993; \url{doi:10.1007/3-540-57273-2_69})]. As one of the corollaries of our main theorem, we get results of \textit{K. Dilcher} [Discrete Math. 145, No. 1--3, 83--93 (1995; Zbl 0834.05005)] and \textit{G. E. Andrews} et al. [SIAM J. Discrete Math. 10, No. 1, 41--56 (1997; Zbl 0871.05049)]. This main theorem involves a surprising new generalization of the divisor function \(\sigma_s(n)\), which we denote by \(\sigma_{s , z}(n)\). Analytic properties of \(\sigma_{s , z}(n)\) are also studied. As a special case of one of our theorem we obtain a result from a recent paper of \textit{K. Bringmann} and \textit{C. Jennings-Shaffer} [Discrete Math. 343, No. 10, Article ID 112019, 6 p. (2020; Zbl 1445.05018)].On the Askey-Wilson polynomials and a \(q\)-beta integralhttps://zbmath.org/1472.050232021-11-25T18:46:10.358925Z"Liu, Zhi-Guo"https://zbmath.org/authors/?q=ai:liu.zhiguo|liu.zhi-guoSummary: A proof of the orthogonality relation for the Askey-Wilson polynomials is given by using a generating function for the Askey-Wilson polynomials and the uniqueness of a rational function expansion. We further use the orthogonality relation for the Askey-Wilson polynomials and a \(q\)-series transformation formula to evaluate a general \(q\)-beta integral with eight parameters. The integrand of this \(q\)-beta integral is the product of two terminating \(_5\phi_4\) series and the value is a \(_{10}\phi_9\) series.Relations between the fractional operators in \(q\)-calculushttps://zbmath.org/1472.050242021-11-25T18:46:10.358925Z"Silvestrov, Sergei"https://zbmath.org/authors/?q=ai:silvestrov.sergei-d"Rajković, Predrag M."https://zbmath.org/authors/?q=ai:rajkovic.predrag-m"Marinković, Sladjana D."https://zbmath.org/authors/?q=ai:marinkovic.sladjana-d"Stanković, Miomir S."https://zbmath.org/authors/?q=ai:stankovic.miomir-sSummary: In this survey paper, we will consider the fractional operators in \(q\)-calculus. Starting from the fractional versions of \(q\)-Pochhammer symbol, we generalize the notions of the fractional \(q\)-integral and \(q\)-derivative by introducing variable lower bound of integration. We discuss their properties, describe relations which connect them, and illustrate notions and results with examples and counterexamples.
For the entire collection see [Zbl 1467.16001].Flag-transitive point-primitive non-symmetric \(2\text{-}(v,k,2)\) designs with alternating soclehttps://zbmath.org/1472.050252021-11-25T18:46:10.358925Z"Liang, Hongxue"https://zbmath.org/authors/?q=ai:liang.hongxue"Zhou, Shenglin"https://zbmath.org/authors/?q=ai:zhou.shenglinSummary: We prove that if \(\mathcal{D}\) is a non-trivial non-symmetric \(2\text{-}(v,k,2)\) design admitting a flag-transitive point-primitive automorphism group \(G\) with \(\mathrm{Soc}(G)=A_n\) for \(n\geq5\), then \(\mathcal{D}\) is a \(2\text{-}(6,3,2)\) or \(2\text{-}(10,4,2)\) design.Constructions of doubly resolvable Steiner quadruple systemshttps://zbmath.org/1472.050262021-11-25T18:46:10.358925Z"Meng, Zhaoping"https://zbmath.org/authors/?q=ai:meng.zhaoping"Zhang, Bin"https://zbmath.org/authors/?q=ai:zhang.bin.2|zhang.bin.1|zhang.bin.3|zhang.bin.4"Wu, Zhanggui"https://zbmath.org/authors/?q=ai:wu.zhangguiA Steiner quadruple system (SQS\((v)\)) is a design with block size 4 in which every set of three points lies in exactly one block. This paper gives two general constructions for doubly resolvable SQS\((v)\)s; one obtains a doubly resolvable SQS\((2v)\) from a doubly resolvable SQS\((v)\), and the other makes use of doubly resolvable candelabra quadruple systems. Using these, doubly resolvable SQS\((v)\)s are obtained for \(m \in \{1,5,7,11,17,26\}\) and \(n\) a positive integer, whenever $v$ is of one of the following forms: $m \cdot 2^n$ with $n \geq 2$, $(m,n) \neq (1,3)$, and $(3m \cdot 2^n) - 16$ with $n \geq 4$, $(m,n) \neq (1,5)$.List colorings count rokudoku-pair squareshttps://zbmath.org/1472.050272021-11-25T18:46:10.358925Z"Carrigan, Braxton"https://zbmath.org/authors/?q=ai:carrigan.braxton"Hammer, James"https://zbmath.org/authors/?q=ai:hammer.james-m"Lorch, John"https://zbmath.org/authors/?q=ai:lorch.john-d"Lorch, Robert"https://zbmath.org/authors/?q=ai:lorch.robert"Owens, Caitlin"https://zbmath.org/authors/?q=ai:owens.caitlinSummary: A rokudoku-pair square is an order-6 sudoku Latin square for both \(2\times 3\) and \(3\times 2\) tiling regions simultaneously. We count the distinct rokudoku-pair squares as well as orbits under the action of a suitable group. Our arguments employ group actions and list colorings of graphs. As an application we determine which rokudoku-pair squares are based on groups.Balancedly splittable orthogonal designs and equiangular tight frameshttps://zbmath.org/1472.050282021-11-25T18:46:10.358925Z"Kharaghani, Hadi"https://zbmath.org/authors/?q=ai:kharaghani.hadi"Pender, Thomas"https://zbmath.org/authors/?q=ai:pender.thomas"Suda, Sho"https://zbmath.org/authors/?q=ai:suda.shoSummary: The concept of balancedly splittable orthogonal designs is introduced along with a recursive construction. As an application, equiangular tight frames over the real, complex, and quaternions meeting the Delsarte-Goethals-Seidel upper bound are obtained.Roman and Vatican crossover designshttps://zbmath.org/1472.050292021-11-25T18:46:10.358925Z"Ollis, M. A."https://zbmath.org/authors/?q=ai:ollis.m-aSummary: Latin squares with a balance property among adjacent pairs of symbols -- being ``Roman'' or ``row-complete'' -- have long been used as uniform crossover designs with the number of treatments, periods and subjects all equal. This has been generalized in two ways: to crossover designs with more subjects and to balance properties at greater distances. We consider both of these simultaneously, introducing and constructing Vatican designs: these have \(\ell t\) subjects, \(t\) periods and treatments, and, for each \(d\) in the range \(1\le d<t\), the number of times that any subject receives treatment \(j\) exactly \(d\) periods after receiving treatment \(i\) is at most \(\ell\). Results include showing the existence of Vatican designs when \(4\le t\le 14\) and \(\ell>1\), and when \(t\in\{3,15\}\) and \(\ell\) is even.Codes, cubes, and graphical designshttps://zbmath.org/1472.050302021-11-25T18:46:10.358925Z"Babecki, Catherine"https://zbmath.org/authors/?q=ai:babecki.catherineSummary: Graphical designs are an extension of spherical designs to functions on graphs. We connect linear codes to graphical designs on cube graphs, and show that the Hamming code in particular is a highly effective graphical design. We show that even in highly structured graphs, graphical designs are distinct from the related concepts of extremal designs, maximum stable sets in distance graphs, and \(t\)-designs on association schemes.The finite matroid-based valuation conjecture is falsehttps://zbmath.org/1472.050312021-11-25T18:46:10.358925Z"Tran, Ngoc Mai"https://zbmath.org/authors/?q=ai:tran.ngoc-maiOn balanced \(( \mathbb{Z}_{4 u} \times \mathbb{Z}_{8 v}, \{4, 5 \}, 1)\) difference packingshttps://zbmath.org/1472.050322021-11-25T18:46:10.358925Z"Zhao, Hengming"https://zbmath.org/authors/?q=ai:zhao.hengming"Qin, Rongcun"https://zbmath.org/authors/?q=ai:qin.rongcun"Wu, Dianhua"https://zbmath.org/authors/?q=ai:wu.dianhuaSummary: Let \(K\) be a set of positive integers and let \(G\) be an additive group. A \((G, K, 1)\) difference packing is a set of subsets of \(G\) with sizes from \(K\) whose list of differences covers every element of \(G\) at most once. It is balanced if the number of blocks of size \(k \in K\) does not depend on \(k\). In this paper, we determine a balanced \(( \mathbb{Z}_{4 u} \times \mathbb{Z}_{8 v}, \{4, 5 \}, 1)\) difference packing of the largest possible size whenever \(uv\) is odd. The corresponding optimal balanced \((4 u, 8 v, \{4, 5 \}, 1)\) optical orthogonal signature pattern codes are also obtained.Concatenation arguments and their applications to polyominoes and polycubeshttps://zbmath.org/1472.050332021-11-25T18:46:10.358925Z"Barequet, Gill"https://zbmath.org/authors/?q=ai:barequet.gill"Ben-Shachar, Gil"https://zbmath.org/authors/?q=ai:ben-shachar.gil"Osegueda, Martha Carolina"https://zbmath.org/authors/?q=ai:osegueda.martha-carolinaSummary: In this paper, we develop a method for setting lower and upper bounds on growth constants of polyominoes and polycubes whose enumerating sequences are so-called quasi sub- or super-multiplicative. The method is based on concatenation arguments, applied directly or recursively. Inter alia, we demonstrate the method on general polycubes, tree polyominoes and polycubes, and convex polyominoes.Existence of a spanning tree having small diameterhttps://zbmath.org/1472.050342021-11-25T18:46:10.358925Z"Egawa, Yoshimi"https://zbmath.org/authors/?q=ai:egawa.yoshimi"Furuya, Michitaka"https://zbmath.org/authors/?q=ai:furuya.michitaka"Matsumura, Hajime"https://zbmath.org/authors/?q=ai:matsumura.hajimeSummary: In this paper, we prove that for a sufficiently large integer \(d\) and a connected graph \(G\), if \(| V(G) | < \frac{ ( d + 2 ) ( \delta ( G ) + 1 )}{ 3} \), then there exists a spanning tree \(T\) of \(G\) such that \(\operatorname{diam}(T) \leq d\).Trestles in the squares of graphshttps://zbmath.org/1472.050352021-11-25T18:46:10.358925Z"Kabela, Adam"https://zbmath.org/authors/?q=ai:kabela.adam"Teska, Jakub"https://zbmath.org/authors/?q=ai:teska.jakubSummary: We show that the square of every connected \(S( K_{1 , 4})\)-free graph satisfying a matching condition has a 2-connected spanning subgraph of maximum degree at most 3. Furthermore, we characterise trees whose square has a 2-connected spanning subgraph of maximum degree at most \(k\). This generalises the results on \(S( K_{1 , 3})\)-free graphs of \textit{G. Hendry} and \textit{W. Vogler} [J. Graph Theory 9, No. 4, 535--537 (1985; Zbl 0664.05038)] and \textit{F. Harary} and \textit{A. Schwenk} [Mathematika 18, 138--140 (1971; Zbl 0221.05052)], respectively.Bipartite biregular Moore graphshttps://zbmath.org/1472.050362021-11-25T18:46:10.358925Z"Araujo-Pardo, G."https://zbmath.org/authors/?q=ai:araujo-pardo.gabriela"Dalfó, C."https://zbmath.org/authors/?q=ai:dalfo.cristina"Fiol, M. A."https://zbmath.org/authors/?q=ai:fiol.miquel-angel"López, N."https://zbmath.org/authors/?q=ai:lopez.nachoSummary: A bipartite graph \(G = (V, E)\) with \(V = V_1 \cup V_2\) is biregular if all the vertices of a stable set \(V_i\) have the same degree \(r_i\) for \(i = 1, 2\). In this paper, we give an improved new Moore bound for an infinite family of such graphs with odd diameter. This problem was introduced in \textit{J. L. A. Yebra} et al. [Ars Comb. 16-A, 131--139 (1983; Zbl 0539.05039)].
Besides, we propose some constructions of bipartite biregular graphs with diameter \(d\) and large number of vertices \(N( r_1, r_2; d)\), together with their spectra. In some cases of diameters \(d = 3\), 4, and 5, the new graphs attaining the Moore bound are unique up to isomorphism.Many cliques with few edges and bounded maximum degreehttps://zbmath.org/1472.050372021-11-25T18:46:10.358925Z"Chakraborti, Debsoumya"https://zbmath.org/authors/?q=ai:chakraborti.debsoumya"Chen, Da Qi"https://zbmath.org/authors/?q=ai:chen.da-qiSummary: Generalized Turán problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem, maximizing the number of cliques of a fixed order in a graph with fixed number of vertices and bounded maximum degree, was recently completely resolved by \textit{Z. Chase} [Adv. Comb. 2020, Paper No. 10, 5 p. (2020; Zbl 1450.05040)]. \textit{R. Kirsch} and \textit{A. J. Radcliffe} [Electron. J. Comb. 26, No. 2, Research Paper P2.36, 23 p. (2019; Zbl 1414.05161)] raised a natural variant of this problem where the number of edges is fixed instead of the number of vertices. In this paper, we determine the maximum number of cliques of a fixed order in a graph with fixed number of edges and bounded maximum degree, resolving a conjecture by Kirsch and Radcliffe [loc. cit.]. We also give a complete characterization of the extremal graphs.Centrality-friendship paradoxes: when our friends are more important than ushttps://zbmath.org/1472.050382021-11-25T18:46:10.358925Z"Higham, Desmond J."https://zbmath.org/authors/?q=ai:higham.desmond-jSummary: The friendship paradox states that, on average, our friends have more friends than we do. In network terms, the average degree over the nodes can never exceed the average degree over the neighbours of nodes. This effect, which is a classic example of sampling bias, has attracted much attention in the social science and network science literature, with variations and extensions of the paradox being defined, tested and interpreted. Here, we show that a version of the paradox holds rigorously for eigenvector centrality: on average, our friends are more important than us. We then consider general matrix-function centrality, including Katz centrality, and give sufficient conditions for the paradox to hold. We also discuss which results can be generalized to the cases of directed and weighted edges. In this way, we add theoretical support for a field that has largely been evolving through empirical testing.On ABC Estrada index of graphshttps://zbmath.org/1472.050392021-11-25T18:46:10.358925Z"Li, Shuchao"https://zbmath.org/authors/?q=ai:li.shuchao"Wang, Lu"https://zbmath.org/authors/?q=ai:wang.lu.4|wang.lu.2|wang.lu.3|wang.lu.1|wang.lu"Zhang, Huihui"https://zbmath.org/authors/?q=ai:zhang.huihuiSummary: Let \(G\) be a graph with vertex set \(V_G = \{ v_1, v_2, \ldots, v_n \}\) and edge set \(E_G\), and let \(d_i\) be the degree of the vertex \(v_i\). The ABC matrix of \(G\) has the value \(\sqrt{ ( d_i + d_j - 2 ) / ( d_i d_j )}\) if \(v_i v_j \in E_G\), and 0 otherwise, as its \((i, j)\)-entry. Let \(\gamma_1, \gamma_2, \ldots, \gamma_n\) be the eigenvalues of the ABC matrix of \(G\) in a non-increasing order. Then the ABC Estrada index of \(G\) is defined as \(\operatorname{EE}_{\mathrm{ABC}}(G) = \sum_{i = 1}^n e^{\gamma_i}\) and the ABC energy of \(G\) is defined as \(\operatorname{E}_{\mathrm{ABC}}(G) = \sum_{i = 1}^n | \gamma_i |\). In this paper, some explicit bounds for the ABC Estrada index of graphs concerning the number of vertices, the number of edges, the maximum degree and the minimum degree, are established. Moreover, some bounds for the ABC Estrada index involving the ABC energy of graphs are also presented. All the corresponding extremal graphs are characterized respectively.Extremal values for the variation of the Randić index of bicyclic graphshttps://zbmath.org/1472.050402021-11-25T18:46:10.358925Z"Lv, Jian-Bo"https://zbmath.org/authors/?q=ai:lv.jianbo"Li, Jianxi"https://zbmath.org/authors/?q=ai:li.jianxiSummary: Let \(G\) be a connected graph of order \(n\). The variation of the Randić index of \(G\) is defined as \[R'(G) = \sum_{uv \in E(G)} \frac{1}{\max\{d(u), d(v)\}},\] where the summation goes over all edges \(uv\) of \(G\) and \(d(u)\) is the degree of the vertex \(u\) in \(G\). In this paper, among all bicyclic graphs of order \(n\), the minimum and maximum values for \(R^\prime\) are determined, respectively.Wiener index and addressing of the total graphhttps://zbmath.org/1472.050412021-11-25T18:46:10.358925Z"Taleshani, M. Gholamnia"https://zbmath.org/authors/?q=ai:taleshani.m-gholamnia"Abbasi, Ahmad"https://zbmath.org/authors/?q=ai:abbasi.ahmadSummary: \textit{R. L. Graham} and \textit{H. O. Pollak} [Lect. Notes Math. 303, 99--110 (1972; Zbl 0251.05123)] showed that the vertices of any connected graph \(G\) can be assigned \(t\)-tuples with entries in \(\{0,a,b\}\), called addresses, such that the distance between any two vertices can be determined from their addresses. In this paper we determine the minimum value of such \(t\), called squashed-cube dimension for the total graph \(T(\Gamma(\mathbb{Z}_{2^{n}p^{m}}))\) where \(n,m \ge 1\) and \(p \ge 3\) is a prime number.Symmetrical 2-extensions of the 3-dimensional grid. I.https://zbmath.org/1472.050422021-11-25T18:46:10.358925Z"Kostousov, Kirill"https://zbmath.org/authors/?q=ai:kostousov.kirill-victorovichSummary: For a positive integer \(d\), a connected graph \(\Gamma\) is a symmetrical 2-extension of the \(d\)-dimensional grid \(\Lambda^d\) if there exists a vertex-transitive group \(G\) of automorphisms of \(\Gamma\) and its imprimitivity system \(\sigma\) with blocks of size 2 such that there exists an isomorphism \(\varphi\) of the quotient graph \(\Gamma/\sigma\) onto \(\Lambda^d\). The tuple \((\Gamma G, \sigma, \varphi)\) with specified components is called a realization of the symmetrical 2-extension \(\Gamma\) of the grid \(\Lambda^d\). Two realizations \((\Gamma_1, G_1, \sigma_1, \varphi_1)\) and \((\Gamma_2, G_2, \sigma_2, \varphi_2)\) are called equivalent if there exists an isomorphism of the graph \(\Gamma_1\) onto \(\Gamma_2\) which maps \(\sigma_1\) onto \(\sigma_2\). \textit{V. I. Trofimov} [Proc. Steklov Inst. Math. 285, S169--S182 (2014; Zbl 1302.05124); translation from Tr. Inst. Mat. Mekh. (Ekaterinburg) 19, No. 3, 290--303 (2013)] proved that, up to equivalence, there are only finitely many realizations of symmetrical 2-extensions of \(\Lambda^d\) for each positive integer \(d\). \textit{E. A. Konovalchik} and \textit{K. V. Kostousov} [``Symmetrical 2-extensions of a 2-dimensional grid. I'', Trudy Inst. Mat. i Mekh. UrO RAN 22, No. 1, 159--179 (2016), \url{http://www.mathnet.ru/links/8e4d57cb5ed95465204b3325cc0e3583/timm1269.pdf}] found all, up to equivalence, realizations of symmetrical 2-extensions of the grid \(\Lambda^2\). In this work we found all, up to equivalence, realizations \(( \Gamma, G, \sigma, \varphi)\) of symmetrical 2-extensions of the grid \(\Lambda^3\) for which only the trivial automorphism of \(\Gamma\) preserves all blocks of \(\sigma \). Namely we prove that there are 5573 such realizations, and that among corresponding graphs \(\Gamma\) there are 5350 pairwise non-isomorphic.Cuts in undirected graphs. Ihttps://zbmath.org/1472.050432021-11-25T18:46:10.358925Z"Sharifov, F."https://zbmath.org/authors/?q=ai:sharifov.firdovsi|sharifov.f-a"Hulianytskyi, L."https://zbmath.org/authors/?q=ai:hulianytskyi.l-f|hulianytskyi.leonidSummary: This part of the paper analyzes new properties of cuts in undirected graphs, presents various models for the maximum cut problem, based on the established correspondence between the cuts in this graph and the specific bases of the extended polymatroid associated with this graph. With respect to the model, formulated as maximization of the convex function on the compact set (extended polymatroid), it was proved that the objective function has the same value at any local and global maxima, i.e., to solve the maximum cut problem, it will suffice to find a base of the extended polymatroid as a local or global maximum of the objective function.Sufficient conditions for planar graphs without 4-cycles and 5-cycles to be 2-degeneratehttps://zbmath.org/1472.050442021-11-25T18:46:10.358925Z"Sittitrai, Pongpat"https://zbmath.org/authors/?q=ai:sittitrai.pongpat"Nakprasit, Kittikorn"https://zbmath.org/authors/?q=ai:nakprasit.kittikornSummary: A graph \(G\) is \(k\)-degenerate if every subgraph of \(G\) has a vertex of degree at most \(k\). It is known that every planar graph of girth 6 (equivalently, without 3-, 4-, and 5-cycles) is 2-degenerate. On the other hand, there exist planar graphs without 4 and 5-cycles such as a truncated dodecahedral graph that are not 2-degenerate. Furthermore, a truncated dodecahedral graph also contains none of 6-, 7-, 8-, and 9-cycles. This motivates us to find sufficient conditions for planar graphs without 4-cycles and 5-cycles to be 2-degenerate.
In this work, we investigate the degeneracy of planar graphs without 4-, 5-, \(j\)-, and \(k\)-cycles where \(j \in \{6, 7, 8, 9 \}\) and \(k \in \{10, 11 \}\). For each \(j \in \{6, 9 \}\), we give an example of a 3-regular planar graph without 4-, 5-, \(j\)-, 10-, and 11-cycles. In contrast, we prove that every planar graph without 4-, 5-, \(j\)-, and \(k\)-cycles is 2-degenerate for each \(j \in \{7, 8 \}\) and \(k \in \{10, 11 \}\).The idemetric property: when most distances are (almost) the samehttps://zbmath.org/1472.050452021-11-25T18:46:10.358925Z"Barmpalias, George"https://zbmath.org/authors/?q=ai:barmpalias.george"Huang, Neng"https://zbmath.org/authors/?q=ai:huang.neng"Lewis-Pye, Andrew"https://zbmath.org/authors/?q=ai:lewis-pye.andrew-e-m"Li, Angsheng"https://zbmath.org/authors/?q=ai:li.angsheng"Li, Xuechen"https://zbmath.org/authors/?q=ai:li.xuechen"Pan, Yicheng"https://zbmath.org/authors/?q=ai:pan.yicheng"Roughgarden, Tim"https://zbmath.org/authors/?q=ai:roughgarden.timSummary: We introduce the \textit{idemetric} property, which formalizes the idea that most nodes in a graph have similar distances between them, and which turns out to be quite standard amongst small-world network models. Modulo reasonable sparsity assumptions, we are then able to show that a strong form of idemetricity is actually equivalent to a very weak expander condition (PUMP). This provides a direct way of providing short proofs that small-world network models such as the Watts-Strogatz model are strongly idemetric (for a wide range of parameters), and also provides further evidence that being idemetric is a common property. We then consider how satisfaction of the idemetric property is relevant to algorithm design. For idemetric graphs, we observe, for example, that a single breadth-first search provides a solution to the all-pairs shortest paths problem, so long as one is prepared to accept paths which are of stretch close to 2 with high probability. Since we are able to show that Kleinberg's model is idemetric, these results contrast nicely with the well known negative results of Kleinberg concerning efficient decentralized algorithms for finding short paths: for precisely the same model as Kleinberg's negative results hold, we are able to show that very efficient (and decentralized) algorithms exist if one allows for reasonable preprocessing. For deterministic distributed routing algorithms we are also able to obtain results proving that less routing information is required for idemetric graphs than in the worst case in order to achieve stretch less than 3 with high probability: while \(\Omega (n^2)\) routing information is required in the worst case for stretch strictly less than 3 on almost all pairs, for idemetric graphs the total routing information required is \(O(n\) log \((n))\).The diameter and radius of radially maximal graphshttps://zbmath.org/1472.050462021-11-25T18:46:10.358925Z"Qiao, Pu"https://zbmath.org/authors/?q=ai:qiao.pu"Zhan, Xingzhi"https://zbmath.org/authors/?q=ai:zhan.xingzhiSummary: A graph is called radially maximal if it is not complete and the addition of any new edge decreases its radius. \textit{F. Harary} and \textit{C. Thomassen} [Math. Proc. Camb. Philos. Soc. 79, 11--18 (1976; Zbl 0319.05128)] proved that the radius \(r\) and diameter \(d\) of any radially maximal graph satisfy \(r\le d\le 2r-2\). \textit{R. D. Dutton} et al. [Linear Algebra Appl. 217, 67--82 (1995; Zbl 0820.05020)] rediscovered this result with a different proof and conjectured that the converse is true, that is, if \(r\) and \(d\) are positive integers satisfying \(r\le d\le 2r-2\), then there exists a radially maximal graph with radius \(r\) and diameter \(d\). We prove this conjecture and a little more.Some results on the \(f\)-chromatic index of graphs whose \(f\)-core has maximum degree 2https://zbmath.org/1472.050472021-11-25T18:46:10.358925Z"Akbari, S."https://zbmath.org/authors/?q=ai:akbari.saieed"Chavooshi, M."https://zbmath.org/authors/?q=ai:chavooshi.m"Ghanbari, M."https://zbmath.org/authors/?q=ai:ghanbari.maryam"Manaviyat, R."https://zbmath.org/authors/?q=ai:manaviyat.raoofeSummary: Let \(G\) be a graph and \(f:V(G)\rightarrow\mathbb{N}\) be a function. An \(f\)-coloring of a graph \(G\) is an edge coloring such that each color appears at each vertex \(v\in V(G)\) at most \(f(v)\) times. The minimum number of colors needed to \(f\)-color \(G\) is called the \(f\)-chromatic index of \(G\) and is denoted by \(\chi^\prime_f(G)\). It was shown that for every graph \(G\), \(\Delta_f(G)\leq\chi^\prime_f(G)\leq\Delta_f(G)+1\), where \(\Delta_f(G)=\max_{v\in V(G)}\lceil\frac{d_G(v)}{f(v)}\rceil\). A graph \(G\) is said to be \(f\)-Class 1 if \(\chi^\prime_f(G)=\Delta_f(G)\), and \(f\)-Class 2, otherwise. Also, \(G_{\Delta_f}\) is the induced subgraph of \(G\) on \(\{v\in V(G):\frac{d_G(v)}{f(v)}=\Delta_f(G)\}\). In this paper, we show that if \(G\) is a connected graph with \(\Delta(G_{\Delta_f})\leq 2\) and \(G\) has an edge cut of size at most \(\Delta_f(G)-2\) which is a star, then \(G\) is \(f\)-Class 1. Also, we prove that if \(G\) is a connected graph and every connected component of \(G_{\Delta_f}\) is a unicyclic graph or a tree and \(G_{\Delta_f}\) is not 2-regular, then \(G\) is \(f\)-Class 1. Moreover, we show that except one graph, every connected claw-free graph \(G\) whose \(f\)-core is 2-regular with a vertex \(v\) such that \(f(v)\neq 1\) is \(f\)-Class 1.Coloring graphs by translates in the circlehttps://zbmath.org/1472.050482021-11-25T18:46:10.358925Z"Candela, Pablo"https://zbmath.org/authors/?q=ai:candela.pablo"Catalá, Carlos"https://zbmath.org/authors/?q=ai:catala.carlos"Hancock, Robert"https://zbmath.org/authors/?q=ai:hancock.robert"Kabela, Adam"https://zbmath.org/authors/?q=ai:kabela.adam"Král', Daniel"https://zbmath.org/authors/?q=ai:kral.daniel"Lamaison, Ander"https://zbmath.org/authors/?q=ai:lamaison.ander"Vena, Lluís"https://zbmath.org/authors/?q=ai:vena.lluisThe chromatic number, denoted by \(\chi(G),\) of a graph, \(G,\) is simply the smallest integer \(k\) of colors assigned to all the vertices in \(G\) such that no two adjacent vertices will share the same color.
To further enrich this research area, some other related notions have also been suggested: the notion of the circular chromatic number of a graph \(g,\) denoted by \(\chi_c(G),\) can be viewed as a mapping that maps all the vertices in \(G\) to unit-length arcs of a circle with a minimum circumstance \(z\) such that adjacent vertices are mapped to internally distinct arcs of such a circle. An independent set of vertices in a graph is a collection of vertices that are not adjacent to each other. The size of such a set is referred to as its weight. Then, the fractional chromatic number of a graph \(G,\) denoted by \(\chi_f(G),\) is defined as the smallest real number \(z\) for which there is an assignment of non-negative weights to all the independents sets of \(G\) such that the sum of their weights is \(z\) and each vertex belongs to independents sets with a total weight of at least 1.
It can be shown that, for each finite and simple graph \(G,\) its circular chromatic number lies between its fractional chromatic number and its chromatic number: \(\chi_f(G) \leq \chi_c(G) \leq \chi(G).\) It is worth pointing out that both inequalities can be strict, and the gap between \(\chi_f(G)\) and \(\chi(G)\) can be arbitrarily large. The vertices of the Kneser graph, \(K(n, k),\) is simply the collection of all the \(k\)-subsets chosen from \(n\) elements, and two vertices are adjacent if the associated \(k\)-subsets are disjoint. It holds that, when \(n \geq 2k,\)
\[
\chi_f(K(n, k)= n/k \leq n-2k+2= \chi_c(K(n, k))=\chi(K(n, k)),
\]
and the inequality is strict for \(n > 2k.\)
Starting from the definition of a quite technical notion of a coloring base of a graph, based on the Borel set, the authors formalize the notion of a gyrocoloring of a graph, denoted by \(\sigma_Z(G),\) and recast the above notions of circular and fractional chromatic numbers in terms of this notion of coloring base, and coin another notion of gyrochromatic number, \(\chi_g(G).\) It turns out that, for each finite and simple graph \(G,\) its gyrochromatic number lies in between its fractional chromatic number and its circular chromatic number:
\[
\chi_f(G) \leq \chi_g(G) \leq \chi_c(G) \leq \chi(G).
\]
Several equivalent definitions of this gyrochromatic numbers, as well as some basic properties, are given. And graphs for which its gyrochromatic number is strictly between the fractional chromatic number and the circular chromatic number are constructed, making use of a special case of the Kneser graph, \(K(2k^3+k k^3).\) It is also shown that some graph does not have its \(\chi_g(G)\)-gyrocoloring, since the associated coloring base does not exist.
Several open problems are suggested, regarding the relationship among the above and other graph coloring notions, to further our understanding of these notions, and their relationship.
This paper is well written organized, but quite technical. For all the details, readers have to consult the original paper, and the references cited within.A branch-and-price algorithm for the minimum sum coloring problemhttps://zbmath.org/1472.050492021-11-25T18:46:10.358925Z"Delle Donne, Diego"https://zbmath.org/authors/?q=ai:donne.diego-delle"Furini, Fabio"https://zbmath.org/authors/?q=ai:furini.fabio"Malaguti, Enrico"https://zbmath.org/authors/?q=ai:malaguti.enrico"Wolfler Calvo, Roberto"https://zbmath.org/authors/?q=ai:wolfler-calvo.robertoSummary: A proper coloring of a given graph is an assignment of a positive integer number (color) to each vertex such that two adjacent vertices receive different colors. This paper studies the Minimum Sum Coloring Problem (MSCP), which asks for finding a proper coloring while minimizing the sum of the colors assigned to the vertices. We propose the first branch-and-price algorithm to solve the MSCP to proven optimality. The newly developed exact approach is based on an Integer Programming (IP) formulation with an exponential number of variables which is tackled by column generation. We present extensive computational experiments, on synthetic and benchmark graphs from the literature, to compare the performance of our newly developed branch-and-price algorithm against three compact IP formulations. On synthetic graphs, our algorithm outperforms the compact formulations in terms of: (i) number of solved instances, (ii) running times and (iii) exit gaps obtained when optimality is not achieved. For the instances, our algorithm is competitive with the best compact formulation and provides very strong dual bounds.Explicit \(\Delta \)-edge-coloring of consecutive levels in a divisor latticehttps://zbmath.org/1472.050502021-11-25T18:46:10.358925Z"Diwan, Ajit A."https://zbmath.org/authors/?q=ai:diwan.ajit-arvindSummary: We show that the explicit 1-factorizations of the middle levels in a Boolean lattice, defined by \textit{D. A. Duffus} et al. [J. Comb. Theory, Ser. A 65, No. 2, 334--342 (1994; Zbl 0795.05111)], and by \textit{H. A. Kierstead} and \textit{W. T. Trotter} [Order 5, No. 2, 163--171 (1988; Zbl 0668.05045)], can both be generalized in a simple way to define explicit \(\Delta \)-edge-colorings of any two consecutive levels in any divisor lattice.Partially normal 5-edge-colorings of cubic graphshttps://zbmath.org/1472.050512021-11-25T18:46:10.358925Z"Jin, Ligang"https://zbmath.org/authors/?q=ai:jin.ligang"Kang, Yingli"https://zbmath.org/authors/?q=ai:kang.yingliThe authors consider a very important problem of normal colorings of cubic graphs. The problem is connected with the very well-known conjecture that every cubic bridgeless graph has normal 5-coloring. In turn, this hypothesis is related with other well-known conjectures. That is why the results presented in the paper are very valuable. The authors establish a lower bound for the number of so named normal edges in partially normal 5-edge-coloring of every bridgeless cubic graph.Corrigendum to: ``A local epsilon version of Reed's conjecture''https://zbmath.org/1472.050522021-11-25T18:46:10.358925Z"Kelly, Tom"https://zbmath.org/authors/?q=ai:kelly.tom"Postle, Luke"https://zbmath.org/authors/?q=ai:postle.lukeSummary: We correct an error that appears in our paper [ibid. 141, 181--222 (2020; Zbl 1430.05038)]. All of the main results remain valid after this correction.The 4-choosability of planar graphs and cycle adjacencyhttps://zbmath.org/1472.050532021-11-25T18:46:10.358925Z"Lin, Juei-Yin"https://zbmath.org/authors/?q=ai:lin.juei-yin"Yang, Chung-Ying"https://zbmath.org/authors/?q=ai:yang.chung-ying"Chang, Gerard Jennhwa"https://zbmath.org/authors/?q=ai:chang.gerard-jennhwaSummary: We study 4-list coloring of planar graphs in this paper, and prove that each planar graph without adjacent 3-cycles, adjacent 4-cycles and small suns is 4-choosable.DP-4-coloring of planar graphs with some restrictions on cycleshttps://zbmath.org/1472.050542021-11-25T18:46:10.358925Z"Li, Rui"https://zbmath.org/authors/?q=ai:li.rui.3|li.rui.1|li.rui.4|li.rui|li.rui.2"Wang, Tao"https://zbmath.org/authors/?q=ai:wang.tao.3|wang.tao|wang.tao.7|wang.tao.1|wang.tao.9|wang.tao.6|wang.tao.8|wang.tao.4|wang.tao.2|wang.tao.5Summary: DP-coloring was introduced by \textit{Z. Dvořák} and \textit{L. Postle} [J. Comb. Theory, Ser. B 129, 38--54 (2018; Zbl 1379.05034)] as a generalization of list coloring. It was originally used to solve a longstanding conjecture by \textit{O. V. Borodin} [Discrete Math. 313, No. 4, 517--539 (2013; Zbl 1259.05042)], stating that every planar graph without cycles of lengths 4 to 8 is 3-choosable. In this paper, we give three sufficient conditions for a planar graph to be DP-4-colorable. Actually all the results (Theorem 1.3, 1.4 and 1.7) are stated in the ``color extendability'' form, and uniformly proved by vertex identification and discharging method.Properly colored cycles in edge-colored complete graphs without monochromatic triangle: a vertex-pancyclic analogous resulthttps://zbmath.org/1472.050552021-11-25T18:46:10.358925Z"Li, Ruonan"https://zbmath.org/authors/?q=ai:li.ruonanSummary: A properly colored cycle (path) in an edge-colored graph is a cycle (path) with consecutive edges assigned distinct colors. A monochromatic triangle is a cycle of length 3 with the edges assigned a same color. It is known that every edge-colored complete graph without monochromatic triangle always contains a properly colored Hamilton path. In this paper, we investigate the existence of properly colored cycles in edge-colored complete graphs when monochromatic triangles are forbidden. We obtain a vertex-pancyclic analogous result combined with a characterization of all the exceptions. As a consequence, we partially confirm a structural conjecture given by \textit{B. Bollobás} and \textit{P. Erdős} [Isr. J. Math. 23, 126--131 (1976; Zbl 0325.05114)] and an algorithmic conjecture given by \textit{G. Gutin} and \textit{E. Jung Kim} [Lect. Notes Comput. Sci. 5420, 200--208 (2009; Zbl 1194.05040)].Extremal problems and results related to Gallai-coloringshttps://zbmath.org/1472.050562021-11-25T18:46:10.358925Z"Li, Xihe"https://zbmath.org/authors/?q=ai:li.xihe"Broersma, Hajo"https://zbmath.org/authors/?q=ai:broersma.hajo-j"Wang, Ligong"https://zbmath.org/authors/?q=ai:wang.ligongSummary: A Gallai-coloring (Gallai-\(k\)-coloring) is an edge-coloring (with colors from \(\{1, 2, \ldots, k \}\)) of a complete graph without rainbow triangles. Given a graph \(H\) and a positive integer \(k\), the \(k\)-colored Gallai-Ramsey number \(G R_k(H)\) is the minimum integer \(n\) such that every Gallai-\(k\)-coloring of the complete graph \(K_n\) contains a monochromatic copy of \(H\). In this paper, we consider two extremal problems related to Gallai-\(k\)-colorings. First, we determine upper and lower bounds for the maximum number of edges that are not contained in any rainbow triangle or monochromatic triangle in a \(k\)-edge-coloring of \(K_n\). Second, for \(n \geq G R_k( K_3)\), we determine upper and lower bounds for the minimum number of monochromatic triangles in a Gallai-\(k\)-coloring of \(K_n\), yielding the exact value for \(k = 3\). Furthermore, we determine the Gallai-Ramsey number \(G R_k( K_4 + e)\) for the graph on five vertices consisting of a \(K_4\) with a pendant edge.On sublinear approximations for the Petersen coloring conjecturehttps://zbmath.org/1472.050572021-11-25T18:46:10.358925Z"Mattiolo, Davide"https://zbmath.org/authors/?q=ai:mattiolo.davide"Mazzuoccolo, Giuseppe"https://zbmath.org/authors/?q=ai:mazzuoccolo.giuseppe"Mkrtchyan, Vahan"https://zbmath.org/authors/?q=ai:mkrtchyan.vahan-vSummary: If \(f:\mathbb{N}\to\mathbb{N}\) is a function, then let us say that \(f\) is sublinear if \[\lim_{n\to+\infty}\frac{f(n)}{n}=0.\] If \(G=(V,E)\) is a cubic graph and \(c:E\to\{1,\dots,k\}\) is a proper \(k\)-edge-coloring of \(G\), then an edge \(e=uv\) of \(G\) is poor (rich) in \(c\), if the edges incident to \(u\) and \(v\) are colored with three (five) colors. An edge is abnormal if it is neither rich nor poor. The Petersen coloring conjecture of \textit{F. Jaeger} [Ars Comb. 20B, 229--244 (1985; Zbl 0593.05031)] states that any bridgeless cubic graph admits a proper 5-edge-coloring \(c\), such that there is no an abnormal edge of \(G\) with respect to \(c\). For a proper 5-edge-coloring \(c\) of \(G\), let \(N_G(c)\) be the set of abnormal edges of \(G\) with respect to \(c\). In this paper we show that (a) The Petersen coloring conjecture is equivalent to the statement that there is a sublinear function \(f:\mathbb{N}\to\mathbb{N}\), such that all bridgeless cubic graphs admit a proper 5-edge-coloring \(c\) with \(|N_G(c)|\le f(|V|)\); (b) for \(k=2,3,4\), the statement that there is a sublinear function \(f:\mathbb{N}\to\mathbb{N}\), such that all (cyclically) \(k\)-edge-connected cubic graphs admit a proper 5-edge-coloring \(c\) with \(|N_G(c)|\le f(|V|)\) is equivalent to the statement that all (cyclically) \(k\)-edge-connected cubic graphs admit a proper 5-edge-coloring \(c\) with \(|N_G(c)|\le 2k+1\).A construction of uniquely colourable graphs with equal colour class sizeshttps://zbmath.org/1472.050582021-11-25T18:46:10.358925Z"Mohr, Samuel"https://zbmath.org/authors/?q=ai:mohr.samuelSummary: A uniquely \(k\)-colourable graph is a graph with exactly one partition of the vertex set into at most \(k\) colour classes. Here, we investigate some constructions of uniquely \(k\)-colourable graphs and give a construction of \(K_k\)-free uniquely \(k\)-colourable graphs with equal colour class sizes.A better upper bound on the chromatic number of (cap, even-hole)-free graphshttps://zbmath.org/1472.050592021-11-25T18:46:10.358925Z"Xu, Yian"https://zbmath.org/authors/?q=ai:xu.yianSummary: A hole is an induced cycle of length at least 4, and an even-hole is a hole of even length. A cap is a graph consisting of a hole and an additional vertex which is adjacent to exactly two adjacent vertices of the hole. \textit{K. Cameron} et al. [ibid. 341, No. 2, 463--473 (2018; Zbl 1376.05132)] proved that every (cap, even-hole)-free graph \(G\) has \(\chi(G) \leqslant \lfloor \frac{ 3}{ 2} \omega(G) \rfloor \), and they also proposed a question stating that if \(\chi(G) \leqslant \lceil \frac{ 5}{ 4} \omega(G) \rceil\) for all (cap, even-hole)-free graphs. \textit{R. Wu} and \textit{B. Xu} [ibid. 342, No. 3, 898--903 (2019; Zbl 1403.05053)] showed that every (cap, even-hole)-free graph \(G\) has \(\chi(G) \leqslant \lceil \frac{ 4}{ 3} \omega(G) \rceil \). In this paper, we improve this upper bound and show that every (cap, even-hole)-free graph has \(\chi(G) \leqslant \lceil \frac{ 9}{ 7} \omega(G) \rceil + 1\).List \(r\)-dynamic coloring of sparse graphshttps://zbmath.org/1472.050602021-11-25T18:46:10.358925Z"Zhu, Junlei"https://zbmath.org/authors/?q=ai:zhu.junlei"Bu, Yuehua"https://zbmath.org/authors/?q=ai:bu.yuehuaAn \(r\)-dynamic coloring of a graph \(G\) is a proper coloring of the vertices of \(G\) such that the neighbors of every vertex receive either at least \(r\) colors or all different colors. The \(r\)-dynamic chromatic number of \(G\), denoted \(\chi_r(G)\) is the minimum number of colors required in an \(r\)-dynamic coloring of \(G\). The list \(r\)-dynamic chromatic number of a graph \(G\), denoted \(\operatorname{ch}_r(G)\), is the list version of the \(r\)-dynamic chromatic number; in other words, each vertex has its own set of possible colors instead of a universal color set. This article proves the following three results regarding the list \(r\)-dynamic chromatic number of a graph \(G\):
\par if \(G\) is planar with girth at least \(5\), then \(\operatorname{ch}_r(G)\leq r+5\) for \(r\geq 15\),
\par if mad\((G)<\frac{10}{3}\), then \(\operatorname{ch}_r(G)\leq r+10\),
\par if mad\((G)<\frac{8}{3}\), then \(\operatorname{ch}_r(G)\leq r+1\) for \(r\geq 14\).
The proofs utilize the popular discharing method.Maximal strongly connected cliques in directed graphs: algorithms and boundshttps://zbmath.org/1472.050612021-11-25T18:46:10.358925Z"Conte, Alessio"https://zbmath.org/authors/?q=ai:conte.alessio"Kanté, Mamadou Moustapha"https://zbmath.org/authors/?q=ai:kante.mamadou-moustapha"Uno, Takeaki"https://zbmath.org/authors/?q=ai:uno.takeaki"Wasa, Kunihiro"https://zbmath.org/authors/?q=ai:wasa.kunihiroSummary: Finding communities in the form of cohesive subgraphs is a fundamental problem in network analysis. In domains that model networks as undirected graphs, communities are generally associated with dense subgraphs, and many community models have been proposed. Maximal cliques are arguably the most widely studied among such models, with early works dating back to the '60s, and a continuous stream of research up to the present. In domains that model networks as directed graphs, several approaches for community detection have been proposed, but there seems to be no clear model of cohesive subgraph, i.e., of what a community should look like. We extend the fundamental model of clique to directed graphs, adding the natural constraint of strong connectivity within the clique.
We consider in this paper the problem of listing all maximal strongly connected cliques in a directed graph. We first investigate the combinatorial properties of strongly connected cliques and use them to prove that every \(n\)-vertex directed graph has at most \(3^{n / 3}\) maximal strongly connected cliques. We then exploit these properties to produce the first algorithms with polynomial delay for enumerating maximal strongly connected cliques: a first algorithm with polynomial delay and exponential space usage, and a second one, based on reverse-search, with higher (still polynomial) delay but which uses linear space.On the unavoidability of oriented treeshttps://zbmath.org/1472.050622021-11-25T18:46:10.358925Z"Dross, François"https://zbmath.org/authors/?q=ai:dross.francois"Havet, Frédéric"https://zbmath.org/authors/?q=ai:havet.fredericSummary: A digraph is \(n\)-unavoidable if it is contained in every tournament of order \(n\). We first prove that every arborescence of order \(n\) with \(k\) leaves is \((n+k-1)\)-unavoidable. We then prove that every oriented tree of order \(n\) \((n\geq 2)\) with \(k\) leaves is \((\frac{3}{2}n+\frac{3}{2}k-2)\)-unavoidable and \((\frac{9}{2}n-\frac{5}{2}k-\frac{9}{2})\)-unavoidable, and thus \((\frac{21}{8}n-\frac{47}{16})\)-unavoidable. Finally, we prove that every oriented tree of order \(n\) with \(k\) leaves is \((n+144k^2-280k+124)\)-unavoidable.Multiplicity and diversity: analysing the optimal solution space of the correlation clustering problem on complete signed graphshttps://zbmath.org/1472.050632021-11-25T18:46:10.358925Z"Arınık, Nejat"https://zbmath.org/authors/?q=ai:arinik.nejat"Figueiredo, Rosa"https://zbmath.org/authors/?q=ai:figueiredo.rosa-m-v"Labatut, Vincent"https://zbmath.org/authors/?q=ai:labatut.vincentSummary: In order to study real-world systems, many applied works model them through signed graphs, that is, graphs whose edges are labelled as either positive or negative. Such a graph is considered as structurally balanced when it can be partitioned into a number of modules, such that positive (respectively negative) edges are located inside (respectively in-between) the modules. When it is not the case, authors look for the closest partition to such balance, a problem called Correlation Clustering (CC). Due to the complexity of the CC problem, the standard approach is to find a single optimal partition and stick to it, even if other optimal or high scoring solutions possibly exist. In this work, we study the space of optimal solutions of the CC problem, on a collection of synthetic complete graphs. We show empirically that under certain conditions, there can be many optimal partitions of a signed graph. Some of these are very different and thus provide distinct perspectives on the system, as illustrated on a small real-world graph. This is an important result, as it implies that one may have to find several, if not all, optimal solutions of the CC problem, in order to properly study the considered system.Constructing cospectral signed graphshttps://zbmath.org/1472.050642021-11-25T18:46:10.358925Z"Belardo, Francesco"https://zbmath.org/authors/?q=ai:belardo.francesco"Brunetti, Maurizio"https://zbmath.org/authors/?q=ai:brunetti.maurizio"Cavaleri, Matteo"https://zbmath.org/authors/?q=ai:cavaleri.matteo"Donno, Alfredo"https://zbmath.org/authors/?q=ai:donno.alfredoSummary: A well-known fact in Spectral Graph Theory is the existence of pairs of cospectral (or isospectral) nonisomorphic graphs, known as PINGS. The work of \textit{A. J. Schwenk} [in: New Direct. Theory Graphs, Proc. Third Ann Arbor Conf., Univ. Michigan 1971, 275--307 (1973; Zbl 0261.05102)] and of \textit{C. D. Godsil} and \textit{B. D. McKay} [Aequationes Math. 25, 257--268 (1982; Zbl 0527.05051)] shed some light on the explanation of the presence of cospectral graphs, and they gave routines to construct PINGS. Here, we consider the Godsil-McKay-type routines developed for graphs, whose adjacency matrices are \((0,1)\)-matrices, to the level of signed graphs, whose adjacency matrices allow the presence of \(-1\)s. We show that, with suitable adaption, such routines can be successfully ported to signed graphs, and we can build pairs of cospectral switching nonisomorphic signed graphs.On the complexity of a linear ordering of weighted directed acyclic graphshttps://zbmath.org/1472.050652021-11-25T18:46:10.358925Z"Shchekalev, M. I."https://zbmath.org/authors/?q=ai:shchekalev.m-i"Bokov, G. V."https://zbmath.org/authors/?q=ai:bokov.grigoriy-v"Kudryavtsev, V. B."https://zbmath.org/authors/?q=ai:kudryavtsev.valerii-borisovichSummary: We consider weighted directed acyclic graphs to whose edges nonnegative integers as weights are assigned. The complexity of a linear ordering of vertices is examined for these graphs in the order of topological sorting. An accurate estimate for the Shannon function of the complexity of the linear ordering problem for weighted directed acyclic graphs is obtained.Fixing number and metric dimension of a zero-divisor graph associated with a ringhttps://zbmath.org/1472.050662021-11-25T18:46:10.358925Z"Ou, Shikun"https://zbmath.org/authors/?q=ai:ou.shikun"Wong, Dein"https://zbmath.org/authors/?q=ai:wong.dein"Tian, Fenglei"https://zbmath.org/authors/?q=ai:tian.fenglei"Zhou, Qi"https://zbmath.org/authors/?q=ai:zhou.qi.1Summary: Let \(R\) be a ring and \(Z(R) = Z_l (R) \cup Z_r (R)\), where \(Z_l (R)\) and \(Z_r (R)\) are the sets of all left and right zero-divisors of \(R\), respectively. The zero-divisor graph of \(R\), denoted by \(\Gamma (R)\), is a simple undirected graph with vertex set \(Z^{\ast} (R) = Z(R) \backslash \{ 0 \}\), and two distinct vertices \(a,b \in Z^{\ast} (R)\) are adjacent if and only if \(ab = 0\) or \(ba = 0\). Let \(n \geq 2\), \(\mathbb{Z}_n\) the ring of integers modulo \(n\), and \(\mathrm{Mat}_n (q)\) the ring of all \(n \times n\) matrices over a finite field of \(q\) elements. In this article, using the technique on characteristic matrices, we give the value of fixing number of \(\Gamma (\prod^n_{i=1} \mathbb{Z}_2 )\). Moreover, we calculate the fixing number and metric dimension of \(\Gamma (\mathbb{Z}_n)\) and \(\Gamma (\mathrm{Mat}_n (q))\).Four-valent oriented graphs of biquasiprimitive typehttps://zbmath.org/1472.050672021-11-25T18:46:10.358925Z"Poznanović, Nemanja"https://zbmath.org/authors/?q=ai:poznanovic.nemanja"Praeger, Cheryl E."https://zbmath.org/authors/?q=ai:praeger.cheryl-eSummary: Let \(\mathcal{OG}(4)\) denote the family of all graph-group pairs \((\Gamma ,G)\) where \(\Gamma\) is 4-valent, connected and \(G\)-oriented (\(G\)-half-arc-transitive). Using a novel application of the structure theorem for biquasiprimitive permutation groups of the second author [Ill. J. Math. 47, No. 1--2, 461--475 (2003; Zbl 1032.20004)], we produce a description of all pairs \((\Gamma , G) \in \mathcal{OG}(4)\) for which every nontrivial normal subgroup of \(G\) has at most two orbits on the vertices of \(\Gamma \), and at least one normal subgroup has two orbits. In particular we show that \(G\) has a unique minimal normal subgroup \(N\) and that \(N \cong T^k\) for a simple group \(T\) and \(k\in \{1,2,4,8\}\). This provides a crucial step towards a general description of the long-studied family \(\mathcal{OG}(4)\) in terms of a normal quotient reduction. We also give several methods for constructing pairs \((\Gamma , G)\) of this type and provide many new infinite families of examples, covering each of the possible structures of the normal subgroup \(N\).A method for enumerating pairwise compatibility graphs with a given number of verticeshttps://zbmath.org/1472.050682021-11-25T18:46:10.358925Z"Azam, Naveed Ahmed"https://zbmath.org/authors/?q=ai:azam.naveed-ahmed"Shurbevski, Aleksandar"https://zbmath.org/authors/?q=ai:shurbevski.aleksandar"Nagamochi, Hiroshi"https://zbmath.org/authors/?q=ai:nagamochi.hiroshiSummary: \textit{N. A. Azam} et al. [``Enumerating all pairwise compatibility graphs with a given number of vertices based on linear programming'', in: Proceedings of 2nd International Workshop on Enumeration Problems and Applications, WEPA, Pisa, Italy. 5--8 (2018)] proposed a method to enumerate all pairwise compatibility graphs (PCGs) with a given number \(n\) of vertices. For a tuple \(( G , T , \sigma , \lambda )\) of a graph \(G\) with \(n\) vertices and a tree \(T\) with \(n\) leaves, a bijection \(\sigma\) between the vertices in \(G\) and the leaves in \(T\), and a bi-partition \(\lambda\) of the set of non-adjacent vertex pairs in \(G\), they formulated two linear programs, LP\(( G , T , \sigma , \lambda )\) and DLP\(( G , T , \sigma , \lambda )\) such that: exactly one of them is feasible; and \(G\) is a PCG if and only if LP\(( G , T , \sigma , \lambda )\) is feasible for some tuple \(( G , T , \sigma , \lambda )\). To reduce the number of graphs \(G\) with \(n\) vertices (resp., tuples) for which the LPs are solved, they excluded PCGs by heuristically generating PCGs (resp., some tuples that contain a sub-tuple \(( G^\prime , T^\prime , \sigma^\prime , \lambda^\prime )\) for \(n = 4\) whose LP\(( G^\prime , T^\prime , \sigma^\prime , \lambda^\prime )\) is infeasible). This paper proposes two improvements in the method: derive a sufficient condition for a graph to be a PCG for a given tree in order to exclude more PCGs; and characterize all sub-tuples \(( G^\prime , T^\prime , \sigma^\prime , \lambda^\prime )\) for \(n = 4\) for which LP\(( G^\prime , T^\prime , \sigma^\prime , \lambda^\prime )\) is infeasible, and enumerate tuples that contain no such sub-tuples by a branch-and-bound algorithm. Experimental results show that our method more efficiently enumerated all PCGs for \(n = 8\).Locally definable vertex set properties are efficiently enumerablehttps://zbmath.org/1472.050692021-11-25T18:46:10.358925Z"Blind, Sarah"https://zbmath.org/authors/?q=ai:blind.sarah"Creignou, Nadia"https://zbmath.org/authors/?q=ai:creignou.nadia"Olive, Frédéric"https://zbmath.org/authors/?q=ai:olive.fredericSummary: We propose a general framework that allows for the study of enumeration of vertex set properties in graphs. We prove that when such a property is locally definable with respect to some order on the set of vertices, then it can be enumerated with linear delay. Our method consists in reducing the considered enumeration problem to the enumeration of paths in directed acyclic graphs. We then apply this general method to enumerate minimal connected dominating sets and maximal irredundant sets in interval graphs and in permutation graphs, as well as maximal irredundant sets in circular-arc graphs and in circular permutation graphs, with linear delay.Efficiently enumerating minimal triangulationshttps://zbmath.org/1472.050702021-11-25T18:46:10.358925Z"Carmeli, Nofar"https://zbmath.org/authors/?q=ai:carmeli.nofar"Kenig, Batya"https://zbmath.org/authors/?q=ai:kenig.batya"Kimelfeld, Benny"https://zbmath.org/authors/?q=ai:kimelfeld.benny"Kröll, Markus"https://zbmath.org/authors/?q=ai:kroll.markusSummary: We present an algorithm that enumerates all the minimal triangulations of a graph in incremental polynomial time. Consequently, we get an algorithm for enumerating all the proper tree decompositions, in incremental polynomial time, where ``proper'' means that the tree decomposition cannot be improved by removing or splitting a bag. The algorithm can incorporate any method for (ordinary, single result) triangulation or tree decomposition, and can serve as an anytime algorithm to improve such a method. We describe an extensive experimental study of an implementation on real data from different fields. Our experiments show that the algorithm improves upon central quality measures over the underlying tree decompositions, and is able to produce a large number of high-quality decompositions.Efficient enumeration of maximal induced bicliqueshttps://zbmath.org/1472.050712021-11-25T18:46:10.358925Z"Hermelin, Danny"https://zbmath.org/authors/?q=ai:hermelin.danny"Manoussakis, George"https://zbmath.org/authors/?q=ai:manoussakis.georgeSummary: Given a graph \(G\) of order \(n\), we consider the problem of enumerating all its maximal induced bicliques. We first propose an algorithm running in time \(\mathcal{O} ( n 3^{n / 3} )\). As the maximum number of maximal induced bicliques of a graph with \(n\) vertices is \(\Theta ( 3^{n / 3} )\), the algorithm is worst-case output size optimal. Then, we prove new bounds on the maximum number of maximal induced bicliques of graphs with respect to their maximum degree \(\Delta\) and degeneracy \(k\), and propose a near-optimal algorithm with enumeration time \(\mathcal{O} ( n k ( \Delta + k ) 3^{\frac{ \Delta + k}{ 3}} )\). Then, we provide output sensitive algorithms for this problem with enumeration time depending only on the maximum degree of the input graph. Since we need to store the bicliques in these algorithms, the space complexity may be exponential. Thus, we show how to modify them so they only require polynomial space, but with a slight time complexity increase.Enumeration of unsensed \(r\)-regular maps on the projective plane and the Klein bottlehttps://zbmath.org/1472.050722021-11-25T18:46:10.358925Z"Krasko, Evgeniy"https://zbmath.org/authors/?q=ai:krasko.evgenii-s"Omelchenko, Alexander"https://zbmath.org/authors/?q=ai:omelchenko.alexander-vSummary: The paper is devoted to the problem of enumerating \(r\)-regular maps on two simplest non-orientable surfaces, the projective plane and the Klein bottle, up to all symmetries (so-called unsensed maps). We obtain general formulas that reduce the problem of counting such maps to the problem of enumerating rooted quotient maps on orbifolds. In addition, we solve the problem of explicitly describing all cyclic orbifolds for such surfaces. We also derive recurrence relations for rooted quotient maps on orbifolds that can be orientable or non-orientable surfaces with branch points and/or boundary components. These results enable us to obtain explicit formulas for the numbers of unsensed maps on the projective plane and the Klein bottle by the number of edges.Efficient enumeration of dominating sets for sparse graphshttps://zbmath.org/1472.050732021-11-25T18:46:10.358925Z"Kurita, Kazuhiro"https://zbmath.org/authors/?q=ai:kurita.kazuhiro"Wasa, Kunihiro"https://zbmath.org/authors/?q=ai:wasa.kunihiro"Arimura, Hiroki"https://zbmath.org/authors/?q=ai:arimura.hiroki"Uno, Takeaki"https://zbmath.org/authors/?q=ai:uno.takeakiSummary: A dominating set \(D\) of a graph \(G\) is a set of vertices such that any vertex in \(G\) is in \(D\) or its neighbor is in \(D\). Enumeration of minimal dominating sets in a graph is one of the central problems in enumeration study since enumeration of minimal dominating sets corresponds to the enumeration of minimal hypergraph transversals. The output-polynomial time enumeration of minimal hypergraph transversals is an interesting open problem. On the other hand, enumeration of dominating sets including non-minimal ones has not been received much attention. In this paper, we address enumeration problems for dominating sets from sparse graphs which are degenerate graphs and graphs with large girth, and we propose two algorithms for solving the problems. The first algorithm enumerates all the dominating sets for a \(k\)-degenerate graph in \(O \left( k\right)\) time per solution using \(O \left( n + m\right)\) space, where \(n\) and \(m\) are respectively the number of vertices and edges in an input graph. That is, the algorithm is optimal for graphs with constant degeneracy such as trees, planar graphs, \(H\)-minor free graphs with some fixed \(H\). The second algorithm enumerates all the dominating sets in constant time per solution for input graphs with girth at least nine.Embedding of recursive circulant \(RC(2^{n},4)\) into circular necklacehttps://zbmath.org/1472.050742021-11-25T18:46:10.358925Z"Mary, R. Stalin"https://zbmath.org/authors/?q=ai:mary.r-stalin"Rajasingh, Indra"https://zbmath.org/authors/?q=ai:rajasingh.indraSummary: In this paper, we compute the wirelength of embedding Recursive Circulants \(RC(2^n,4)\) into Circular Necklace. Further, we identify a set of edges \(S\) in \(RC(2^n,4)\) such that the wirelength of embedding \(RC(2^n,4)\setminus S\) into circular necklace is minimum.Minimizing the number of edges in \(\mathcal{C}_{\geq r} \)-saturated graphshttps://zbmath.org/1472.050752021-11-25T18:46:10.358925Z"Ma, Yue"https://zbmath.org/authors/?q=ai:ma.yue"Hou, Xinmin"https://zbmath.org/authors/?q=ai:hou.xinmin"Hei, Doudou"https://zbmath.org/authors/?q=ai:hei.doudou"Gao, Jun"https://zbmath.org/authors/?q=ai:gao.junSummary: Given a family of graphs \(\mathcal{F} \), a graph \(G\) is said to be \(\mathcal{F} \)-saturated if \(G\) does not contain a copy of \(F\) as a subgraph for any \(F \in \mathcal{F} \), but the addition of any edge \(e \notin E(G)\) creates at least one copy of some \(F \in \mathcal{F}\) within \(G\). The minimum size of an \(\mathcal{F} \)-saturated graph on \(n\) vertices is called the saturation number, denoted by \(\operatorname{sat}(n, \mathcal{F})\). Let \(\mathcal{C}_{\geq r}\) be the family of cycles of length at least \(r\). \textit{M. Ferrara} et al. [J. Graph Theory 71, No. 3--4, 416--434 (2012; Zbl 1283.05231)] gave lower and upper bounds of \(\operatorname{sat}(n, C_{\geq r})\) and determined the exact values of \(\operatorname{sat}(n, C_{\geq r})\) for \(3 \leq r \leq 5\). In this paper, we determine the exact value of \(\operatorname{sat}(n, \mathcal{C}_{\geq r})\) for \(r = 6\) and \(28 \leq \frac{ n}{ 2} \leq r \leq n\) and give new upper and lower bounds for the other cases.On Turán numbers of the complete 4-graphshttps://zbmath.org/1472.050762021-11-25T18:46:10.358925Z"Sidorenko, Alexander"https://zbmath.org/authors/?q=ai:sidorenko.alexanderSummary: The Turán number \(T(n, \alpha + 1, r)\) is the minimum number of edges in an \(n\)-vertex \(r\)-graph whose independence number does not exceed \(\alpha \). For each \(r \geq 2\), there exists \(t_\ast(r)\) such that \(T(n, \alpha + 1, r) = t_\ast(r) n^r \alpha^{1 - r}(1 + o(1))\) as \(\alpha / r \to \infty\) and \(n / \alpha \to \infty \). It is known that \(t_\ast(2) = 1 / 2\), and the conjectured value of \(t_\ast(3)\) is \(2/3\). We prove that \(t_\ast(4) < 0.706335\).A note on the Turán number of disjoint union of wheelshttps://zbmath.org/1472.050772021-11-25T18:46:10.358925Z"Xiao, Chuanqi"https://zbmath.org/authors/?q=ai:xiao.chuanqi"Zamora, Oscar"https://zbmath.org/authors/?q=ai:zamora.oscarSummary: The Turán number of a graph \(H\), \(\operatorname{ex}(n, H)\), is the maximum number of edges in a graph on \(n\) vertices which does not have \(H\) as a subgraph. A wheel \(W_n\) is an \(n\)-vertex graph formed by connecting a single vertex to all vertices of a cycle \(C_{n - 1} \). Let \(m W_{2 k + 1}\) (\(k \geq 3\)) denote the graph defined by taking \(m\) vertex disjoint copies of \(W_{2 k + 1} \). For sufficiently large \(n\), we determine the Turán number and all extremal graphs for \(m W_{2 k + 1}\) (\(k \geq 3\)). Let \(\mathcal{W}^h\) be the family of graphs obtain by the disjoint union of a finite number of wheels, such that, the number of even wheels in the union is \(h\), (\(h \geq 1\)). For any \(W \in \mathcal{W}^h\), we also provide the Turán number and all extremal graphs for \(W\), when \(n\) is sufficiently large.Types of embedded graphs and their Tutte polynomialshttps://zbmath.org/1472.050782021-11-25T18:46:10.358925Z"Huggett, Stephen"https://zbmath.org/authors/?q=ai:huggett.stephen-a"Moffatt, Iain"https://zbmath.org/authors/?q=ai:moffatt.iainSummary: We take an elementary and systematic approach to the problem of extending the Tutte polynomial to the setting of embedded graphs. Four notions of embedded graphs arise naturally when considering deletion and contraction operations on graphs on surfaces. We give a description of each class in terms of coloured ribbon graphs. We then identify a universal deletion-contraction invariant (i.e., a `Tutte polynomial') for each class. We relate these to graph polynomials in the literature, including the Bollobás-Riordan, Krushkal and Las Vergnas polynomials, and give state-sum formulations, duality relations, deleton-contraction relations, and quasi-tree expansions for each of them.Independence polynomials and Alexander-Conway polynomials of plumbing linkshttps://zbmath.org/1472.050792021-11-25T18:46:10.358925Z"Stoimenow, A."https://zbmath.org/authors/?q=ai:stoimenow.alexanderSummary: We use the Chudnovsky-Seymour Real Root Theorem for independence polynomials to obtain some statements about the coefficients and roots of the Alexander and Conway polynomial of some types of plumbing links, addressing conjectures of Fox, Hoste and Liechti.Extremal graphs for blow-ups of keyringshttps://zbmath.org/1472.050802021-11-25T18:46:10.358925Z"Ni, Zhenyu"https://zbmath.org/authors/?q=ai:ni.zhenyu"Kang, Liying"https://zbmath.org/authors/?q=ai:kang.liying"Shan, Erfang"https://zbmath.org/authors/?q=ai:shan.erfang"Zhu, Hui"https://zbmath.org/authors/?q=ai:zhu.huiAuthors' abstract: The blow-up of a graph \(H\) is the graph obtained from replacing each edge in \(H\) by a clique of the same size where the new vertices of the cliques are all different. Given a graph \(H\) and a positive integer \(n\), the extremal number, ex \((n, H)\), is the maximum number of edges in a graph on \(n\) vertices that does not contain \(H\) as a subgraph. A keyring \(C_s(k)\) is a \((k+s)\)-edge graph obtained from a cycle of length \(k\) by appending \(s\) leaves to one of its vertices. This paper determines the extremal number and finds the extremal graphs for the blow-ups of keyrings \(C_s(k)\) \((k\geq 3,s\geq 1)\) when \(n\) is sufficiently large. For special cases when \(k=0\) or \(s=0\), the extremal number of the blow-ups of the graph \(C_s(0)\) (a star) has been determined by \textit{P. Erdős} et al. [J. Comb. Theory, Ser. B 64, No. 1, 89--100 (1995; Zbl 0822.05036)] and \textit{G. Chen} et al. [ibid. 89, No. 2, 159--171 (2003; Zbl 1031.05069)], while the extremal number and extremal graphs for the blow-ups of the graph \(C_0(k)\) (a cycle) when \(n\) is sufficiently large has been determined by \textit{H. Liu} [Electron. J. Comb. 20, No. 1, Research Paper P65, 16 p. (2013; Zbl 1266.05074)].Beyond non-backtracking: non-cycling network centrality measureshttps://zbmath.org/1472.050812021-11-25T18:46:10.358925Z"Arrigo, Francesca"https://zbmath.org/authors/?q=ai:arrigo.francesca"Higham, Desmond J."https://zbmath.org/authors/?q=ai:higham.desmond-j"Noferini, Vanni"https://zbmath.org/authors/?q=ai:noferini.vanniSummary: Walks around a graph are studied in a wide range of fields, from graph theory and stochastic analysis to theoretical computer science and physics. In many cases it is of interest to focus on non-backtracking walks; those that do not immediately revisit their previous location. In the network science context, imposing a non-backtracking constraint on traditional walk-based node centrality measures is known to offer tangible benefits. Here, we use the Hashimoto matrix construction to characterize, generalize and study such non-backtracking centrality measures. We then devise a recursive extension that systematically removes triangles, squares and, generally, all cycles up to a given length. By characterizing the spectral radius of appropriate matrix power series, we explore how the universality results on the limiting behaviour of classical walk-based centrality measures extend to these non-cycling cases. We also demonstrate that the new recursive construction gives rise to practical centrality measures that can be applied to large-scale networks.Classes of cubic graphs containing cycles of integer-power lengthshttps://zbmath.org/1472.050822021-11-25T18:46:10.358925Z"Couch, P. J."https://zbmath.org/authors/?q=ai:couch.pj"Daniel, B. D."https://zbmath.org/authors/?q=ai:daniel.b-d"Wright, W. Paul"https://zbmath.org/authors/?q=ai:wright.w-paulP. Erdős and A. Gyárfás conjectured in 1995 that every graph with minimum degree three has a cycle of length \(2k\) for some integer \(k > 1\). Caro has asked the related question of whether every such graph has a cycle whose length is a non-trivial power of some natural number. Main theorem of this article is as follows: Theorem. Suppose that \(G\) is an almost claw-free graph with minimum degree 3. Then \(G\) has a cycle of length \(a^k\) for some integers \(a \geq 2\) and \(k \geq 2\).Maximum density of vertex-induced perfect cycles and paths in the hypercubehttps://zbmath.org/1472.050832021-11-25T18:46:10.358925Z"Goldwasser, John"https://zbmath.org/authors/?q=ai:goldwasser.john-l"Hansen, Ryan"https://zbmath.org/authors/?q=ai:hansen.ryanSummary: Let \(H\) and \(K\) be subsets of the vertex set \(V( Q_d)\) of the \(d\)-cube \(Q_d\) (we call \(H\) and \(K\) configurations in \(Q_d)\). We say \(K\) is an exact copy of \(H\) if there is an automorphism of \(Q_d\) which sends \(H\) to \(K\). If \(d\) is a positive integer and \(H\) is a configuration in \(Q_d\), we define \(\lambda(H, d)\) to be the limit as \(n\) goes to infinity of the maximum fraction, over all subsets \(S\) of \(V( Q_n)\), of sub-\(d\)-cubes of \(Q_n\) whose intersection with \(S\) is an exact copy of \(H\). We determine \(\lambda( C_8, 4)\) and \(\lambda( P_4, 3)\) where \(C_8\) is a ``perfect'' 8-cycle in \(Q_4\) and \(P_4\) is a ``perfect'' path with 4 vertices in \(Q_3\), and make conjectures about \(\lambda( C_{2 d}, d)\) and \(\lambda( P_{d + 1}, d)\) for larger values of \(d\). In our proofs there are connections with counting the number of sequences with certain properties and with the inducibility of certain small graphs. In particular, we needed to determine the inducibility of two vertex disjoint edges in the family of bipartite graphs.Erdős-Hajnal-type results for monotone pathshttps://zbmath.org/1472.050842021-11-25T18:46:10.358925Z"Pach, János"https://zbmath.org/authors/?q=ai:pach.janos"Tomon, István"https://zbmath.org/authors/?q=ai:tomon.istvanSummary: An ordered graph is a graph with a linear ordering on its vertex set. We prove that for every positive integer \(k\), there exists a constant \(c_k>0\) such that any ordered graph \(G\) on \(n\) vertices with the property that neither \(G\) nor its complement contains an induced monotone path of size \(k\), has either a clique or an independent set of size at least \(n^{c_k}\). This strengthens a result of \textit{N. Bousquet} et al. [ibid. 113, 261--264 (2015; Zbl 1315.05077)], who proved the analogous result for unordered graphs.
A key idea of the above paper was to show that any unordered graph on \(n\) vertices that does not contain an induced path of size \(k\), and whose maximum degree is at most \(c(k)n\) for some small \(c(k)>0\), contains two disjoint linear size subsets with no edge between them. This approach fails for ordered graphs, because the analogous statement is false for \(k\geq 3\), by a construction of Fox. We provide some further examples showing that this statement also fails for ordered graphs avoiding other ordered trees.Compact cactus representations of all non-trivial min-cutshttps://zbmath.org/1472.050852021-11-25T18:46:10.358925Z"Lo, On-Hei S."https://zbmath.org/authors/?q=ai:lo.on-hei-solomon"Schmidt, Jens M."https://zbmath.org/authors/?q=ai:schmidt.jens-m"Thorup, Mikkel"https://zbmath.org/authors/?q=ai:thorup.mikkelSummary: Recently, \textit{K.-I. Kawarabayashi} and \textit{M. Thorup} [J. ACM 66, No. 1, Article No. 4, 50 p. (2019; Zbl 1426.68217)] presented the first deterministic edge-connectivity recognition algorithm in near-linear time. A crucial step in their algorithm uses the existence of vertex subsets of a simple graph \(G\) on \(n\) vertices whose contractions leave a multigraph with \(\widetilde{O} ( n / \delta )\) vertices and \(\widetilde{O} ( n )\) edges that preserves all non-trivial min-cuts of \(G\), where \(\delta\) is the minimum degree of \(G\) and \(\widetilde{O}\) hides logarithmic factors.
We present a simple argument that improves this contraction-based sparsifier by eliminating the poly-logarithmic factors, that is, we show a contraction-based sparsification that leaves \(O ( n / \delta )\) vertices and \(O ( n )\) edges, preserves all non-trivial min-cuts and can be computed in near-linear time \(\widetilde{O} ( m )\), where \(m\) is the number of edges of \(G\). We also obtain that every simple graph has \(O ( ( n / \delta )^2 )\) non-trivial min-cuts.
Our approach allows to represent all non-trivial min-cuts of a graph by a cactus representation, whose cactus graph has \(O ( n / \delta )\) vertices. Moreover, this cactus representation can be derived directly from the standard cactus representation of all min-cuts in linear time. We apply this compact structure to show that all min-cuts can be explicitly listed in \(\widetilde{O} ( m ) + O ( n^2 / \delta )\) time for every simple graph, which improves the previous best time bound \(O ( n m )\) given by \textit{D. Gusfield} and \textit{D. Naor} [Algorithmica 10, No. 1, 64--89 (1993; Zbl 0781.90087)].On Hamiltonian cycles in balanced \(k\)-partite graphshttps://zbmath.org/1472.050862021-11-25T18:46:10.358925Z"DeBiasio, Louis"https://zbmath.org/authors/?q=ai:debiasio.louis"Spanier, Nicholas"https://zbmath.org/authors/?q=ai:spanier.nicholasSummary: For all integers \(k\) with \(k \geq 2\), if \(G\) is a balanced \(k\)-partite graph on \(n \geq 3\) vertices with minimum degree at least
\[\lceil \frac{ n}{ 2} \rceil + \lfloor \frac{ n + 2}{ 2 \lceil \frac{ k + 1}{ 2} \rceil} \rfloor - \frac{ n}{ k} = \begin{cases} \lceil \frac{ n}{ 2} \rceil + \lfloor \frac{ n + 2}{ k + 1} \rfloor - \frac{ n}{ k} &: k \text{ odd } \\ \frac{ n}{ 2} + \lfloor \frac{ n + 2}{ k + 2} \rfloor - \frac{ n}{ k} &: k \text{ even } \end{cases},\]
then \(G\) has a Hamiltonian cycle unless 4 divides \(n\) and \(k \in \{2, \frac{ n}{ 2} \} \). In the case where 4 divides \(n\) and \(k \in \{2, \frac{ n}{ 2} \} \), we can characterize the graphs which do not have a Hamiltonian cycle and see that \(\lceil \frac{ n}{ 2} \rceil + \lfloor \frac{ n + 2}{ 2 \lceil \frac{ k + 1}{ 2} \rceil} \rfloor - \frac{ n}{ k} + 1\) suffices. This result is tight for all \(k \geq 2\) and \(n \geq 3\) divisible by \(k\).Catlin's reduced graphs with small ordershttps://zbmath.org/1472.050872021-11-25T18:46:10.358925Z"Lai, Hong-Jian"https://zbmath.org/authors/?q=ai:lai.hong-jian"Wang, Keke"https://zbmath.org/authors/?q=ai:wang.keke"Xie, Xiaowei"https://zbmath.org/authors/?q=ai:xie.xiaowei"Zhan, Mingquan"https://zbmath.org/authors/?q=ai:zhan.mingquanSummary: A graph is supereulerian if it has a spanning closed trail. \textit{P. A. Catlin} [Congr. Numerantium 76, 173--181 (1990; Zbl 0862.05058)] raised the problem of determining the reduced nonsupereulerian graphs with small orders, as such results are of particular importance in the study of Eulerian subgraphs and Hamiltonian line graphs. We determine all reduced graphs with order at most 14 and with few vertices of degree 2, extending former results of \textit{W.-G. Chen} and \textit{Z.-H. Chen} [J. Comb. Math. Comb. Comput. 96, 41--63 (2016; Zbl 1351.05136)]. \textit{D. Bauer} [Congr. Numerantium 49, 11--18 (1985; Zbl 0621.05021)] proposed the problems of determining best possible sufficient conditions on minimum degree of a simple graph (or a simple bipartite graph, respectively) \(G\) to ensure that its line graph \(L(G)\) is Hamiltonian. These problems have been settled by \textit{P. A. Catlin} and \textit{H.-J. Lai} [J. Comb. Theory, Ser. B 66, No. 1, 123--139 (1996; Zbl 0839.05070)], respectively. As an application of our main results, we prove the following for a connected simple graph \(G\) on \(n\) vertices:
\par i) If \(\delta (G) \geq \frac{n}{10}\), then for sufficiently large \(n, L(G)\) is Hamilton-connected if and only if both \(\kappa (G) \geq 3\) and \(G\) is not nontrivially contractible to the Wagner graph.
\par ii) If \(G\) is bipartite and \(\delta (G) > \frac{n}{20}\), then for sufficiently large \(n, L(G)\) is Hamilton-connected if and only if both \(\kappa (G) \geq 3\) and \(G\) is not nontrivially contractible to the Wagner graph.Sharp thresholds for nonlinear Hamiltonian cycles in hypergraphshttps://zbmath.org/1472.050882021-11-25T18:46:10.358925Z"Narayanan, Bhargav"https://zbmath.org/authors/?q=ai:narayanan.bhargav-p"Schacht, Mathias"https://zbmath.org/authors/?q=ai:schacht.mathiasFor positive integers \(r>l\), an \(r\)-uniform hypergraph is an \(l\)-cycle if there exists a cyclic ordering of its vertices such that each of its edges consists of \(r\) consecutive vertices, and such that every pair of consecutive edges (in the natural ordering of the edges) intersect in precisely \(l\) vertices; such cycles are said to be linear when \(l=1\), and nonlinear when \(l>1\). The sharp threshold for nonlinear Hamiltonian cycles is determined, together with an explicit formula for \(r>l>1\). This resolves several questions raised by \textit{A. Dudek} and \textit{A. Frieze} [Electron. J. Comb. 18, No. 1, Research Paper P48, 14 p. (2011; Zbl 1218.05174)].New methods to attack the Buratti-Horak-Rosa conjecturehttps://zbmath.org/1472.050892021-11-25T18:46:10.358925Z"Ollis, M. A."https://zbmath.org/authors/?q=ai:ollis.m-a"Pasotti, Anita"https://zbmath.org/authors/?q=ai:pasotti.anita"Pellegrini, Marco A."https://zbmath.org/authors/?q=ai:pellegrini.marco-antonio"Schmitt, John R."https://zbmath.org/authors/?q=ai:schmitt.john-rSummary: The conjecture, still widely open, posed by Marco Buratti, Peter Horak and Alex Rosa states that a list \(L\) of \(v - 1\) positive integers not exceeding \(\lfloor \frac{ v}{ 2} \rfloor\) is the list of edge-lengths of a suitable Hamiltonian path of the complete graph with vertex-set \(\{0, 1, \ldots, v - 1 \}\) if and only if, for every divisor \(d\) of \(v\), the number of multiples of \(d\) appearing in \(L\) is at most \(v - d\). In this paper we present new methods that are based on linear realizations and can be applied to prove the validity of this conjecture for a vast choice of lists. As example of their flexibility, we consider lists whose underlying set is one of the following: \(\{x, y, x + y \}\), \(\{1, 2, 3, 4 \}\), \(\{1, 2, 4, \ldots, 2 x \}\), \(\{1, 2, 4, \ldots, 2 x, 2 x + 1 \}\). We also consider lists with many consecutive elements.A note on equitable Hamiltonian cycleshttps://zbmath.org/1472.050902021-11-25T18:46:10.358925Z"Ophelders, Tim"https://zbmath.org/authors/?q=ai:ophelders.tim"Lambers, Roel"https://zbmath.org/authors/?q=ai:lambers.roel"Spieksma, Frits C. R."https://zbmath.org/authors/?q=ai:spieksma.frits-c-r"Vredeveld, Tjark"https://zbmath.org/authors/?q=ai:vredeveld.tjarkSummary: Given a complete graph with an even number of vertices, and with each edge colored with one of two colors (say red or blue), an equitable Hamiltonian cycle is a Hamiltonian cycle that can be decomposed into two perfect matchings such that both perfect matchings have the same number of red edges. We show that, for any coloring of the edges, in any complete graph on at least 6 vertices, an equitable Hamiltonian cycle exists.Noncorona graphs with strong anti-reciprocal eigenvalue propertyhttps://zbmath.org/1472.050912021-11-25T18:46:10.358925Z"Ahmad, Uzma"https://zbmath.org/authors/?q=ai:ahmad.uzma"Hameed, Saira"https://zbmath.org/authors/?q=ai:hameed.saira"Jabeen, Shaista"https://zbmath.org/authors/?q=ai:jabeen.shaistaSummary: Let \(G\) be a graph having a unique perfect matching and \(A(G)\) be the adjacency matrix of \(G. \, G\) is said to have the strong anti-reciprocal eigenvalue property (property (-SR)) if for each eigenvalue \(\lambda\) of \(A(G)\), its reciprocal \(-1/ \lambda\) is also an eigenvalue of \(A(G)\), with the same multiplicity. In this article, seven classes of noncorona graphs with property (-SR) are obtained.Brouwer type conjecture for the eigenvalues of distance signless Laplacian matrix of a graphhttps://zbmath.org/1472.050922021-11-25T18:46:10.358925Z"Alhevaz, A."https://zbmath.org/authors/?q=ai:alhevaz.abdollah"Baghipur, M."https://zbmath.org/authors/?q=ai:baghipur.maryam"Ganie, Hilal A."https://zbmath.org/authors/?q=ai:ganie.hilal-ahmad"Pirzada, S."https://zbmath.org/authors/?q=ai:pirzada.shariefuddinSummary: Let \(G\) be a simple connected graph with \(n\) vertices, \(m\) edges and having distance signless Laplacian eigenvalues \(\rho_1 \geq \rho_2 \geq \dots \geq \rho_n \geq 0\). For \(1 \leq k \leq n\), let \(M_k (G) = \sum^k_{i=1} \rho_i\) and \(N_k (G) = \sum^{k-1}_{i=0} \rho_{n-i}\) be respectively the sum of \(k\)-largest distance signless Laplacian eigenvalues and the sum of \(k\)-smallest distance signless Laplacian eigenvalues of \(G\). In this paper, we obtain the bounds for \(M_k (G)\) and \(N_k (G)\) in terms of the number of vertices \(n\) and the transmission \(\sigma (G)\) of the graph \(G\). We propose a Brouwer-type conjecture for \(M_k (G)\) and show that it holds for graphs of diameter one and graphs of diameter two for all \(k\). As a consequence, we observe that the conjecture holds for threshold graphs and split graphs (of diameter two). We also show that it holds for \(k=n-1\) and \(n\) for all graphs and for some \(k\) for \(r\)-transmission regular graphs.On distance and Laplacian matrices of trees with matrix weightshttps://zbmath.org/1472.050932021-11-25T18:46:10.358925Z"Atik, Fouzul"https://zbmath.org/authors/?q=ai:atik.fouzul"Kannan, M. Rajesh"https://zbmath.org/authors/?q=ai:rajesh-kannan.m"Bapat, Ravindra B."https://zbmath.org/authors/?q=ai:bapat.ravindra-bhalchandraSummary: The distance matrix of a simple connected graph \(G\) is \(D(G)=(d_{ij})\), where \(d_{ij}\) is the distance between the vertices \(i\) and \(j\) in \(G\). We consider a weighted tree \(T\) on \(n\) vertices with edge weights are square matrices of the same size. The distance \(d_{ij}\) between the vertices \(i\) and \(j\) is the sum of the weight matrices of the edges in the unique path from \(i\) to \(j\). In this article, we establish a characterization for the trees in terms of the rank of (matrix) weighted Laplacian matrix associated with it. We present a necessary and sufficient condition for the distance matrix \(D\), with matrix weights, to be invertible and the formula for the inverse of \(D\), if it exists. Then we study some of the properties of the distance matrices of matrix weighted trees in connection with the Laplacian matrices, incidence matrices, and \(g\)-inverses. Finally, we derive an interlacing inequality for the eigenvalues of distance and Laplacian matrices for the case of positive definite matrix weights.Principal eigenvectors of general hypergraphshttps://zbmath.org/1472.050942021-11-25T18:46:10.358925Z"Cardoso, Kauê"https://zbmath.org/authors/?q=ai:cardoso.kaue"Trevisan, Vilmar"https://zbmath.org/authors/?q=ai:trevisan.vilmarSummary: In this paper, we obtain bounds for the extreme entries of the principal eigenvector of a hypergraph; these bounds are computed using the spectral radius and some classical parameters such as maximum and minimum degree. We also study inequalities involving the ratio and difference between the two extreme entries of this vector.On the signless Laplacian spectral radius of \(K_{s,t}\)-minor free graphshttps://zbmath.org/1472.050952021-11-25T18:46:10.358925Z"Chen, Ming-Zhu"https://zbmath.org/authors/?q=ai:chen.mingzhu"Zhang, Xiao-Dong"https://zbmath.org/authors/?q=ai:zhang.xiaodongSummary: In this paper, we prove that if \(G\) is a \(K_{2,t}\)-minor free graph of order \(n \geq t^2 + 4t + 1\) with \(t \geq 3\), the signless Laplacian spectral radius \(q(G) \leq \frac{1}{2} (n+2t+2+ \sqrt{(n-2t+2)^2 +8t-8})\) with equality if and only if \(n \equiv 1 \pmod t\) and \(G = F_{2,t} (n)\), where \(F_{s,t} (n) := K_{s-1} \vee (p \cdot K_t \cup K_r)\) for \(n-s+1= pt+r\) and \(0 \leq r < t\). In particular, if \(t=3\) and \(n \geq 22\), then \(F_{2,3} (n)\) is the unique \(K_{2,3}\)-minor free graph of order \(n\) with the maximum signless Laplacian spectral radius. In addition, \(F_{3,3} (n)\) is the unique extremal graph with the maximum signless Laplacian spectral radius among all \(K_{3,3}\)-minor free graphs of order \(n \geq 1186\).A bound on the spectral radius of graphs in terms of their Zagreb indiceshttps://zbmath.org/1472.050962021-11-25T18:46:10.358925Z"Feng, Lihua"https://zbmath.org/authors/?q=ai:feng.lihua"Lu, Lu"https://zbmath.org/authors/?q=ai:lu.lu"Réti, Tamás"https://zbmath.org/authors/?q=ai:reti.tamas"Stevanović, Dragan"https://zbmath.org/authors/?q=ai:stevanovic.draganLet \(G=(V,E)\) be a connected and simple graph with \(n\) vertices, \(m\) edges and adjacency matrix \(A\). Denote by \(\lambda_1\geq \dots \geq \lambda_n\) the eigenvalues of \(A\). An eigenvalue is called main if its eigenspace is not perpendicular to the all-one vector \(j\). \textit{E. M. Hagos} [Linear Algebra Appl. 356, No. 1--3, 103--111 (2002; Zbl 1015.05051)] proved that the number of main eigenvalues of \(G\) equals the largest \(k\) such that \(j,Aj,\dots,A^{k-1}j\) are linearly independent.
For a vertex \(v\in V\), denote by \(d_v\) its degree. The first Zagreb index of \(G\) is defined as \[ M_1=\sum_{v\in V}d_v^2=j^TA^2j, \] and the second Zagreb index of \(G\) is defined as: \[ M_2=\sum_{uv\in E}d_ud_v=\frac{j^TA^3j}{2}. \]
Let \(\alpha=\frac{2(nM_2-mM_1)}{nM_1-4m^2}\) and \(\beta=\frac{M_1^2-4mM_2}{nM_1-4m^2}\). The main result of this paper is that \[ \lambda_1\geq \frac{\alpha+\sqrt{\alpha^2+4\beta}}{2}, \] with equality if and only if \(G\) has exactly two main eigenvalues. Explicit comparisons with other bounds on \(\lambda_1\) are also provided in the paper.Graphs cospectral with \(\operatorname{NU}(n + 1,q^2)\), \(n \neq 3\)https://zbmath.org/1472.050972021-11-25T18:46:10.358925Z"Ihringer, Ferdinand"https://zbmath.org/authors/?q=ai:ihringer.ferdinand"Pavese, Francesco"https://zbmath.org/authors/?q=ai:pavese.francesco"Smaldore, Valentino"https://zbmath.org/authors/?q=ai:smaldore.valentinoSummary: Let \(\mathcal{H}(n, q^2)\) be a non-degenerate Hermitian variety of \(\operatorname{PG}(n, q^2)\), \(n \geq 2\). Let \(\operatorname{NU}(n + 1, q^2)\) be the graph whose vertices are the points of \(\operatorname{PG}(n, q^2) \setminus \mathcal{H}(n, q^2)\) and two vertices \(P_1\), \(P_2\) are adjacent if the line joining \(P_1\) and \(P_2\) is tangent to \(\mathcal{H}(n, q^2)\). Then \(\operatorname{NU}(n + 1, q^2)\) is a strongly regular graph. In this paper we show that \(\operatorname{NU}(n + 1, q^2)\), \(n \neq 3\), is not determined by its spectrum.Distance spectral radii of \(k\)-uniform hypertrees with given parametershttps://zbmath.org/1472.050982021-11-25T18:46:10.358925Z"Liu, Xiangxiang"https://zbmath.org/authors/?q=ai:liu.xiangxiang"Wang, Ligong"https://zbmath.org/authors/?q=ai:wang.ligong"Li, Xihe"https://zbmath.org/authors/?q=ai:li.xiheSummary: The distance spectral radius of a connected hypergraph is the largest eigenvalue of its distance matrix. In this paper, we obtain the \(k\)-uniform hypertree with minimum distance spectral radius among all \(k\)-uniform hypertrees with \(n\) vertices and matching number \(r\). We also characterize the \(k\)-uniform hypertree with minimum distance spectral radius among all \(k\)-uniform hypertrees with \(n\) vertices and independence number \(\alpha\).Computation of general Randić polynomial and general Randić energy of some graphshttps://zbmath.org/1472.050992021-11-25T18:46:10.358925Z"Ramane, Harishchandra S."https://zbmath.org/authors/?q=ai:ramane.harishchandra-s"Gudodagi, Gouramma A."https://zbmath.org/authors/?q=ai:gudodagi.gouramma-aThe general Randić matrix of a graph \(G\) of order \(n\), denoted by \(\operatorname{GR}(G)\), is the \(n \times n\) matrix whose \((i, j)\)-entry is \((d_{i}d_{j})^{\alpha}\), \(\alpha \in \mathbb{R}\), if the vertices \(v_i\) and \(v_j\) are adjacent and \(0\) otherwise, where \(d_{i}\) is the degree of \(v_{i}\) for \(i = 1, \ldots, n\). The general Randić polynomial of \(G\) is simply the characteristic polynomial of \(\operatorname{GR}(G)\), and the general Randić energy \(\operatorname{EGR}(G)\) of \(G\) is defined as the sum of the absolute values of the eigenvalues of \(\operatorname{GR}(G)\). In the particular case when \(\alpha = -\frac{1}{2}\), \(\operatorname{EGR}(G)\) is the usual Randić energy of \(G\). In this paper, the authors compute the general Randić polynomial and the general Randić energy of paths, cycles, complete graphs, complete bipartite graphs, friendship graphs and Dutch windmill graphs.A conjecture on the eigenvalues of threshold graphshttps://zbmath.org/1472.051002021-11-25T18:46:10.358925Z"Tura, Fernando C."https://zbmath.org/authors/?q=ai:tura.fernando-colmanA simple graph \(G = (V, E)\) is called a threshold graph if there exists a function \(w: V \rightarrow [0,\infty)\) and a non-negative real number \(t\) such that \(uv \in E\) if and only if \(w(u) + w(v) \ge t\). A simple graph is called an anti-regular graph if only two vertices of it have equal degrees. It is known that, up to isomorphism, there is exactly one connected anti-regular graph on \(n\) vertices, which is denoted as \(A_n\). Anti-regular graphs form a subfamily of the family of threshold graphs. In [\textit{C. O. Aguilar} et al., Linear Algebra Appl. 557, 84--104 (2018; Zbl 1396.05064)], it was conjectured that for each \(n\), \(A_n\) has the smallest positive eigenvalue and the largest negative eigenvalue less than \(-1\) among all threshold graphs on \(n\) vertices. In [\textit{C. O. Aguilar} et al., ibid. 588, 210--223 (2020; Zbl 1437.05128)], this conjecture was proved for all threshold graphs on \(n\) vertices except for \(n - 2\) critical cases where the interlacing method fails. In this paper, the author deals with these cases and thereby completes the proof of this conjecture.Nullity of a graph in terms of path cover numberhttps://zbmath.org/1472.051012021-11-25T18:46:10.358925Z"Wang, Long"https://zbmath.org/authors/?q=ai:wang.long.1|wang.longSummary: The nullity of a graph \(G\), written as \(\eta (G)\), is defined to be the multiplicity of zero eigenvalues of its adjacency matrix. A path cover of \(G\) is a set of vertex-disjoint induced paths in \(G\) such that every vertex of \(G\) is a vertex of one of the paths. The path cover number of \(G\), written as \(\rho (G)\), is the minimum size of a path cover of \(G\). In this article, we prove that \(\eta (G) \leq \rho (G)\) for a connected graph \(G\) in which every block is a cycle or a clique. Furthermore, if every block of \(G\) has at least three vertices, then \(\eta (G) = \rho (G)\) if and only if every block of \(G\) is a cycle of size a multiple of 4.Generalized spectral characterization of mixed graphshttps://zbmath.org/1472.051022021-11-25T18:46:10.358925Z"Wang, Wei"https://zbmath.org/authors/?q=ai:wang.wei.26"Qiu, Lihong"https://zbmath.org/authors/?q=ai:qiu.lihong"Qian, Jianguo"https://zbmath.org/authors/?q=ai:qian.jianguo"Wang, Wei"https://zbmath.org/authors/?q=ai:wang.wei.5A mixed graph \(G\) is said to be strongly determined by its generalized Hermitian spectrum (abbreviated SHDGS), if, up to isomorphism, \(G\) is the unique mixed graph that is cospectral with \(G\) w.r.t. the generalized Hermitian spectrum. The authors conjecture that every such graph is SHDGS and prove that, for any mixed graph \(H\) that is cospectral with \(G\) w.r.t. the generalized Hermitian spectrum, there exists a Gaussian rational unitary matrix \(U\) with \(Ue = e\) such that \(U^\ast A(G)U = A(H)\) and \((1+i)U\) is a Gaussian integral matrix. The authors verify the conjecture in two extremal cases when \(G\) is either an undirected graph or a self-converse oriented graph. Consequently, the authors prove that all directed paths of even order are SHDGS.Bounds for the spectral radius and energy of extended adjacency matrix of graphshttps://zbmath.org/1472.051032021-11-25T18:46:10.358925Z"Wang, Zhao"https://zbmath.org/authors/?q=ai:wang.zhao"Mao, Yaping"https://zbmath.org/authors/?q=ai:mao.yaping"Furtula, Boris"https://zbmath.org/authors/?q=ai:furtula.boris"Wang, Xu"https://zbmath.org/authors/?q=ai:wang.xu|wang.xu.3|wang.xu.2|wang.xu.5|wang.xu.1|wang.xu.4Summary: An extend adjacency matrix of a graph \((A_{ex})\) was introduced decades ago as a precursor for developing a few quite useful molecular topological descriptors. The spectral radius \((\eta_1)\) of the extended adjacency matrix and the extended energy of a graph \((\mathcal{E}_{ex})\) have been successfully utilized in QSPR/QSAR investigations. Here, the \(\eta_1\) and \(\mathcal{E}_{ex}\) have been further mathematically analyzed. Several sharp upper bounds for the \(\mathcal{E}_{ex}\) are obtained. In addition, the Nordhaus-Gaddum-type results for the \(\eta_1\) and \(\mathcal{E}_{ex}\) are presented.Bounds on the nullity, the \(H\)-rank and the Hermitian energy of a mixed graphhttps://zbmath.org/1472.051042021-11-25T18:46:10.358925Z"Wei, Wei"https://zbmath.org/authors/?q=ai:wei.wei.6|wei.wei.5|wei.wei.2|wei.wei.3|wei.wei.7|wei.wei.4"Li, Shuchao"https://zbmath.org/authors/?q=ai:li.shuchao"Ma, Hongping"https://zbmath.org/authors/?q=ai:ma.hongpingSummary: Given an \(n\)-vertex graph \(G\) with maximum degree \(\Delta\), the mixed graph \(D_G\) is constructed from \(G\) by orienting some of its edges, where \(G\) is called the underlying graph of \(D_G\). Let \(r_H (D_G)\) be the \(H\)-rank of \(D_G\) and let \(\alpha (G)\) be the independence number of \(G\). In this paper, we firstly determine the maximum nullity of \(n\)-vertex mixed graphs with maximum degree \(\Delta\). The corresponding extremal graphs are identified. We secondly establish upper and lower bounds on \(r_H (D_G) + \alpha (G), r_H (D_G)-\alpha (G), r_H (D_G)/\alpha (G)\), respectively. Finally, we characterize the relationship between the Hermitian energy and the \(H\)-rank of a mixed graph.Further results on the least \(Q\)-eigenvalue of a graph with fixed domination numberhttps://zbmath.org/1472.051052021-11-25T18:46:10.358925Z"Yu, Guanglong"https://zbmath.org/authors/?q=ai:yu.guanglong"Wu, Yarong"https://zbmath.org/authors/?q=ai:wu.yarong"Zhai, Mingqing"https://zbmath.org/authors/?q=ai:zhai.mingqingSummary: In this paper, we proceed on determining the minimum \(q_{\min}\) among the connected nonbipartite graphs on \(n \geq 5\) vertices and with domination number \(\frac{n+1}{3} < \gamma \leq \frac{n-1}{2}\). Further results obtained are as follows:
\par i) among all nonbipartite connected graph of order \(n \geq 5\) and with domination number \(\frac{n-1}{2}\), the minimum \(q_{\min}\) is completely determined;
\par ii) among all nonbipartite graphs of order \(n \geq 5\), with odd-girth \(g_o \leq 5\) and domination number at least \(\frac{n+1}{3} < \gamma \leq \frac{n-2}{2}\), the minimum \(q_{\min}\) is completely determined.A note on spectral radius and degree deviation in graphshttps://zbmath.org/1472.051062021-11-25T18:46:10.358925Z"Zhang, Wenqian"https://zbmath.org/authors/?q=ai:zhang.wenqianThe study of spectral radius of irregular graphs is an interesting topic. Let \(G\) be a graph on \(n\) vertices with \(m\) edges, and \(d_1, d_2, \ldots, d_n\) be the degrees of the vertices in \(G\). Set \[ s(G) = \sum_{1\leq i\leq n}\left|d_i - \frac{2m}{n}\right|, \] which is called the degree deviation. Let \(\lambda_1(G)\) denote the largest eigenvalue of the adjacency matrix of \(G\). \textit{V. Nikiforov} [Linear Algebra Appl. 414, No. 1, 347--360 (2006; Zbl 1092.05046)] proved that \[ \frac{s^2(G)}{2n^2\sqrt{2m}}\leq \lambda_1(G)- \frac{2m}{n} \le \sqrt{s(G)}, \] and conjectured that \[ \lambda_1(G)- \frac{2m}{n} \le \sqrt{\frac{1}{2}s(G)}, \] if \(n\) and \(m\) are large enough.
In this paper, the author prove that \[ \lambda_1(G)- \frac{2m}{n} \le \sqrt{\frac{9}{10}s(G)}, \] which improves the result in [loc. cit.]. Moreover, the author confirm that the above conjecture is valid if \(G\) is bipartite.Decompositions of complete graphs and complete bipartite graphs into bipartite cubic graphs of order at most 12https://zbmath.org/1472.051072021-11-25T18:46:10.358925Z"Adams, Peter"https://zbmath.org/authors/?q=ai:adams.peter-j"Bunge, Ryan C."https://zbmath.org/authors/?q=ai:bunge.ryan-c"Eggleton, Roger B."https://zbmath.org/authors/?q=ai:eggleton.roger-b"El-Zanati, Saad I."https://zbmath.org/authors/?q=ai:el-zanati.saad-i-el-zanati"Odabaşı, U."https://zbmath.org/authors/?q=ai:odabasi.ugur"Wannasit, Wannasiri"https://zbmath.org/authors/?q=ai:wannasit.wannasiriSummary: There are ten bipartite cubic graphs of order \(n\ge 12\). For each such graph \(G\) we give necessary and sufficient conditions for the existence of decompositions of \(K_n\) and of \(K_{m,n}\) into copies of \(G\).On the balanceability of some graph classeshttps://zbmath.org/1472.051082021-11-25T18:46:10.358925Z"Dailly, Antoine"https://zbmath.org/authors/?q=ai:dailly.antoine"Hansberg, Adriana"https://zbmath.org/authors/?q=ai:hansberg.adriana"Ventura, Denae"https://zbmath.org/authors/?q=ai:ventura.denaeSummary: Given a graph \(G\), a 2-coloring of the edges of \(K_n\) is said to contain a balanced copy of \(G\) if we can find a copy of \(G\) such that half of its edges are in each color class. If, for every sufficiently large \(n\), there exists an integer \(k\) such that every 2-coloring of \(K_n\) with more than \(k\) edges in each color class contains a balanced copy of \(G\), then we say that \(G\) is balanceable. Balanceability was introduced by \textit{Y. Caro} et al. [Graphs Comb. 35, No. 4, 855--865 (2019; Zbl 1416.05186); ``Unavoidable chromatic patterns in 2-colorings of the complete graph'', Preprint, \url{arXiv:1810.12375}], who also gave a structural characterization of balanceable graphs.
In this paper, we extend the study of balanceability by finding new sufficient conditions for a graph to be balanceable or not. We use those conditions to fully characterize the balanceability of graph classes such as rectangular and triangular grids, as well as a special class of circulant graphs.The anti-Ramsey numbers of \(C_3\) and \(C_4\) in complete \(r\)-partite graphshttps://zbmath.org/1472.051092021-11-25T18:46:10.358925Z"Fang, Chunqiu"https://zbmath.org/authors/?q=ai:fang.chunqiu"Győri, Ervin"https://zbmath.org/authors/?q=ai:gyori.ervin"Li, Binlong"https://zbmath.org/authors/?q=ai:li.binlong"Xiao, Jimeng"https://zbmath.org/authors/?q=ai:xiao.jimengSummary: A subgraph of an edge-colored graph is rainbow, if all of its edges have different colors. For a graph \(G\) and a family \(\mathcal{H}\) of graphs, the anti-Ramsey number \(\operatorname{ar}(G, \mathcal{H})\) is the maximum number \(k\) such that there exists an edge-coloring of \(G\) with exactly \(k\) colors without rainbow copy of any graph in \(\mathcal{H} \). In this paper, we study the anti-Ramsey numbers of \(C_3\) and \(C_4\) in complete \(r\)-partite graphs. For \(r \geq 3\) and \(n_1 \geq n_2 \geq \cdots \geq n_r \geq 1\), we determine \(\operatorname{ar}( K_{n_1 , n_2 , \ldots , n_r}, \{ C_3, C_4 \})\), \(\operatorname{ar}( K_{n_1 , n_2 , \ldots , n_r}, C_3)\) and \(\operatorname{ar}( K_{n_1 , n_2 , \ldots , n_r}, C_4)\).The game chromatic number of corona of two graphshttps://zbmath.org/1472.051102021-11-25T18:46:10.358925Z"Alagammai, R."https://zbmath.org/authors/?q=ai:alagammai.r"Vijayalakshmi, V."https://zbmath.org/authors/?q=ai:vijayalakshmi.vIn this article, the authors study the game chromatic number of two graphs for certain choices of graphs.
The game chromatic number \(\chi_g(G)\) is defined as the least number of colors in the color set \(X\) for which the first player has a winning strategy in the following two-person coloring game on \(G\).
Alternately, both players properly color a vertex of \(G\) with a color from \(X\). The first player wins if all vertices of \(G\) are colored at the end. The second player wins if at any stage of the game, there is an uncolored vertex \(v\) such that for every \(x \in X\), \(v\) has a neighbour colored with \(x.\)
The corona of two graphs \(G\) and \(H\), having order \(n\) and \(m\) respectively, is the graph \(G \circ H\) formed from one copy of \(G\) and \(n\) copies of \(H\) where the \(i^{\text{th}}\) vertex of \(G\) is adjacent to every vertex in the \(i^{\text{th}}\) copy of \(H\).
Noting that \(G \circ H\) is a union (on the same vertex set) of \(G \cup nH\) and \(n S_{m+1}\), as a corollary of Theorem \(2\) in [\textit{D. J. Guan} and \textit{X. Zhu}, J. Graph Theory 30, No. 1, 67--70 (1999; Zbl 0929.05032)], the authors derive that \(\chi_g( G \circ H) \le \max\{\Delta(G), \Delta(H) \}+2\) as one of their main results.
Note that as a corollary, \(\chi_g(P_n \circ P_m) \le 4\). The authors prove that this is tight if \(n \ge 6\) and \(m=2\), as well as when \(n \ge 2\) and \(m \ge 3\) by proving that the second player has a winning strategy in these cases if \(X\) contains only three colors.
Other examples they study, are \(\chi_g(P_n \circ C_m)\), \(\chi_g(P_n \circ K_{a,b})\), \(\chi_g(P_n \circ W_m)\), \(\chi_g(S_{m+1} \circ P_n)\), \(\chi_g(K_n \circ K_n)\) and \(\chi_g(K_n \circ K_{n,n})\). These are (for sufficiently large parameters) equal to 4, 4, 5, 4, \(n+1\) and \(n+1\) respectively.Correction to: ``Properties of a \(q\)-analogue of zero forcing''https://zbmath.org/1472.051112021-11-25T18:46:10.358925Z"Butler, Steve"https://zbmath.org/authors/?q=ai:butler.steve"Erickson, Craig"https://zbmath.org/authors/?q=ai:erickson.craig"Fallat, Shaun"https://zbmath.org/authors/?q=ai:fallat.shaun-m"Hall, H. Tracy"https://zbmath.org/authors/?q=ai:hall.h-tracy"Kroschel, Brenda"https://zbmath.org/authors/?q=ai:kroschel.brenda-k"Lin, Jephian C.-H."https://zbmath.org/authors/?q=ai:lin.jephian-chin-hung"Shader, Bryan"https://zbmath.org/authors/?q=ai:shader.bryan-l"Warnberg, Nathan"https://zbmath.org/authors/?q=ai:warnberg.nathan"Yang, Boting"https://zbmath.org/authors/?q=ai:yang.botingFrom the text: The LaTeX control sequence \(\setminus\)\(\deg\) is interpreted differently in the authors' and the publisher's LaTeX setting. The authors' intention is deg, as the degree of a vertex on a graph; however, it becomes a circle, as the degree for temperature, in the publisher's environment. As a consequence, the mistakes occur whenever the macro \(\setminus\)\(\deg\) is used. The original article [the authors, ibid. 36, No. 5, 1401--1419 (2020; Zbl 1458.05166)] has been corrected.Integral flow and cycle chip-firing on graphshttps://zbmath.org/1472.051122021-11-25T18:46:10.358925Z"Dochtermann, Anton"https://zbmath.org/authors/?q=ai:dochtermann.anton"Meyers, Eli"https://zbmath.org/authors/?q=ai:meyers.eli"Samavedam, Rahgav"https://zbmath.org/authors/?q=ai:samavedam.rahgav"Yi, Alex"https://zbmath.org/authors/?q=ai:yi.alexSummary: Motivated by the notion of chip-firing on the dual graph of a planar graph, we consider `integral flow chip-firing' on an arbitrary graph \(G\). The chip-firing rule is governed by \(\mathcal{L}^\ast(G)\), the dual Laplacian of \(G\) determined by choosing a basis for the lattice of integral flows on \(G\). We show that any graph admits such a basis so that \(\mathcal{L}^\ast(G)\) is an M-matrix, leading to a firing rule on these basis elements that is avalanche finite. This follows from a more general result on bases of integral lattices that may be of independent interest. Our results provide a notion of \(z\)-superstable flow configurations that are in bijection with the set of spanning trees of \(G\). We show that for planar graphs, as well as for the graphs \(K_5\) and \(K_{3,3}\), one can find such a flow M-basis that consists of cycles of the underlying graph. We consider the question for arbitrary graphs and address some open questions.The simplex geometry of graphshttps://zbmath.org/1472.051132021-11-25T18:46:10.358925Z"Devriendt, Karel"https://zbmath.org/authors/?q=ai:devriendt.karel"Van Mieghem, Piet"https://zbmath.org/authors/?q=ai:van-mieghem.pietSummary: Graphs are a central object of study in various scientific fields, such as discrete mathematics, theoretical computer science and network science. These graphs are typically studied using combinatorial, algebraic or probabilistic methods, each of which highlights the properties of graphs in a unique way. Here, we discuss a novel approach to study graphs: the simplex geometry (a simplex is a generalized triangle). This perspective, proposed by \textit{M. Fiedler} [Matrices and graphs in geometry. Cambridge: Cambridge University Press (2011; Zbl 1225.51017)], introduces techniques from (simplex) geometry into the field of graph theory and conversely, via an exact correspondence. We introduce this graph-simplex correspondence, identify a number of basic connections between graph characteristics and simplex properties, and suggest some applications as example.Separation dimension and degreehttps://zbmath.org/1472.051142021-11-25T18:46:10.358925Z"Scott, Alex"https://zbmath.org/authors/?q=ai:scott.alexander-d"Wood, David R."https://zbmath.org/authors/?q=ai:wood.david-ronaldSummary: The separation dimension of a graph \(G\) is the minimum positive integer \(d\) for which there is an embedding of \(G\) into \(\mathbb{R}^d\), such that every pair of disjoint edges are separated by some axis-parallel hyperplane. We prove a conjecture of \textit{N. Alon} et al. [SIAM J. Discrete Math. 29, No. 1, 59--64 (2015; Zbl 1327.05245)] by showing that every graph with maximum degree \(\Delta\) has separation dimension less than \(20\Delta\), which is best possible up to a constant factor. We also prove that graphs with separation dimension 3 have bounded average degree and bounded chromatic number, partially resolving an open problem by \textit{N. Alon} et al. [J. Graph Theory 89, No. 1, 14--25 (2018; Zbl 1398.05138)].On problems of \(\mathcal{CF}\)-connected graphs for \({K}_{{m,n}} \)https://zbmath.org/1472.051152021-11-25T18:46:10.358925Z"Staš, Michal"https://zbmath.org/authors/?q=ai:stas.michal"Valiska, Juraj"https://zbmath.org/authors/?q=ai:valiska.jurajSummary: A connected graph \(G\) is \(\mathcal{CF} \)-connected if there is a path between every pair of vertices with no crossing on its edges for each optimal drawing of \(G\). We conjecture that a complete bipartite graph \(K_{m,n}\) is \(\mathcal{CF} \)-connected if and only if it does not contain a subgraph of \(K_{3,6}\) or \(K_{4,4}\). We establish the validity of this conjecture for all complete bipartite graphs \(K_{m,n}\) for any \(m,n\) with \(\min \{m,n\}\leq 6\), and conditionally for \(m,n\geq 7\) on the assumption of Zarankiewicz's conjecture that \(\operatorname{cr}(K_{m,n})=\left\lfloor \frac{m}{2} \right \rfloor \left \lfloor \frac{m-1}{2} \right \rfloor \left \lfloor \frac{n}{2} \right \rfloor \left \lfloor \frac{n-1}{2} \right \rfloor \).Inner-outer curvatures, Ollivier-Ricci curvature and volume growth of graphshttps://zbmath.org/1472.051162021-11-25T18:46:10.358925Z"Adriani, Andrea"https://zbmath.org/authors/?q=ai:adriani.andrea"Setti, Alberto G."https://zbmath.org/authors/?q=ai:setti.alberto-gSummary: We are concerned with the study of different notions of curvature on graphs. We show that if a graph has stronger inner-outer curvature growth than a model graph, then it has faster volume growth too. We also study the relationships of volume growth with other kind of curvatures, such as the Ollivier-Ricci curvature.Sampling hypergraphs with given degreeshttps://zbmath.org/1472.051172021-11-25T18:46:10.358925Z"Dyer, Martin"https://zbmath.org/authors/?q=ai:dyer.martin-e"Greenhill, Catherine"https://zbmath.org/authors/?q=ai:greenhill.catherine-s"Kleer, Pieter"https://zbmath.org/authors/?q=ai:kleer.pieter"Ross, James"https://zbmath.org/authors/?q=ai:ross.james-lance|ross.james-e|ross.james-b"Stougie, Leen"https://zbmath.org/authors/?q=ai:stougie.leenSummary: There is a well-known connection between hypergraphs and bipartite graphs, obtained by treating the incidence matrix of the hypergraph as the biadjacency matrix of a bipartite graph. We use this connection to describe and analyse a rejection sampling algorithm for sampling simple uniform hypergraphs with a given degree sequence. Our algorithm uses, as a black box, an algorithm \(\mathcal{A}\) for sampling bipartite graphs with given degrees, uniformly or nearly uniformly, in (expected) polynomial time. The expected runtime of the hypergraph sampling algorithm depends on the (expected) runtime of the bipartite graph sampling algorithm \(\mathcal{A} \), and the probability that a uniformly random bipartite graph with given degrees corresponds to a simple hypergraph. We give some conditions on the hypergraph degree sequence which guarantee that this probability is bounded below by a positive constant.Normalized Laplace operators for hypergraphs with real coefficientshttps://zbmath.org/1472.051182021-11-25T18:46:10.358925Z"Jost, Jürgen"https://zbmath.org/authors/?q=ai:jost.jurgen"Mulas, Raffaella"https://zbmath.org/authors/?q=ai:mulas.raffaellaSummary: Chemical hypergraphs and their associated normalized Laplace operators are generalized and studied in the case where each vertex-hyperedge incidence has a real coefficient. We systematically study the effect of symmetries of a hypergraph on the spectrum of the Laplacian.Proof of the Brown-Erdős-Sós conjecture in groupshttps://zbmath.org/1472.051192021-11-25T18:46:10.358925Z"Nenadov, Rajko"https://zbmath.org/authors/?q=ai:nenadov.rajko"Sudakov, Benny"https://zbmath.org/authors/?q=ai:sudakov.benny"Tyomkyn, Mykhaylo"https://zbmath.org/authors/?q=ai:tyomkyn.mykhayloSummary: The conjecture of \textit{V. T. Sós} et al. [Period. Math. Hung. 3, 221--228 (1973; Zbl 0269.05111)] states that, for any \(k\geq 3\), if a 3-uniform hypergraph \(H\) with \(n\) vertices does not contain a set of \(k+3\) vertices spanning at least \(k\) edges then it has \(o(n^2)\) edges. The case \(k=3\) of this conjecture is the celebrated (6, 3)-theorem of \textit{I. Z. Ruzsa} and \textit{E. Szemerédi} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. János Bolyai 18, 939--945 (1978; Zbl 0393.05031)] which implies Roth's theorem on 3-term arithmetic progressions in dense sets of integers. \textit{J. Solymosi} [Comb. Probab. Comput. 24, No. 4, 680--686 (2015; Zbl 1371.05302)] observed that, in order to prove the conjecture, one can assume that \(H\) consists of triples \((a,b,ab)\) of some finite quasigroup \(\Gamma\). Since this problem remains open for all \(k\geq 4\), he further proposed to study triple systems coming from finite groups. In this case he proved that the conjecture holds also for \(k=4\). Here we completely resolve the Brown-Erdős-Sós conjecture for all finite groups and values of \(k\). Moreover, we prove that the hypergraphs coming from groups contain sets of size \(\Theta(\sqrt k)\) which span \(k\) edges. This is best possible and goes far beyond the conjecture.The second largest spectral radii of uniform hypertrees with given size of matchinghttps://zbmath.org/1472.051202021-11-25T18:46:10.358925Z"Su, Li"https://zbmath.org/authors/?q=ai:su.li"Kang, Liying"https://zbmath.org/authors/?q=ai:kang.liying"Li, Honghai"https://zbmath.org/authors/?q=ai:li.honghai"Shan, Erfang"https://zbmath.org/authors/?q=ai:shan.erfangSummary: In this paper, we almost completely determine the hypertrees with the second largest spectral radius among all hypertrees with \(m\) edges and given size of matching by using the matching polynomials and ordering of hypertrees.Complexity and characterization aspects of edge-related domination for graphshttps://zbmath.org/1472.051212021-11-25T18:46:10.358925Z"Pan, Zhuo"https://zbmath.org/authors/?q=ai:pan.zhuo"Li, Xianyue"https://zbmath.org/authors/?q=ai:li.xianyue"Xu, Shou-Jun"https://zbmath.org/authors/?q=ai:xu.shoujunLet \(G=(V,E)\) be a connected graph. A subset \(F\) of \(E\) is an edge dominating set (resp. a total edge dominating set) if every edge in \(E \setminus F\) (resp. in \(E\)) is adjacent to at least one edge in \(F\). The minimum cardinality of an edge dominating set (resp. a total edge dominating set) of \(G\) is the edge domination number (resp. total edge domination number) of \(G\), denoted by \(\gamma^\prime(G)\) (resp. \(\gamma_t^\prime(G)\)). A semitotal edge dominating set is an edge dominating set \(S\) such that, for every edge \(e\) in \(S\), there exists such an edge \(e^\prime\) in \(S\) that \(e\) either is adjacent to \(e^\prime\) or shares a common neighbor edge with \(e^\prime\). The semitotal edge domination number, denoted by \(\gamma_{st}^\prime (G)\), is the minimum cardinality of a semitotal edge dominating set of \(G\). It is obvious from definitions that \(\gamma'(G)\le \gamma_{st}^\prime (G) \le \gamma_t^\prime(G)\).
In this paper, the authors first prove that the problem of deciding whether \(\gamma^\prime(G) = \gamma_{st}^\prime (G)\) is NP-hard in planar bipartite graphs with maximum degree 4, and then prove that the problem of deciding whether \(\gamma^\prime(G) = \gamma_{t}^\prime (G)\) is NP-hard in planar graphs with maximum degree 4, respectively. Furthermore, they characterize trees with equal edge domination and semitotal edge domination numbers.Maximal independent sets and regularity of graphshttps://zbmath.org/1472.051222021-11-25T18:46:10.358925Z"Trung, Tran Nam"https://zbmath.org/authors/?q=ai:trung.tran-namA general stochastic matching model on multigraphshttps://zbmath.org/1472.051232021-11-25T18:46:10.358925Z"Begeot, Jocelyn"https://zbmath.org/authors/?q=ai:begeot.jocelyn"Marcovici, Irène"https://zbmath.org/authors/?q=ai:marcovici.irene"Moyal, Pascal"https://zbmath.org/authors/?q=ai:moyal.pascal"Rahme, Youssef"https://zbmath.org/authors/?q=ai:rahme.youssefSummary: We extend the general stochastic matching model on graphs introduced in \textit{J. Mairesse} and \textit{P. Moyal} [J. Appl. Probab. 53, No. 4, 1064--1077 (2016; Zbl 1356.60147)], to matching models on multigraphs, that is, graphs with self-loops. The evolution of the model can be described by a discrete time Markov chain whose positive recurrence is investigated. Necessary and sufficient stability conditions are provided, together with the explicit form of the stationary probability in the case where the matching policy is `First Come, First Matched'.Short proofs on the structure of general partition, equistable and triangle graphshttps://zbmath.org/1472.051242021-11-25T18:46:10.358925Z"Cerioli, Márcia R."https://zbmath.org/authors/?q=ai:cerioli.marcia-r"Martins, Taísa"https://zbmath.org/authors/?q=ai:martins.taisa-lSummary: While presenting a combinatorial point of view to the class of equistable graphs, \textit{Š. Miklavič} and \textit{M. Milanič} [ibid. 159, No. 11, 1148--1159 (2011; Zbl 1223.05235)] pointed out the inclusions among the classes of equistable, general partition and triangle graphs. \textit{J. Orlin} [Ann. Discrete Math. 1, 415--419 (1977; Zbl 0361.05039)] conjectured that the first two classes were equivalent, but \textit{M. Milanič} and \textit{N. Trotignon} [J. Graph Theory 84, No. 4, 536--551 (2017; Zbl 1359.05054)] showed that the three classes are all distinct, leading to the following hierarchy: general partition graphs \(\subset\) equistable graphs \(\subset\) triangle graphs.
In this paper, we solve an open problem from \textit{C. Anbeek} et al. [Australas. J. Comb. 16, 285--293 (1997; Zbl 0884.05076)] by showing that all the above classes are equivalent when restricted to planar graphs. Additionally, we present short proofs on some structural properties of the general partition, equistable and triangle classes. In particular, we show that the general partition and triangle classes are both closed under the operations of substitution, induction and contraction of modules which allow us to recover several results in the literature.Disjoint odd cycles in cubic solid brickshttps://zbmath.org/1472.051252021-11-25T18:46:10.358925Z"Chen, Guantao"https://zbmath.org/authors/?q=ai:chen.guantao"Feng, Xing"https://zbmath.org/authors/?q=ai:feng.xing"Lu, Fuliang"https://zbmath.org/authors/?q=ai:lu.fuliang"Zhang, Lianzhu"https://zbmath.org/authors/?q=ai:zhang.lianzhuThe paper under review dispute the conjecture of \textit{C. L. Lucchesi} et al. [SIAM J. Discrete Math. 32, No. 2, 1478--1504 (2018; Zbl 1395.05135)] that no cubic solid brick contains two vertex-disjoint odd cycles.
An edge cut of a graph \(G\) can be defined as the set of edges between a vertex subset \(X\) and \(V(G)\backslash X\), and denoted by \(\partial(X)\). Further, let \(G/X\) and \(G/\overline{X}\) be obtained from \(G\) by contracting \(X\) and \(\overline{X}\), respectively, and call them \(\partial (X)\)-contractions of \(G\). A connected graph with at least two vertices is matching covered if each of its edges is contained in some perfect matching. Let \(G\) be a matching covered graph. \(\partial(X)\) is called a separating cut if both \(\partial (X)\)-contractions of \(G\) are also matching covered and is called a tight cut if \(| \partial (X) \cap M| = 1\) for each perfect matching \(M\) of \(G\). (And hence a tight cut is always a separating cut.)
A trivial edge cut \(\partial(X)\) is one that with \(|X|=1\) or \(|\overline{X}|=1\). A matching covered graph without nontrivial tight cuts is called a brick, if it is nonbipartite and a brace if it is bipartite.
A matching covered graph is solid if each of its separating cuts is tight. It can be shown that every bipartite matching covered graph is solid. In the nonbipartite case, researches focus on solid bricks, which are nonbipartite matching covered graph containing no nontrivial separating cuts.
\textit{T. M. M. de Carvalho} et al. [Int. J. Pure Appl. Math. 15, No. 3, 319--332 (2004; Zbl 1078.30046), Theorem 3.5] presented a proof of a theorem of Reed and Wakabayashi that a brick \(G\) is nonsolid if and only if there exist two vertex-disjoint odd cycles \(C_1\) and \(C_2\) such that \(G - V(C 1 \cup C 2)\) has a perfect matching. Consequently, every brick with no two vertex-disjoint odd cycles is solid. Lucchesi et al. [loc. cit.] constructed infinite families of solid bricks containing two vertex-disjoint odd cycles, none of which is cubic. Thus they conjectured that no cubic solid brick contains two vertex-disjoint odd cycles. In this work, the authors construct an infinite family of graphs showing that this conjecture fails, and show that the minimum counterexample is unique, which has 12 vertices.Perfect matchings and Hamiltonicity in the Cartesian product of cycleshttps://zbmath.org/1472.051262021-11-25T18:46:10.358925Z"Gauci, John Baptist"https://zbmath.org/authors/?q=ai:gauci.john-baptist"Zerafa, Jean Paul"https://zbmath.org/authors/?q=ai:zerafa.jean-paulSummary: If every perfect matching of a graph \(G\) extends to a Hamiltonian cycle, we shall say that \(G\) has the PMH-property -- a concept first studied in the 1970s by \textit{M. Las Vergnas} [Problèmes de couplages et problèmes hamiltoniens en thèorie des graphes. University of Paris 6, Paris (PhD Thesis) (1972)] and \textit{R. Häggkvist} [in: Graph theory and related topics. Proceedings of the conference held in honour of Professor W. T. Tutte on the occasion of his sixtieth birthday, University of Waterloo, July 5-9, 1977. New York - San Francisco - London: Academic Press, A Subsidiary of Harcourt Brace Jovanovich, Publishers. 219--231 (1979; Zbl 0462.05047)]. A pairing of a graph \(G\) is a perfect matching of the complete graph having the same vertex set as \(G\). A somewhat stronger property than the PMH-property is the following. A graph \(G\) has the PH-property if every pairing of \(G\) can be extended to a Hamiltonian cycle of the underlying complete graph using only edges from \(G\). The name for the latter property was coined in [\textit{A. Alahmadi} et al., Discrete Math. Theor. Comput. Sci. 17, No. 1, 241--254 (2015; Zbl 1311.05157)]; however, this was not the first time this property was studied. \textit{J. Fink} [J. Comb. Theory, Ser. B 97, No. 6, 1074--1076 (2007; Zbl 1126.05080)] proved that every \(n\)-dimensional hypercube, for \(n\ge 2\), has the PH-property. After characterising all the cubic graphs having the PH-property, Alahmadi et al. [loc. cit.] attempt to characterise all 4-regular graphs having the same property by posing the following problem: for which values of \(p\) and \(q\) does the Cartesian product \(C_p\square C_q\) of two cycles on \(p\) and \(q\) vertices have the PH-property? We here show that this only happens when both \(p\) and \(q\) are equal to four, namely for \(C_4\square C_4 \), the 4-dimensional hypercube. For all other values, we show that \(C_p\square C_q\) does not even admit the PMH-property.Characterizing \(\mathcal{P}_{\geqslant 2} \)-factor and \(\mathcal{P}_{\geqslant 2} \)-factor covered graphs with respect to the size or the spectral radiushttps://zbmath.org/1472.051272021-11-25T18:46:10.358925Z"Li, Shuchao"https://zbmath.org/authors/?q=ai:li.shuchao"Miao, Shujing"https://zbmath.org/authors/?q=ai:miao.shujingSummary: A \(\mathcal{P}_{\geqslant k} \)-factor \((k \geqslant 2)\) of a graph \(G\) is a spanning subgraph of \(G\) in which each component is a path of order at least \(k\). A graph \(G\) is called a \(\mathcal{P}_{\geqslant k} \)-factor covered graph if for each edge \(e\) of \(G\), there is a \(\mathcal{P}_{\geqslant k} \)-factor covering \(e\). In this paper, we first establish two lower bounds on the size of a graph \(G\), in which one bound guarantees that \(G\) contains a \(\mathcal{P}_{\geqslant 2} \)-factor, the other bound ensures that the graph \(G\) is a \(\mathcal{P}_{\geqslant 2} \)-factor covered graph. Then we establish two lower bounds on the spectral radius of a graph \(G\), in which one bound guarantees that the graph \(G\) has a \(\mathcal{P}_{\geqslant 2} \)-factor, the other bound ensures that the graph \(G\) is a \(\mathcal{P}_{\geqslant 2} \)-factor covered graph. Furthermore, we construct some extremal graphs to show all the bounds obtained in this contribution are best possible.Decomposition of complete graphs into arbitrary treeshttps://zbmath.org/1472.051282021-11-25T18:46:10.358925Z"Sethuraman, G."https://zbmath.org/authors/?q=ai:sethuraman.g"Murugan, V."https://zbmath.org/authors/?q=ai:murugan.veerapazham|murugan.v-velIn this paper, the authors deal with tree-decompositions of complete graphs. A classical conjecture in the area is one by Ringel, which states that for each \(m\geq 1\) and a tree \(T\) with \(m\) edges, the complete graph \(K_{2m+1}\) can be decomposed into \(2m+1\) copies of \(T\). In this paper, the authors go further and conjecture that:
\par Conjecture: For each \(m\geq 1\) and trees \(T_1\), \(T_2\) with \(|E(T_1)|=|E(T_2)|=m\), the complete graph \(K_{4m+1}\) can be decomposed into copies of \(T_1\) and \(T_2\).
\par In order to support their conjecture, in the paper, the authors prove the following result which is the main one:
\par Theorem: For any \(m\geq 1\) and any tree \(T\) with \(|E(T)|=m\), the complete graph \(K_{4cm+1}\) can be decomposed into copies of \(T\) and \(T_0\). Here, \(c\) is any positive constant and \(T_0\) is the path on \(m\) edges or the star on \(m\) edges.On null 3-hypergraphshttps://zbmath.org/1472.051292021-11-25T18:46:10.358925Z"Frosini, Andrea"https://zbmath.org/authors/?q=ai:frosini.andrea"Kocay, William L."https://zbmath.org/authors/?q=ai:kocay.william-l"Palma, Giulia"https://zbmath.org/authors/?q=ai:palma.giulia"Tarsissi, Lama"https://zbmath.org/authors/?q=ai:tarsissi.lamaSummary: Given a 3-uniform hypergraph \(H\) consisting of a set \(V\) of vertices, and \(T \subseteq \binom{V}{3}\) triples, a null labelling is an assignment of \(\pm 1\) to the triples such that each vertex is contained in an equal number of triples labelled \(+ 1\) and \(- 1\). Thus, the signed degree of each vertex is zero. A necessary condition for a null labelling is that the degree of every vertex of \(H\) is even. The Null Labelling Problem is to determine whether \(H\) has a null labelling. It is proved that this problem is NP-complete. Computer enumerations suggest that most hypergraphs which satisfy the necessary condition do have a null labelling. Some constructions are given which produce hypergraphs satisfying the necessary condition, but which do not have a null labelling. A self complementary 3-hypergraph with this property is also constructed.\(L(1, 2)\)-labeling numbers on square of cycleshttps://zbmath.org/1472.051302021-11-25T18:46:10.358925Z"Liu, Le"https://zbmath.org/authors/?q=ai:liu.le"Wu, Qiong"https://zbmath.org/authors/?q=ai:wu.qiongAn \(L(k_1,k_2,\dots,k_p)\)-labelling of a graph \(G=(V,E)\) is a function \(f:V\to\mathbb{N}\cup\{0\}\) such that, for any \(u,v\in V\), and \(i=1,2,\dots,p\), \(|f(u)-f(v)|\ge k_i\) whenever the distance \(d(u,v)=i\); the span of \(f\) is the maximum difference between labels assigned by \(f\); the \(L(1,d)\)-labelling number of \(G\), \(\lambda_{1,d}(G)\), is the minimum span of all \(L(1,d)\)-labellings of \(G\) [\textit{J. R. Griggs} and \textit{R. K. Yeh}, SIAM J. Discrete Math. 5, No. 4, 586--595 (1992; Zbl 0767.05080); \textit{X. T. Jin} and \textit{R. K. Yeh}, Nav. Res. Logist. 52, No. 2, 159--164 (2005; Zbl 1063.05118)]. If \(C_n\) represents the undirected cycle on \(n\) vertices, then \(C_n^2\) represents its square, on the same vertex-set, where any two vertices \(u,v\) are adjacent in \(C_n^2\) if \(d(u,v)\le2\) in \(C_n\). The authors prove Theorem 2.1: \(\lambda_{1,2}(C_n^2)=n-1\) for positive integers \(n\) if \(3\le n\le 9\); Theorem 2.2: \(\lambda_{1,2}(C_n^2)=5\) for positive integers \(n\ge12\) such that \(n\equiv 0\pmod 6\); Theorem 2.3: \(\lambda_{1,2}(C_{10}^2)=8\); Theorem 2.4: \(\lambda_{1,2}(C_{11}^2)=6\); Theorem 2.6: \(\lambda_{1,2}(C_{15}^2)=7=\lambda_{1,2}(C_{16}^2)\); Theorem 2.7: \(\lambda_{1,2}(C_{n}^2)=6\), for \(n\ge13; n\ne15,16\); \(n\not\equiv0\pmod 6\).Classification of tetravalent distance magic circulant graphshttps://zbmath.org/1472.051312021-11-25T18:46:10.358925Z"Miklavič, Štefko"https://zbmath.org/authors/?q=ai:miklavic.stefko"Šparl, Primož"https://zbmath.org/authors/?q=ai:sparl.primozSummary: Let \({\Gamma} = (V, E)\) be a graph of order \(n\). A distance magic labeling of \(\Gamma\) is a bijection \(\ell : V \to \{1, 2, \ldots, n \}\) for which there exists a positive integer \(k\) such that \(\sum_{x \in N ( u )} \ell(x) = k\) for all vertices \(u \in V\), where \(N(u)\) is the neighborhood of \(u\). A graph is said to be distance magic if it admits a distance magic labeling.
In this paper we classify all connected tetravalent distance magic circulants, that is Cayley graphs \(\operatorname{Cay}( \mathbb{Z}_n; S)\) where \(S = \{\pm a, \pm b \}\) for some \(1 \leq a < b < n / 2\) with \(\gcd(n, a, b) = 1\). As a consequence we solve an open problem posed by \textit{S. Cichacz} and \textit{D. Froncek} [ibid. 339, No. 1, 84--94 (2016; Zbl 1322.05120)].Thresholds versus fractional expectation-thresholdshttps://zbmath.org/1472.051322021-11-25T18:46:10.358925Z"Frankston, Keith"https://zbmath.org/authors/?q=ai:frankston.keith"Kahn, Jeff"https://zbmath.org/authors/?q=ai:kahn.jeff-d"Narayanan, Bhargav"https://zbmath.org/authors/?q=ai:narayanan.bhargav-p"Park, Jinyoung"https://zbmath.org/authors/?q=ai:park.jinyoungSummary: Proving a conjecture of \textit{M. Talagrand} [in: Proceedings of the 42nd annual ACM symposium on theory of computing, STOC '10. Cambridge, MA, USA, June 5--8, 2010. New York, NY: Association for Computing Machinery (ACM). 13--36 (2010; Zbl 1293.60014)], a fractional version of the ``expectation-threshold'' conjecture of \textit{J. Kahn} and \textit{G. Kalai} [Comb. Probab. Comput. 16, No. 3, 495--502 (2007; Zbl 1118.05093)], we show that \(p_c(\mathcal{F})=O(q_f(\mathcal{F})\log\ell(\mathcal{F}))\) for any increasing family \(\mathcal{F}\) on a finite set \(X\), where \(p_c(\mathcal{F})\) and \(q_f(\mathcal{F})\) are the threshold and ``fractional expectation-threshold'' of \(\mathcal{F}\), and \(\ell(\mathcal{F})\) is the maximum size of a minimal member of \(\mathcal{F}\). This easily implies several heretofore difficult results and conjectures in probabilistic combinatorics, including thresholds for perfect hypergraph matchings [\textit{A. Johansson} et al., Random Struct. Algorithms 33, No. 1, 1--28 (2008; Zbl 1146.05040)], bounded degree spanning trees [\textit{R. Montgomery}, Adv. Math. 356, Article ID 106793, 92 p. (2019; Zbl 1421.05080)], and bounded degree graphs (new). We also resolve (and vastly extend) the ``axial'' version of the random multi-dimensional assignment problem (earlier considered by Martin-Mézard-Rivoire and \textit{A. Frieze} and \textit{G. B. Sorkin} [Random Struct. Algorithms 46, No. 1, 160--196 (2015; Zbl 1347.60141)]). Our approach builds on a recent breakthrough of \textit{R. Alweiss} et al. [in: Proceedings of the 52nd annual ACM SIGACT symposium on theory of computing, STOC '20, Chicago, IL, USA, June 22--26, 2020. New York, NY: Association for Computing Machinery (ACM). 624--630 (2020; Zbl 07298275)] on the Erdő-Rado ``Sunflower Conjecture''.Directed random geometric graphshttps://zbmath.org/1472.051332021-11-25T18:46:10.358925Z"Michel, Jesse"https://zbmath.org/authors/?q=ai:michel.jesse"Reddy, Sushruth"https://zbmath.org/authors/?q=ai:reddy.sushruth"Shah, Rikhav"https://zbmath.org/authors/?q=ai:shah.rikhav"Silwal, Sandeep"https://zbmath.org/authors/?q=ai:silwal.sandeep"Movassagh, Ramis"https://zbmath.org/authors/?q=ai:movassagh.ramisSummary: Many real-world networks are intrinsically directed. Such networks include activation of genes, hyperlinks on the internet and the network of followers on Twitter among many others. The challenge, however, is to create a network model that has many of the properties of real-world networks such as power-law degree distributions and the small-world property. To meet these challenges, we introduce the Directed Random Geometric Graph (DRGG) model, which is an extension of the random geometric graph model. We prove that it is scale-free with respect to the indegree distribution, has binomial outdegree distribution, has a high clustering coefficient, has few edges and is likely small-world. These are some of the main features of aforementioned real-world networks. We also empirically observed that word association networks have many of the theoretical properties of the DRGG model.On the isomorphisms between evolution algebras of graphs and random walkshttps://zbmath.org/1472.051342021-11-25T18:46:10.358925Z"Cadavid, Paula"https://zbmath.org/authors/?q=ai:cadavid.paula"Rodiño Montoya, Mary Luz"https://zbmath.org/authors/?q=ai:rodino-montoya.mary-luz"Rodriguez, Pablo M."https://zbmath.org/authors/?q=ai:rodriguez.pablo-mSummary: Evolution algebras are non-associative algebras inspired from biological phenomena, with applications to or connections with different mathematical fields. There are two natural ways to define an evolution algebra associated to a given graph. While one takes into account only the adjacencies of the graph, the other includes probabilities related to the symmetric random walk on the same graph. In this work we state new properties related to the relation between these algebras, which is one of the open problems in the interplay between evolution algebras and graphs. On the one hand, we show that for any graph both algebras are strongly isotopic. On the other hand, we provide conditions under which these algebras are or are not isomorphic. For the case of finite non-singular graphs we provide a complete description of the problem, while for the case of finite singular graphs we state a conjecture supported by examples and partial results. The case of graphs with an infinite number of vertices is also discussed. As a sideline of our work, we revisit a result existing in the literature about the identification of the automorphism group of an evolution algebra, and we give an improved version of it.Expected hitting times for random walks on the diamond hierarchical graphs involving some classical parametershttps://zbmath.org/1472.051352021-11-25T18:46:10.358925Z"Guo, Ziliang"https://zbmath.org/authors/?q=ai:guo.ziliang"Li, Shuchao"https://zbmath.org/authors/?q=ai:li.shuchao"Liu, Xin"https://zbmath.org/authors/?q=ai:liu.xin.5|liu.xin.4|liu.xin.3|liu.xin.2|liu.xin|liu.xin.1"Mei, Xiaoling"https://zbmath.org/authors/?q=ai:mei.xiaolingSummary: Given a connected graph \(G\), the diamond hierarchical graph \(S(G)\) is formed by adding two new vertices \(v_e, w_e\) for each edge \(e= uv\) in \(G\) and then deleting edge \(e\) and adding in edges \(uv_e\), \(uw_e\) and \(v_e v\), \(w_e v\). In this paper, the eigenvalues and eigenvectors of the probability transition matrix of a random walks on \(S_G\) are completely provided at first. Then the expected hitting time between any two vertices of \(S_G\) is determined in regards to those of \(G\). Finally, as applications, the connection between the Kemeny's constant (resp. degree-Kirchhoff index, spanning tree number) of \(S_G\) and \(G\) is established. Furthermore, the resistance distance between any two vertices of \(S_G\) is given compared to those of \(G\).A low discrepancy sequence on graphshttps://zbmath.org/1472.051362021-11-25T18:46:10.358925Z"Cloninger, A."https://zbmath.org/authors/?q=ai:cloninger.alexander"Mhaskar, H. N."https://zbmath.org/authors/?q=ai:mhaskar.hrushikesh-nSummary: Many applications such as election forecasting, environmental monitoring, health policy, and graph based machine learning require taking expectation of functions defined on the vertices of a graph. We describe a construction of a sampling scheme analogous to the so called Leja points in complex potential theory that can be proved to give low discrepancy estimates for the approximation of the expected value by the impirical expected value based on these points. In contrast to classical potential theory where the kernel is fixed and the equilibrium distribution depends upon the kernel, we fix a probability distribution and construct a kernel (which represents the graph structure) for which the equilibrium distribution is the given probability distribution. Our estimates do not depend upon the size of the graph.Lower bound for the cost of connecting tree with given vertex degree sequencehttps://zbmath.org/1472.051372021-11-25T18:46:10.358925Z"Goubko, Mikhail"https://zbmath.org/authors/?q=ai:goubko.mikhail-v"Kuznetsov, Alexander"https://zbmath.org/authors/?q=ai:kuznetsov.alexander-i|kuznetsov.alexander-gennadevich|kuznetsov.aleksandr-vladimirovich|kuznetsov.alexander-g|kuznetsov.aleksandr-petrovich|kuznetsov.alexander-m|kuznetsov.alexander-aSummary: The optimal connecting network problem generalizes many models of structure optimization known from the literature, including communication and transport network topology design, graph cut and graph clustering, etc. For the case of connecting trees with the given sequence of vertex degrees the cost of the optimal tree is shown to be bounded from below by the solution of a semidefinite optimization program with bilinear matrix inequality constraints, which is reduced to the solution of a series of convex programs with linear matrix inequality constraints. The proposed lower-bound estimate is used to construct several heuristic algorithms and to evaluate their quality on a variety of generated and real-life datasets.Network comparison and the within-ensemble graph distancehttps://zbmath.org/1472.051382021-11-25T18:46:10.358925Z"Hartle, Harrison"https://zbmath.org/authors/?q=ai:hartle.harrison"Klein, Brennan"https://zbmath.org/authors/?q=ai:klein.brennan"McCabe, Stefan"https://zbmath.org/authors/?q=ai:mccabe.stefan"Daniels, Alexander"https://zbmath.org/authors/?q=ai:daniels.alexander"St-Onge, Guillaume"https://zbmath.org/authors/?q=ai:st-onge.guillaume"Murphy, Charles"https://zbmath.org/authors/?q=ai:murphy.charles"Hébert-Dufresne, Laurent"https://zbmath.org/authors/?q=ai:hebert-dufresne.laurentSummary: Quantifying the differences between networks is a challenging and ever-present problem in network science. In recent years, a multitude of diverse, \textit{ad hoc} solutions to this problem have been introduced. Here, we propose that simple and well-understood ensembles of random networks -- such as Erdős-Rényi graphs, random geometric graphs, Watts-Strogatz graphs, the configuration model and preferential attachment networks -- are natural benchmarks for network comparison methods. Moreover, we show that the expected distance between two networks independently sampled from a generative model is a useful property that encapsulates many key features of that model. To illustrate our results, we calculate this \textit{within-ensemble graph distance} and related quantities for classic network models (and several parameterizations thereof) using 20 distance measures commonly used to compare graphs. The within-ensemble graph distance provides a new framework for developers of graph distances to better understand their creations and for practitioners to better choose an appropriate tool for their particular task.The degree analysis of an inhomogeneous growing network with two types of verticeshttps://zbmath.org/1472.051392021-11-25T18:46:10.358925Z"Huang, Huilin"https://zbmath.org/authors/?q=ai:huang.huilinSummary: We consider an inhomogeneous growing network with two types of vertices. The degree sequences of two different types of vertices are investigated, respectively. We not only prove that the asymptotical degree distribution of type \(s\) for this process is power law with exponent \(2 + \left(\left(1 + \delta\right) q_s + \beta \left(1 - q_s\right)\right) / \alpha q_s\), but also give the strong law of large numbers for degree sequences of two different types of vertices by using a different method instead of Azuma's inequality. Then we determine asymptotically the joint probability distribution of degree for pairs of adjacent vertices with the same type and with different types, respectively.Symmetric core and spanning trails in directed networkshttps://zbmath.org/1472.051402021-11-25T18:46:10.358925Z"Liu, Juan"https://zbmath.org/authors/?q=ai:liu.juan"Yang, Hong"https://zbmath.org/authors/?q=ai:yang.hong"Lai, Hong-Jian"https://zbmath.org/authors/?q=ai:lai.hong-jian"Zhang, Xindong"https://zbmath.org/authors/?q=ai:zhang.xindongSummary: A digraph \(D\) is supereulerian if \(D\) has a spanning closed trail, and is strongly trail-connected if for any pair of vertices \(u, v \in V(D)\), \(D\) has a spanning \((u, v)\)-trail and a spanning \((v, u)\)-trail. The symmetric core \(J = J(D)\) of a digraph \(D\) is a spanning subdigraph of \(D\) with \(A(J)\) consisting of all symmetric arcs in \(D\). Let \(J_1, J_2, \dots, J_{k ( D )}\) be the connected symmetric components of \(J\) and define \(\lambda_0(D) = \min \{\lambda( J_i) : 1 \leq i \leq k(D) \}\). We prove that the contraction \(D^\prime = D / J\) can be used to predict the existence of spanning trails in \(D\). It is known that if \(k(D) \leq 2\), then \(D\) has a spanning closed trail. In particular, each of the following holds for a strong digraph \(D\) with \(k(D) \geq 3\).
\par i) If \(\lambda_0(D) \geq k(D) - 2\), then \(D\) has a spanning trail if and only if \(D^\prime\) has a spanning trail.
\par ii) If \(\lambda_0(D) \geq k(D) - 1\), then \(D\) is supereulerian if and only if \(D^\prime\) is supereulerian.
\par iii) If \(\lambda_0(D) \geq k(D)\), then \(D\) is strongly trail-connected if and only if \(D^\prime\) is strongly trail-connected.On the von Neumann entropy of graphshttps://zbmath.org/1472.051412021-11-25T18:46:10.358925Z"Minello, Giorgia"https://zbmath.org/authors/?q=ai:minello.giorgia"Rossi, Luca"https://zbmath.org/authors/?q=ai:rossi.luca.1"Torsello, Andrea"https://zbmath.org/authors/?q=ai:torsello.andreaSummary: The von Neumann entropy of a graph is a spectral complexity measure that has recently found applications in complex networks analysis and pattern recognition. Two variants of the von Neumann entropy exist based on the graph Laplacian and normalized graph Laplacian, respectively. Due to its computational complexity, previous works have proposed to approximate the von Neumann entropy, effectively reducing it to the computation of simple node degree statistics. Unfortunately, a number of issues surrounding the von Neumann entropy remain unsolved to date, including the interpretation of this spectral measure in terms of structural patterns, understanding the relation between its two variants and evaluating the quality of the corresponding approximations. In this article, we aim to answer these questions by first analysing and comparing the quadratic approximations of the two variants and then performing an extensive set of experiments on both synthetic and real-world graphs. We find that (1) the two entropies lead to the emergence of similar structures, but with some significant differences; (2) the correlation between them ranges from weakly positive to strongly negative, depending on the topology of the underlying graph; (3) the quadratic approximations fail to capture the presence of non-trivial structural patterns that seem to influence the value of the exact entropies; and (4) the quality of the approximations, as well as which variant of the von Neumann entropy is better approximated, depends on the topology of the underlying graph.Characterization of robustness and resilience in graphs: a mini-reviewhttps://zbmath.org/1472.051422021-11-25T18:46:10.358925Z"Schaeffer, S. E."https://zbmath.org/authors/?q=ai:schaeffer.satu-elisa"Valdés, V."https://zbmath.org/authors/?q=ai:valdes.v"Figols, J."https://zbmath.org/authors/?q=ai:figols.j"Bachmann, I."https://zbmath.org/authors/?q=ai:bachmann.ivana"Morales, F."https://zbmath.org/authors/?q=ai:morales.fernando-a"Bustos-Jiménez, J."https://zbmath.org/authors/?q=ai:bustos-jimenez.javierSummary: We briefly survey recent proposals that seek to capture in numerical terms the resilience and the robustness of a graph. After a brief introduction and the establishment of notation and terminology, we catalogue characterizations proposed in journal articles published within the last two decades. We then describe some of the numerous application areas for such characterizations. We experiment with implementations of numerous characteristics on several graph-generation models, after which we conclude with a discussion of open problems and future directions.Networks beyond pairwise interactions: structure and dynamicshttps://zbmath.org/1472.051432021-11-25T18:46:10.358925Z"Battiston, Federico"https://zbmath.org/authors/?q=ai:battiston.federico"Cencetti, Giulia"https://zbmath.org/authors/?q=ai:cencetti.giulia"Iacopini, Iacopo"https://zbmath.org/authors/?q=ai:iacopini.iacopo"Latora, Vito"https://zbmath.org/authors/?q=ai:latora.vito"Lucas, Maxime"https://zbmath.org/authors/?q=ai:lucas.maxime"Patania, Alice"https://zbmath.org/authors/?q=ai:patania.alice"Young, Jean-Gabriel"https://zbmath.org/authors/?q=ai:young.jean-gabriel"Petri, Giovanni"https://zbmath.org/authors/?q=ai:petri.giovanniSummary: The complexity of many biological, social and technological systems stems from the richness of the interactions among their units. Over the past decades, a variety of complex systems has been successfully described as networks whose interacting pairs of nodes are connected by links. Yet, from human communications to chemical reactions and ecological systems, interactions can often occur in groups of three or more nodes and cannot be described simply in terms of dyads. Until recently little attention has been devoted to the higher-order architecture of real complex systems. However, a mounting body of evidence is showing that taking the higher-order structure of these systems into account can enhance our modeling capacities and help us understand and predict their dynamical behavior. Here we present a complete overview of the emerging field of networks beyond pairwise interactions. We discuss how to represent higher-order interactions and introduce the different frameworks used to describe higher-order systems, highlighting the links between the existing concepts and representations. We review the measures designed to characterize the structure of these systems and the models proposed to generate synthetic structures, such as random and growing bipartite graphs, hypergraphs and simplicial complexes. We introduce the rapidly growing research on higher-order dynamical systems and dynamical topology, discussing the relations between higher-order interactions and collective behavior. We focus in particular on new emergent phenomena characterizing dynamical processes, such as diffusion, synchronization, spreading, social dynamics and games, when extended beyond pairwise interactions. We conclude with a summary of empirical applications, and an outlook on current modeling and conceptual frontiers.On the combinatorics of some Origami models and related graphshttps://zbmath.org/1472.051442021-11-25T18:46:10.358925Z"Lengvárszky, Zsolt"https://zbmath.org/authors/?q=ai:lengvarszky.zsolt"Smith, John"https://zbmath.org/authors/?q=ai:smith.john-l|smith.john-a-s|smith.john-d|smith.john-w|smith.john-howard|smith.john-r|smith.john-miles|smith.john-mGiven a graph \(G\), an assignment is a pair of functions \(t^-: V(G) \rightarrow \mathbb{N}\) and \(t^+: V(G) \rightarrow \mathbb{N}\). Given an assignment \(M = (t^{-}, t^{+})\) of a graph \(G\), the authors say that \(G\) is \(M\)-super-duper if there exist an orientation of the edges of \(G\) such that \(d^+(x) = t^+(x)\) and \(d^-(x) = t^-(x)\) for all \(x \in V(G)\). Clearly a necessary condition that \(M\) needs to satisfy for \(G\) to be \(M\)-super-duper is that \(M\) should be degree-preserving, meaning that \(t^+(x) + t^-(x) = d(x)\) for all \(x \in V(G)\). An assignment is balanced if \(|t^+(x) - t^-(x)| \leq 1\) for all \(x \in V(G)\). The authors say that a graph \(G\) is super-duper if \(G\) is \(M\)-super-duper for all its degree-preserving balanced assignments \(M\).
The authors are motivated by applications to the study of a family of origami patterns. The studied patterns are created by assembling \(n\) copies of a polyhedron \(B\), one copy in each vertex of an \(n\)-vertex polyhedron \(A\). It turns out that such a construction is possible only if the graph associated to the polyhedron \(B\) is \(M\)-super-duper for an assignment \(M\) which depends on \(A\).
Motivated by this connection, the authors study in detail assignments in polyhedral graphs. They characterise the assignments which can be achieved in the cube and the dodecahedron. The authors find certain obstructions which cannot appear as subgraphs in general super-duper graphs. They use this to show that there is at most a finite number of super-duper \(3\)-regular graphs, with improved bounds in the case of \(3\)-regular polyhedral graphs. Finally, they also study \(M\)-super-duper assignments in \(k\)-regular graphs for larger \(k\).The Ramsey property for operator spaces and noncommutative Choquet simpliceshttps://zbmath.org/1472.051452021-11-25T18:46:10.358925Z"Bartošová, Dana"https://zbmath.org/authors/?q=ai:bartosova.dana"López-Abad, Jordi"https://zbmath.org/authors/?q=ai:lopez-abad.jordi"Lupini, Martino"https://zbmath.org/authors/?q=ai:lupini.martino"Mbombo, Brice"https://zbmath.org/authors/?q=ai:mbombo.brice-rSummary: The noncommutative Gurarij space \(\mathbb{NG} \), initially defined by \textit{T. Oikhberg} [Arch. Math. 86, No. 4, 356--364 (2006; Zbl 1119.46045)], is a canonical object in the theory of operator spaces. As the Fraïssé limit of the class of finite-dimensional nuclear operator spaces, it can be seen as the noncommutative analogue of the classical Gurarij Banach space. In this paper, we prove that the automorphism group of \(\mathbb{NG}\) is extremely amenable, i.e. any of its actions on compact spaces has a fixed point. The proof relies on the Dual Ramsey Theorem, and a version of the Kechris-Pestov-Todorcevic correspondence in the setting of operator spaces.
Recent work of \textit{K. R. Davidson} and \textit{M. Kennedy} [``Noncommutative Choquet theory'', Preprint, \url{arXiv:1905.08436}], building on previous work of Arveson, Effros, Farenick, Webster, and Winkler, among others, shows that nuclear operator systems can be seen as the noncommutative analogue of Choquet simplices. The analogue of the Poulsen simplex in this context is the matrix state space \(\mathbb{NP}\) of the Fraïssé limit \(A(\mathbb{NP})\) of the class of finite-dimensional nuclear operator systems. We show that the canonical action of the automorphism group of \(\mathbb{NP}\) on the compact set \(\mathbb{NP}_1\) of unital linear functionals on \(A( \mathbb{NP})\) is minimal and it factors onto any minimal action, whence providing a description of the universal minimal flow of \(\operatorname{Aut} (\mathbb{NP})\).Translation invariant filters and van der Waerden's theoremhttps://zbmath.org/1472.051462021-11-25T18:46:10.358925Z"Di Nasso, Mauro"https://zbmath.org/authors/?q=ai:di-nasso.mauroSummary: We present a self-contained proof of a strong version of van der Waerden's Theorem. By using translation invariant filters that are maximal with respect to inclusion, a simple inductive argument shows the existence of ``piecewise syndetically''-many monochromatic arithmetic progressions of any length \(k\) in every finite coloring of the natural numbers. All the presented constructions are constructive in nature, in the sense that the involved maximal filters are defined by recurrence on suitable countable algebras of sets. No use of the axiom of choice or of Zorn's Lemma is needed.
For the entire collection see [Zbl 1431.11004].Combinatorial anti-concentration inequalities, with applicationshttps://zbmath.org/1472.051472021-11-25T18:46:10.358925Z"Fox, Jacob"https://zbmath.org/authors/?q=ai:fox.jacob"Kwan, Matthew"https://zbmath.org/authors/?q=ai:kwan.matthew"Sauermann, Lisa"https://zbmath.org/authors/?q=ai:sauermann.lisaSummary: We prove several different anti-concentration inequalities for functions of independent Bernoulli-distributed random variables. First, motivated by a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn, we prove some ``Poisson-type'' anti-concentration theorems that give bounds of the form \(1/e+o(1)\) for the point probabilities of certain polynomials. Second, we prove an anti-concentration inequality for polynomials with nonnegative coefficients which extends the classical Erdős-Littlewood-Offord theorem and improves a theorem of \textit{R. Meka} et al. [Theory Comput. 12, Paper No. 11, 17 p. (2016; Zbl 1392.68193)] for polynomials of this type. As an application, we prove some new anti-concentration bounds for subgraph counts in random graphs.An extension of Stanley's chromatic symmetric function to binary delta-matroidshttps://zbmath.org/1472.051482021-11-25T18:46:10.358925Z"Nenasheva, M."https://zbmath.org/authors/?q=ai:nenasheva.m"Zhukov, V."https://zbmath.org/authors/?q=ai:zhukov.v-a.1|zhukov.v-o|zhukov.vadim-g|zhukov.vladimir-v|zhukov.v-d|zhukov.vitalii-vladimirovich|zhukov.v-e|zhukov.v-t|zhukov.victor-p|zhukov.vyacheslav|zhukov.v-iSummary: Stanley's symmetrized chromatic polynomial is a generalization of the ordinary chromatic polynomial to a graph invariant with values in a ring of polynomials in infinitely many variables. The ordinary chromatic polynomial is a specialization of Stanley's one.
To each orientable embedded graph with a single vertex, a simple graph is associated, which is called the intersection graph of the embedded graph. As a result, we can define Stanley's symmetrized chromatic polynomial for any orientable embedded graph with a single vertex. Our goal is to extend Stanley's chromatic polynomial to embedded graphs with arbitrary number of vertices, and not necessarily orientable. In contrast to well-known extensions of, say, the Tutte polynomial from abstract to embedded graphs [\textit{C. Chun} et al., J. Comb. Theory, Ser. A 167, 7--59 (2019; Zbl 1417.05103)], our extension is based not on the structure of the underlying abstract graph and the additional information about the embedding. Instead, we consider the binary delta-matroid associated to an embedded graph and define the extended Stanley chromatic polynomial as an invariant of binary delta-matroids. We show that, similarly to Stanley's symmetrized chromatic polynomial of graphs, which satisfies 4-term relations for simple graphs, the polynomial that we introduce satisfies the 4-term relations for binary delta-matroids [\textit{S. Lando} and \textit{V. Zhukov}, Mosc. Math. J. 17, No. 4, 741--755 (2017; Zbl 1414.05067)].
For graphs, Stanley's chromatic function produces a knot invariant by means of the correspondence between simple graphs and knots. Analogously we may interpret the suggested extension as an invariant of links, using the correspondence between binary delta-matroids and links.Differential posets and restriction in critical groupshttps://zbmath.org/1472.051492021-11-25T18:46:10.358925Z"Agarwal, Ayush"https://zbmath.org/authors/?q=ai:agarwal.ayush"Gaetz, Christian"https://zbmath.org/authors/?q=ai:gaetz.christianSummary: In recent work, \textit{G. Benkart} et al. [Math. Z. 290, No. 1--2, 615--648 (2018; Zbl 1448.17015)] defined the critical group of a faithful representation of a finite group \(G\), which is analogous to the critical group of a graph. In this paper we study maps between critical groups induced by injective group homomorphisms and in particular the map induced by restriction of the representation to a subgroup. We show that in the abelian group case the critical groups are isomorphic to the critical groups of a certain Cayley graph and that the restriction map corresponds to a graph covering map. We also show that when \(G\) is an element in a differential tower of groups, as introduced by [\textit{A. Miller} and \textit{V. Reiner}, Order 26, No. 3, 197--228 (2009; Zbl 1228.05096)], critical groups of certain representations are closely related to words of up-down maps in the associated differential poset. We use this to generalize an explicit formula for the critical group of the permutation representation of \(\mathfrak{S}_n\) given by the second author, and to enumerate the factors in such critical groups.Generalisations of the Harer-Zagier recursion for 1-point functionshttps://zbmath.org/1472.051502021-11-25T18:46:10.358925Z"Chaudhuri, Anupam"https://zbmath.org/authors/?q=ai:chaudhuri.anupam"Do, Norman"https://zbmath.org/authors/?q=ai:do.normanSummary: \textit{J. Harer} and \textit{D. Zagier} [Invent. Math. 85, 457--485 (1986; Zbl 0616.14017)] proved a recursion to enumerate gluings of a \(2d\)-gon that result in an orientable genus \(g\) surface, in their work on Euler characteristics of moduli spaces of curves. Analogous results have been discovered for other enumerative problems, so it is natural to pose the following question: how large is the family of problems for which these so-called 1-point recursions exist? In this paper, we prove the existence of 1-point recursions for a class of enumerative problems that have Schur function expansions. In particular, we recover the Harer-Zagier recursion, but our methodology also applies to the enumeration of dessins d'enfant, to Bousquet-Mélou-Schaeffer numbers, to monotone Hurwitz numbers, and more. On the other hand, we prove that there is no 1-point recursion that governs single Hurwitz numbers. Our results are effective in the sense that one can explicitly compute particular instances of 1-point recursions, and we provide several examples. We conclude the paper with a brief discussion and a conjecture relating 1-point recursions to the theory of topological recursion.Higher Specht bases for generalizations of the coinvariant ringhttps://zbmath.org/1472.051512021-11-25T18:46:10.358925Z"Gillespie, M."https://zbmath.org/authors/?q=ai:gillespie.maria-monks|gillespie.m-i"Rhoades, B."https://zbmath.org/authors/?q=ai:rhoades.billy-e|rhoades.brendonThe authors provide an extension of the higher Specht basis for the classical coinvariant ring \(R_n\), consisting of the higher Specht polynomials due to \textit{S. Ariki} et al. [Hiroshima Math. J. 27, No. 1, 177--188 (1997; Zbl 0886.20009)], to the generalized coinvariant rings \(R_{n,k}\) introduced in [\textit{J. Haglund} et al., Adv. Math. 329, 851--915 (2018; Zbl 1384.05043)]. They give a conjectured higher Specht basis for the Garsia-Procesi modules \(R_{\mu}\) [\textit{A. M. Garsia} and \textit{C. Procesi}, Adv. Math. 94, No. 1, 82--138 (1992; Zbl 0797.20012)], and provide a proof of the conjecture in the case of two-row partition shapes \(\mu\). Subsequently, a higher Specht basis for an infinite subfamily of the modules \(R_{n,k,\mu}\) defined by \textit{S. T. Griffin} [Trans. Am. Math. Soc. 374, No. 4, 2609--2660 (2021; Zbl 1459.05340)], which are a common generalization of \(R_{n,k}\) and \(R_\mu\), is constructed.Crystal structures for symmetric Grothendieck polynomialshttps://zbmath.org/1472.051522021-11-25T18:46:10.358925Z"Monical, Cara"https://zbmath.org/authors/?q=ai:monical.cara"Pechenik, Oliver"https://zbmath.org/authors/?q=ai:pechenik.oliver"Scrimshaw, Travis"https://zbmath.org/authors/?q=ai:scrimshaw.travisSummary: The symmetric Grothendieck polynomials representing Schubert classes in the K theory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type \(A_n\) crystal structure on these tableaux. This crystal yields a new combinatorial formula for decomposing symmetric Grothendieck polynomials into Schur polynomials. For single-columns and single-rows, we give a new combinatorial interpretation of Lascoux polynomials (K-analogs of Demazure characters) by constructing a K-theoretic analog of crystals with an appropriate analog of a Demazure crystal. We relate our crystal structure to combinatorial models using excited Young diagrams, Gelfand-Tsetlin patterns via the 5-vertex model, and biwords via Hecke insertion to compute symmetric Grothendieck polynomials.Crystal for stable Grothendieck polynomialshttps://zbmath.org/1472.051532021-11-25T18:46:10.358925Z"Morse, Jennifer"https://zbmath.org/authors/?q=ai:morse.jennifer"Pan, Jianping"https://zbmath.org/authors/?q=ai:pan.jianping"Poh, Wencin"https://zbmath.org/authors/?q=ai:poh.wencin"Schilling, Anne"https://zbmath.org/authors/?q=ai:schilling.anneA crystal is a directed graph whose encoded information mirror that of the highest weight theory of a root system. Their importance relies on that it reduces problems about representations of Kac-Moody Lie algebras to analogous problems but in a purely combinatorial context; and conversely. References introducing crystals are, from a combinatorial point of view, [\textit{D. Bump} and \textit{A. Schilling}, Crystal bases. Representations and combinatorics. Hackensack, NJ: World Scientific (2017; Zbl 1440.17001); \textit{P. Hersh} and \textit{C. Lenart}, Math. Z. 286, No. 3--4, 1435--1464 (2017; Zbl 1371.05315)]; from the algebraic side, see [\textit{J. Hong} and \textit{S.-J. Kang}, Introduction to quantum groups and crystal bases. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 1134.17007)].
Here, the authors associate a type A crystal on the set of \(321\)-avoiding Hecke factorizations; for an expanded version of the content of this paper see [\textit{J. Morse} et al., Electron. J. Comb. 27, No. 2, Research Paper P2.29, 48 p. (2020; Zbl 1441.05237)]. More references: [\textit{M. Albert} et al., Eur. J. Comb. 78, 44--72 (2019; Zbl 1414.05004); \textit{M. Bóna}, Combinatorics of permutations. Boca Raton, FL: CRC Press (2012; Zbl 1255.05001)]. The authors also define a new insertion from decreasing factorizations to pairs of semistandard Yount tableaux, and prove several properties; in particular, this new insertion intertwines with the crystal operators. Everything is related with the combinatorics of Young tableaux.
Additional references: [\textit{M. Gillespie} et al., Algebr. Comb. 3, No. 3, 693--725 (2020; Zbl 1443.05183); \textit{Y.-T. Oh} and \textit{E. Park}, Electron. J. Comb. 26, No. 4, Research Paper P4.39, 19 p. (2019; Zbl 1428.05329); \textit{S. Assaf} and \textit{E. K. Oğuz}, Sémin. Lothar. Comb. 80B, 80B.26, 12 p. (2018; Zbl 1411.05272); \textit{J.-H. Kwon}, Handb. Algebra 6, 473--504 (2009; Zbl 1221.17017); \textit{G. Benkart} and \textit{S.-J. Kang}, Adv. Stud. Pure Math. 28, 21--54 (2000; Zbl 1027.17009); \textit{T. H. Baker}, Prog. Math. 191, 1--48 (2000; Zbl 0974.05080); \textit{G. Cliff}, J. Algebra 202, No. 1, 10--35 (1998; Zbl 0969.17010); \textit{P. Littelmann}, J. Algebra 175, No. 1, 65--87 (1995; Zbl 0831.17004); \textit{A. Puskás}, Assoc. Women Math. Ser. 16, 333--362 (2019; Zbl 1416.05300); \textit{V. I. Danilov} et al., Algebra 2013, Article ID 483949, 14 p. (2013; Zbl 1326.05045); \textit{T. Lam} and \textit{P. Pylyavskyy}, Sel. Math., New Ser. 19, No. 1, 173--235 (2013; Zbl 1260.05043); \textit{V. Genz} et al., Sel. Math., New Ser. 27, No. 4, Paper No. 67, 45 p. (2021; Zbl 07383344); \textit{N. Jacon}, Electron. J. Comb. 28, No. 2, Research Paper P2.21, 16 p. (2021; Zbl 07356164); \textit{T. Shoji} and \textit{Z. Zhou}, J. Algebra 569, 67--110 (2021; Zbl 07286477)].Alcove paths and Gelfand-Tsetlin patternshttps://zbmath.org/1472.051542021-11-25T18:46:10.358925Z"Watanabe, Hideya"https://zbmath.org/authors/?q=ai:watanabe.hideya"Yamamura, Keita"https://zbmath.org/authors/?q=ai:yamamura.keitaSummary: In their study of the equivariant K-theory of the generalized flag varieties \(G/P\), where \(G\) is a complex semisimple Lie group, and \(P\) is a parabolic subgroup of \(G\), \textit{C. Lenart} and \textit{A. Postnikov} [Trans. Am. Math. Soc. 360, No. 8, 4349--4381 (2008; Zbl 1211.17021)] introduced a combinatorial tool, called the alcove path model. It provides a model for the highest weight crystals with dominant integral highest weights, generalizing the model by semistandard Young tableaux. In this paper, we prove a simple and explicit formula describing the crystal isomorphism between the alcove path model and the Gelfand-Tsetlin pattern model for type \(A\).The Bruhat order, the lookup conjecture and spiral Schubert varieties of type \(\tilde{A}_2\)https://zbmath.org/1472.051552021-11-25T18:46:10.358925Z"Graham, William"https://zbmath.org/authors/?q=ai:graham.william-a"Li, Wenjing"https://zbmath.org/authors/?q=ai:li.wenjingSummary: Although the Bruhat order on a Weyl group is closely related to the singularities of the Schubert varieties for the corresponding Kac-Moody group, it can be difficult to use this information to prove general theorems. This paper uses the action of the affine Weyl group of type \(\tilde{A}_2\) on a Euclidean space \(V \cong \mathbb{R}^2\) to study the Bruhat order on \(W\). We believe that these methods can be used to study the Bruhat order on arbitrary affine Weyl groups. Our motivation for this study was to extend the lookup conjecture of \textit{B. D. Boe} and \textit{W. Graham} [Am. J. Math. 125, No. 2, 317--356 (2003; Zbl 1074.14045)] (which is a conjectural simplification of the Carrell-Peterson criterion (see [\textit{J. B. Carrell}, Proc. Symp. Pure Math. 56, 53--61 (1994; Zbl 0818.14020)]) for rational smoothness) to type \(\tilde{A}_2\). Computational evidence suggests that the only Schubert varieties in type \(\tilde{A}_2\) where the ``nontrivial'' case of the lookup conjecture occurs are the spiral Schubert varieties, and as a step towards the lookup conjecture, we prove it for a spiral Schubert variety \(X ( w )\) of type \(\tilde{A}_2\). The proof uses descriptions we obtain of the elements \(x \leq w\) and of the rationally smooth locus of \(X ( w )\) in terms of the \(W\)-action on \(V\). As a consequence we describe the maximal nonrationally smooth points of \(X ( w )\). The results of this paper are used in a sequel to describe the smooth locus of \(X ( w )\), which is different from the rationally smooth locus.The Möbius function of \(\mathrm{PSL}(3, 2^p)\) for any prime \(p\)https://zbmath.org/1472.051562021-11-25T18:46:10.358925Z"Borello, Martino"https://zbmath.org/authors/?q=ai:borello.martino"Dalla Volta, Francesca"https://zbmath.org/authors/?q=ai:dalla-volta.francesca"Zini, Giovanni"https://zbmath.org/authors/?q=ai:zini.giovanniSchurity of the wedge product of association schemes and generalized wreath product of permutation groupshttps://zbmath.org/1472.051572021-11-25T18:46:10.358925Z"Bagherian, Javad"https://zbmath.org/authors/?q=ai:bagherian.javadSummary: The generalized wreath product of permutation groups was introduced by \textit{S. Evdokimov} and \textit{I. Ponomarenko} [Eur. J. Comb. 30, No. 6, 1456--1476 (2009; Zbl 1228.05311); St. Petersbg. Math. J. 24, No. 3, 431--460 (2013; Zbl 1278.20002); translation from Algebra Anal. 24, No. 3, 84--127 (2012)] in order to study the schurity problem for S-rings over cyclic groups. In this paper we construct the generalized wreath product of permutation groups by a method entirely different from Evdokimov and Ponomarenko's construction. Then we give a necessary and sufficient condition for the wedge product of schurian association schemes coming from the generalized wreath product of permutation groups.A lower bound for the discriminant of polynomials related to Chebyshev polynomialshttps://zbmath.org/1472.051582021-11-25T18:46:10.358925Z"Filipovski, Slobodan"https://zbmath.org/authors/?q=ai:filipovski.slobodanThe discriminant of a polynomial \(f=a\prod_{i=0}^m (x-\alpha_i)\) is defined by \[D(f)=a^{2m-2}\prod_{1\leq i<j\leq m} (\alpha_j-\alpha_i)^2.\] The discriminant is a useful tool to get information about the roots of a high-degree polynomial.
In this paper, the author gives a lower bound for the discriminant of the polynomials \(\{G_{k,i}\}\) defined by \(G_{k,0}(x)=1\), \(G_{k,1}(x)=x+1\) and then recursively \[G_{k,i+2}(x)=xG_{k,i+1}(x)-(k-1)G_{k,i}(x).\]
These polynomials have been used for the study of graphs and their girth (this is the length of the shortest circuit in the graph), specially due to its relation with the Moore Bound \[M_{d,k}=\begin{cases} 1+d\frac{(d-1)^{k-1}-1}{d-2}& d>2\\
2k+1& d=2 \end{cases},\] which relates the degree, the order, the diameter and the girth of a graph. The discriminant of these polynomials have been used to prove the existence of graphs with certain degree or girth as in [\textit{C. Delorme} and \textit{G. Pineda-Villavicencio}, Electron. J. Comb. 17, No. 1, Research Paper R143, 25 p. (2010; Zbl 1204.05043)].
While all of this is addressed in the second section of the paper and the author suggests a path to proof his main result in a graph-theoretical way, most of the document is devoted to proof the result in a more elementary way. The main idea is to use orthogonality of some polynomials related to \(\{G_{k,i}\}\) and from there stablish a relation through simple inequalities with the discriminant of \(G_{k,d}\). While the notation could be a little heavy, the results are quite simple to follow, leading to the following bound for the discriminant: \[D(G_{k,d})>d^d(k-2)\left[\sqrt{k(k-1)^2-2}\right]^{d-2}.\]Two characterizations of the grid graphshttps://zbmath.org/1472.051592021-11-25T18:46:10.358925Z"Gebremichel, Brhane"https://zbmath.org/authors/?q=ai:gebremichel.brhane"Cao, Meng-Yue"https://zbmath.org/authors/?q=ai:cao.mengyue"Koolen, Jack H."https://zbmath.org/authors/?q=ai:koolen.jack-hSummary: In this paper we give two characterizations of the \(p \times q\)-grid graphs as co-edge-regular graphs with four distinct eigenvalues.Idempotent systems and character algebrashttps://zbmath.org/1472.051602021-11-25T18:46:10.358925Z"Nomura, Kazumasa"https://zbmath.org/authors/?q=ai:nomura.kazumasa"Terwilliger, Paul"https://zbmath.org/authors/?q=ai:terwilliger.paul-mSummary: We recently introduced the notion of an idempotent system. This linear algebraic object is motivated by the structure of an association scheme. There is a type of idempotent system, said to be symmetric. In the present paper we classify up to isomorphism the idempotent systems and the symmetric idempotent systems. We also describe how symmetric idempotent systems are related to character algebras.Bialgebra structures on table algebrashttps://zbmath.org/1472.051612021-11-25T18:46:10.358925Z"Singh, Gurmail"https://zbmath.org/authors/?q=ai:singh.gurmailSummary: A bialgebra is both a unital associative algebra and a coassociative coalgebra \(A\) over a field \(K\) whose algebra and coalgebra structures satisfy certain compatibility conditions. A table algebra \(A\) is an algebra over \(\mathbb{C}\) with distinguish basis \(\mathcal{B}\) and involution \(\ast\) that satisfies certain conditions. In this paper, we establish a process to find bialgebra structures on commutative table algebras and using that we find bialgebra structures on table algebras of dimensions 2 and 3.Möbius and coboundary polynomials for matroidshttps://zbmath.org/1472.051622021-11-25T18:46:10.358925Z"Johnsen, Trygve"https://zbmath.org/authors/?q=ai:johnsen.trygve"Verdure, Hugues"https://zbmath.org/authors/?q=ai:verdure.huguesSummary: We study how some coefficients of two-variable coboundary polynomials can be derived from Betti numbers of Stanley-Reisner rings. We also explain how the connection with these Stanley-Reisner rings forces the coefficients of the two-variable coboundary polynomials and Möbius polynomials to satisfy certain universal equations.Dimension of CPT posetshttps://zbmath.org/1472.060022021-11-25T18:46:10.358925Z"Majumder, Atrayee"https://zbmath.org/authors/?q=ai:majumder.atrayee"Mathew, Rogers"https://zbmath.org/authors/?q=ai:mathew.rogers"Rajendraprasad, Deepak"https://zbmath.org/authors/?q=ai:rajendraprasad.deepakThe dimension of a poset \((X, \le)\), \(\dim X\), is the minimum cardinality of a collection \((\le_i: i \in I)\) of linear orders on \(X\) such that \(x \le y\) iff \(x \le_i y\) for all \(i \in I\); such a collection is called a realizer of \((X, \le)\). A CPT poset is a poset \((X, \le\)) which is isomorphic to a poset (called a CPT model of \(X\)) consisting of paths of some tree (the host tree) and ordered by inclusion. The main theorem gives an upper estimation for the dimension of a CPT poset modeled by the set of all paths of a rooted tree where every internal vertex has exactly \(k\) children. An upper estimation of \(\dim X\) for a poset \(X\) admitting a CPT model in a host tree of a given radius and maximum degree is obtained as a corollary. The proof of the theorem gives an algorithm for constructing a realizer, under inclusion relation, for the poset consisting of all 1-element and 2-element subsets of \(\{1,2,\dots,n\}\) .Purely chromatic and hyper chromatic latticeshttps://zbmath.org/1472.060042021-11-25T18:46:10.358925Z"Pawar, Madhukar M."https://zbmath.org/authors/?q=ai:pawar.madhukar-m"Wadile, Amit S."https://zbmath.org/authors/?q=ai:wadile.amit-sIn this paper, the authors introduce the concepts of a hyper chromatic, a critically chromatic and a purely chromatic lattice. Some characterizations for purely chromatic and hyper chromatic dismantlable lattices are proved. The following results are proved.
Theorem 1. A nontrivial lattice of finite length is a purely 2-chromatic lattice if and only if it is modular.
Theorem 2. A dismantlable lattice is a hyper chromatic lattice if and only if it is 2-chromatic and nonmodular.
Some relationships between the chromatic number of two lattices and their linear sum, vertical sum and adjunct are proved.Fifty years of Kurepa's \(!n\) hypothesishttps://zbmath.org/1472.110032021-11-25T18:46:10.358925Z"Mijajlović, Žarko"https://zbmath.org/authors/?q=ai:mijajlovic.zarkoSummary: \textit{Đ. Kurepa} [Math. Balk. 1, 147--153 (1971; Zbl 0224.10009)] formulated in 1971 the so called left factorial hypothesis. This hypothesis is still open, despite much efforts to solve it. Here we give some historical notes and review the current status of the hypothesis. Using probabilistic model we also estimated sums of Kurepa reminders and discussed in details finite Kurepa trees.On a congruence conjecture of Swisherhttps://zbmath.org/1472.110152021-11-25T18:46:10.358925Z"He, Bing"https://zbmath.org/authors/?q=ai:he.bing|he.bing.1|he.bing.4|he.bing.2|he.bing.3Summary: A congruence on a conjecture of van Hamme is established. This result confirms a particular case of a congruence conjecture of \textit{H. Swisher} [Res. Math. Sci. 2, Paper No. 18, 21 p. (2015; Zbl 1337.33005)].Supercongruences for sums involving Domb numbershttps://zbmath.org/1472.110212021-11-25T18:46:10.358925Z"Liu, Ji-Cai"https://zbmath.org/authors/?q=ai:liu.jicaiSummary: We prove some supercongruence and divisibility results on sums involving Domb numbers, which confirm four conjectures of Z.-W. Sun and Z.-H. Sun. For instance, by using a transformation formula due to Chan and Zudilin, we show that for any prime \(p\geq 5\),
\[
\sum\limits_{k=0}^{p-1}\frac{3k+1}{(-32)^k}\operatorname{Domb}(k)\equiv (-1)^{\frac{p-1}{2}}p+p^3E_{p-3}\pmod{p^4},
\]
which is regarded as a \(p\)-adic analogue of the interesting formula for \(1/\pi\) due to Rogers:
\[
\sum\limits_{k=0}^\infty\frac{3k+1}{(-32)^k}\operatorname{Domb}(k)=\frac{2}{\pi}.
\]
Here \(\operatorname{Domb}(n)\) and \(E_n\) are the famous Domb numbers and Euler numbers.Supercongruences and binary quadratic formshttps://zbmath.org/1472.110262021-11-25T18:46:10.358925Z"Sun, Zhi-Hong"https://zbmath.org/authors/?q=ai:sun.zhihongSummary: Let \(p > 3\) be a prime, and let \(a,b\) be two rational \(p\)-adic integers. We present general congruences for \(\sum_{k=0}^{p-1}\binom{a}{k}\binom{-1-a}{k}\frac{p}{k+b}\pmod{p^2} \). Let \(\{D_n\}\) be the Domb numbers given by \(D_n=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}{k}\binom{2n-2k}{n-k} \). We also prove that
\[\sum_{n=0}^{p-1}\frac{D_n}{16^n}\equiv \sum_{n=0}^{p-1}\frac{D_n}{4^n} \equiv \begin{cases} 4x^2-2p\pmod{p^2} &\text{if } 3\mid p-1\text{ and so }p=x^2+3y^2,\\
0\pmod{p^2} &\text{if }p\equiv 2\pmod 3,\end{cases}\]
which was conjectured by Z. W. Sun.Proof of two congruences concerning Legendre polynomialshttps://zbmath.org/1472.110272021-11-25T18:46:10.358925Z"Wang, Chen"https://zbmath.org/authors/?q=ai:wang.chen"Xia, Wei"https://zbmath.org/authors/?q=ai:xia.wei.3|xia.wei.2|xia.wei|xia.wei.1Summary: The Legendre polynomials \(P_n(x)\) are defined by
\[
P_n(x)=\sum\limits_{k=0}^n \binom{n+k}{k}\binom{n}{k}\left(\frac{x-1}{2}\right)^k\quad (n=0,1,2,\ldots).
\]
In this paper, we prove two congruences concerning Legendre polynomials. For any prime \(p>3\), by using the symbolic summation package Sigma, we show that
\[
\sum\limits_{k=0}^{p-1}(2k+1)P_k(-5)^3\equiv p-\frac{10}{3}p^2q_p(2)\pmod{p^3},
\]
where \(q_p(2)=(2^{p-1}-1)/p\) is the Fermat quotient. This confirms a conjecture of Z.-W. Sun. Furthermore, we prove the following congruence which was conjectured by V. J. W. Guo
\begin{align*}
& \sum\limits_{k=0}^{p-1}(-1)^k(2k+1)P_k(2x+1)^4\\
\equiv p & \sum\limits_{k=0}^{(p-1)/2}(-1)^k\binom{2k}{k}^2(x^2+x)^k(2x+1)^{2k}\pmod{p^3},
\end{align*}
where \(p\) is an odd prime and \(x\) is an integer.Factorization theorems for relatively prime divisor sums, GCD sums and generalized Ramanujan sumshttps://zbmath.org/1472.110382021-11-25T18:46:10.358925Z"Mousavi, Hamed"https://zbmath.org/authors/?q=ai:mousavi.hamed"Schmidt, Maxie D."https://zbmath.org/authors/?q=ai:schmidt.maxie-dThe authors give new factorization theorems for Lambert series which allow them to express formal generating functions for special sums as invertible matrix transformations involving partition functions.
There are many formulas in the paper which look complicated to the reviewer but the authors of the article believe that the formulas will be useful in the future. Note also that there are two typographical errors on the second page of the paper. First, the last term in equation (1) should be
\[ \sum_{\substack{1<d\leq x \\ 1<(d,x)<d}}f(d). \]
Secondly, the term \(f(15)\) should be added to the right-hand side of the equation for \(\tilde{S_f}(24)\). There might be more errors on the other pages of the paper.Permutative numbershttps://zbmath.org/1472.110462021-11-25T18:46:10.358925Z"Lucht, Lutz G."https://zbmath.org/authors/?q=ai:lucht.lutz-gerhard"Motzer, Renate"https://zbmath.org/authors/?q=ai:motzer.renateZusammenfassung: Zur Untersuchung des 2017/18 von \textit{H. Hischer} [GDM Mitteil. 105, 29--30 (2018)] kommunizierten Vorkommens von 10-stelligen Dezimalzahlen, die sämtliche Ziffern \(1,2,\ldots,9\) enthalten und zu denen es einen ganzzahligen Teiler \(>1\) derart gibt, daß die Division nur ihre Ziffernreihenfolge verändert, wird hier der Begriff der permutativen Zahl eingeführt. Dieser Beitrag liefert einige allgemeine Ergebnisse über Existenz und Konstruktion permutativer Zahlen und weist auf offene Probleme hin.Efficient arithmetic regularity and removal lemmas for induced bipartite patternshttps://zbmath.org/1472.110542021-11-25T18:46:10.358925Z"Alon, Noga"https://zbmath.org/authors/?q=ai:alon.noga"Fox, Jacob"https://zbmath.org/authors/?q=ai:fox.jacob"Zhao, Yufei"https://zbmath.org/authors/?q=ai:zhao.yufeiSummary: Let \(G\) be an abelian group of bounded exponent and \(A \subseteq G\). We show that if the collection of translates of \(A\) has VC dimension at most \(d\), then for every \(\varepsilon > 0\) there is a subgroup \(H\) of \(G\) of index at most \(\varepsilon^{-d-o(1)}\) such that one can add or delete at most \(\varepsilon|G|\) elements to/from \(A\) to make it a union of \(H\)-cosets.
We also establish a removal lemma with polynomial bounds, with applications to property testing, for induced bipartite patterns in a finite abelian group with bounded exponent.Counting on Euler and Bernoulli number identitieshttps://zbmath.org/1472.110732021-11-25T18:46:10.358925Z"Benjamin, Arthur T."https://zbmath.org/authors/?q=ai:benjamin.arthur-t"Lentfer, John"https://zbmath.org/authors/?q=ai:lentfer.john"Martinez, Thomas C."https://zbmath.org/authors/?q=ai:martinez.thomas-cLet \(\{E_n\}\) and \(\{B_n\}\) be the Euler numbers and Bernoulli numbers, respectively. In the paper under review, the authors give a combinatorial proof of the identity \[B_{2n}=\frac{2n}{2^{2n}(2^{2n}-1)}\sum_{j=0}^{n-1}\binom{2n-1}{2j}E_{2j}\ (n\ge 1).\]Hankel continued fractions and Hankel determinants of the Euler numbershttps://zbmath.org/1472.110742021-11-25T18:46:10.358925Z"Han, Guo-Niu"https://zbmath.org/authors/?q=ai:han.guo-niuThe author presents the Hankel continued fraction expansion and Hankel determinants of the Euler numbers, thus bringing together known results for the even (secant numbers) and odd Euler numbers (tangent numbers). The proofs are based on Flajolet's continued fraction for permutation statistics and combinatorial interpretations of the Euler number. The continued fraction expansion also leads to a \(q\)-analogue of the Euler numbers.A generalization of primitive sets and a conjecture of Erdőshttps://zbmath.org/1472.110772021-11-25T18:46:10.358925Z"Chan, Tsz Ho"https://zbmath.org/authors/?q=ai:chan.tsz-ho"Lichtman, Jared Duker"https://zbmath.org/authors/?q=ai:lichtman.jared-duker"Pomerance, Carl"https://zbmath.org/authors/?q=ai:pomerance.carlA set of integers greater than 1 is primitive, if no element of the set divides another element of the set. Erdős proved in 1935 that there is a number \(K\) such that for every primitive set \(A\), \(\sum_{n\in A} 1/n \leq K\). In 1988 Erdős conjectured that the least upper bound \(K\) is realized by the set of primes. There is a generalization of primitivity: a set \(A\) of integers greater than 1 with \(|A|\geq k+1\) is called \(k\)-primitive, if no element of \(A\) divides the product of \(k\) distinct other elements of \(A\). The paper under review shows that the sum of reciprocals of the elements of any 2-primitive set up to any bound \(n\) is at most the sum of reciprocals of primes up to the bound \(n\), a nice analogue of the conjecture of Erdős.The Schur degree of additive setshttps://zbmath.org/1472.110792021-11-25T18:46:10.358925Z"Eliahou, S."https://zbmath.org/authors/?q=ai:eliahou.shalom"Revuelta, M. P."https://zbmath.org/authors/?q=ai:revuelta.m-pastoraSummary: Let \(( G,+)\) be an abelian group. A subset of \(G\) is sumfree if it contains no elements \(x,y,z\) such that \(x + y = z\). We extend this concept by introducing the \textit{Schur degree} of a subset of \(G\), where Schur degree 1 corresponds to sumfree. The classical inequality \(S(n) \leq R_n ( 3 ) - 2\), between the Schur number \(S(n)\) and the Ramsey number \(R_n(3) = R( 3, \ldots, 3)\), is shown to remain valid in a wider context, involving the Schur degree of certain subsets of \(G\). Recursive upper bounds are known for \(R_n(3)\) but not for \(S(n)\) so far. We formulate a conjecture which, if true, would fill this gap. Indeed, our study of the Schur degree leads us to conjecture \(S(n) \leq n(S(n - 1) + 1)\) for all \(n \geq 2\). If true, it would yield substantially better upper bounds on the Schur numbers, e.g. \(S(6) \leq 966\) conjecturally, whereas all is known so far is \(536 \leq S(6) \leq 1836\).Zero-sum analogues of van der Waerden's theorem on arithmetic progressionshttps://zbmath.org/1472.110802021-11-25T18:46:10.358925Z"Robertson, Aaron"https://zbmath.org/authors/?q=ai:robertson.aaronSummary: Let \(r\) and \(k\) be positive integers with \(r \mid k\). Denote by \(z(k; r)\) the minimum integer such that every coloring \(\chi : [ 1, z(k; r)] \to \{ 0, 1, \dots , r - 1 \}\) admits a \(k\)-term arithmetic progression \(a, a + d, \dots , a + (k - 1) d\) with \(\sum^{k -1}_{j=0} \chi (a + jd) \equiv 0 \pmod r\). We investigate these numbers as well as a ``mixed'' monochromatic/zero-sum analogue. We also present an interesting reciprocity between the van der Waerden numbers and \(z(k; r)\).Weighted partitions and generalized \(r\)-Lah numbershttps://zbmath.org/1472.110812021-11-25T18:46:10.358925Z"Belbachir, Hacène"https://zbmath.org/authors/?q=ai:belbachir.hacene"Belkhir, Amine"https://zbmath.org/authors/?q=ai:belkhir.amine"Bousbaa, Imad-Eddine"https://zbmath.org/authors/?q=ai:bousbaa.imad-eddineSummary: Using weighted ordered partitions, we provide a new combinatorial interpretation for the two-parameter polynomial generalization of the \(r\)-Lah numbers. Moreover, by the inclusion-exclusion principle, we give combinatorial proofs for an explicit formula and some combinatorial properties. Finally, we provide an expression involving symmetric functions.A new approach to the \(r\)-Whitney numbers by using combinatorial differential calculushttps://zbmath.org/1472.110842021-11-25T18:46:10.358925Z"Méndez, Miguel A."https://zbmath.org/authors/?q=ai:mendez.miguel-a"Ramírez, José L."https://zbmath.org/authors/?q=ai:ramirez.jose-luisSummary: In the present article we introduce two new combinatorial interpretations of the \(r\)-Whitney numbers of the second kind obtained from the combinatorics of the differential operators associated to the grammar \(G := \{y \rightarrow yx^m, x \rightarrow x\}\). By specializing \(m = 1\) we obtain also a new combinatorial interpretation of the \(r\)-Stirling numbers of the second kind. Again, by specializing to the case \(r = 0\) we introduce a new generalization of the Stirling number of the second kind and through them a binomial type family of polynomials that generalizes Touchard's polynomials. Moreover, we recover several known identities involving the \(r\)-Dowling polynomials and the \(r\)-Whitney numbers using the combinatorial differential calculus. We construct a family of posets that generalize the classical Dowling lattices. The \(r\)-Withney numbers of the first kind are obtained as the sum of the Möbius function over elements of a given rank. Finally, we prove that the \(r\)-Dowling polynomials are a Sheffer family relative to the generalized Touchard binomial family, study their umbral inverses, and introduce \([m]\)-Stirling numbers of the first kind. From the relation between umbral calculus and the Riordan matrices we give several new combinatorial identitiesOn diagonal equations over finite fields via walks in NEPS of graphshttps://zbmath.org/1472.111012021-11-25T18:46:10.358925Z"Videla, Denis E."https://zbmath.org/authors/?q=ai:videla.denis-eSummary: We obtain an explicit combinatorial formula for the number of solutions \((x_1,\ldots,x_r)\in(\mathbb{F}_{p^{ab}})^r\) to the diagonal equation \(x_1^k+\cdots+x_r^k=\alpha\) over the finite field \(\mathbb{F}_{p^{ab}}\), with \(k=\frac{p^{ab}-1}{b(p^a-1)}\) and \(b>1\), by using the number of \(r\)-walks in NEPS of complete graphs.The Dedekind eta function and D'Arcais-type polynomialshttps://zbmath.org/1472.111222021-11-25T18:46:10.358925Z"Heim, Bernhard"https://zbmath.org/authors/?q=ai:heim.bernhard-ernst"Neuhauser, Markus"https://zbmath.org/authors/?q=ai:neuhauser.markusSummary: D'Arcais-type polynomials encode growth and non-vanishing properties of the coefficients of powers of the Dedekind eta function. They also include associated Laguerre polynomials. We prove growth conditions and apply them to the representation theory of complex simple Lie algebras and to the theory of partitions, in the direction of the Nekrasov-Okounkov hook length formula. We generalize and extend results of \textit{B. Kostant} [Invent. Math. 158, No. 1, 181--226 (2004; Zbl 1076.17002)] and \textit{G.-N. Han} [Ann. Inst. Fourier 60, No. 1, 1--29 (2010; Zbl 1215.05013)].Analysis of generalized continued fraction algorithms over polynomialshttps://zbmath.org/1472.112072021-11-25T18:46:10.358925Z"Berthé, Valérie"https://zbmath.org/authors/?q=ai:berthe.valerie"Nakada, Hitoshi"https://zbmath.org/authors/?q=ai:nakada.hitoshi"Natsui, Rie"https://zbmath.org/authors/?q=ai:natsui.rie"Vallée, Brigitte"https://zbmath.org/authors/?q=ai:vallee.brigitteIts well knew that the greatest common divisor (gcd) computation for univariate polynomials is a basic operation in computer algebra and Euclid's algorithm completely solves the problem of gcd computation for two entries. However, there does not exist a canonical generalization of Euclid's algorithm when working with at least three entries. Three generalized Euclidean algorithms for polynomials with coefficients in a finite field, inspired by classical multidimensional continued fraction maps, namely the Jacobi-Perron, the Brun, and the fully subtractive maps were chosen and compared in this paper. The two-dimensional versions of the Jacobi-Perron, the Brun, and the fully subtractive algorithms are associated with continued fraction maps. A unified framework for these algorithms and their associated continued fraction maps are provided. The convergence of the continued fraction maps was discussed. The bivariate generating functions are the main tool of the study. This enables in particular to exhibit asymptotic Gaussian laws. The various costs for the gcd algorithms, including the number of iterations and two versions of the bit-complexity, corresponding to two representations of polynomials analyzed in the paper. The associated two-dimensional continued fraction maps are studied and the invariance and the ergodicity of the Haar measure are proved. The authors obtain corresponding estimates for the costs of truncated trajectories under the action of these continued fraction maps and are compared the two models (gcd algorithms and their associated continued fraction maps).Improved dispersion bounds for modified Fibonacci latticeshttps://zbmath.org/1472.112152021-11-25T18:46:10.358925Z"Kritzinger, Ralph"https://zbmath.org/authors/?q=ai:kritzinger.ralph"Wiart, Jaspar"https://zbmath.org/authors/?q=ai:wiart.jasparConsider point sets \(\mathcal P\) in the unit square \([0,1]^2\) consisting of \(N\) not necessarily distinct elements and let \(\mathcal B\) be the set of all axes-parallel boxes in the unit square. The authors study dispersion of \(\mathcal P\) defined as \( \text{disp }(\mathcal P):=\sup_{B\in\mathcal B, B\cap\mathcal P=\emptyset}\lambda(B) \), where \(\lambda(B)\) denotes the area of the box \(B\). Known previous results: \textit{A. Dumitrescu} and \textit{M. Jiang} [Algorithmica 66, No. 2, 225--248 (2013; Zbl 1262.68186)]
proved that \( \text{disp }(\mathcal P)\ge\max(1/(N+1),5/(4(N+5))). \) \textit{S. Breneis} and \textit{A. Hinrichs} [Radon Ser. Comput. Appl. Math. 26, 117--132 (2020; Zbl 1471.11050)] constructed points with small dispersion using the Fibonacci numbers \(F_m\). The dispersion of the Fibonacci lattice \( \mathcal F_m=\{(k/F_m,\{kF_{m-2}/F_m\}):k=0,1,2,\dots,F_m-1\} \) is given by \( \text{disp }\mathcal F_m=(2(F_m-1)/F^2_m) \) for \(m\ge8\), where \(\{x\}\) denotes the fractional part of \(x\). Then the limit \(\lim_{m\to\infty}|\mathcal F_m|\text{disp }(\mathcal F_m)=2\). The central result of this paper is replacing the limit 2 by a smaller constant 1.894427\dots for the modified Fibonacci lattice \(\widetilde{\mathcal F}_m\) defined in the following way: \par \(\pi(k)=kF_{m-2}\pmod {F_m}\), \par \(s(i)=(\sqrt{5}+1)/2 \text{ if } \pi(i)<\pi(i+1) \text{ and } 1 \text{ otherwise }\), \par \(L=\sum_{i=0}^{F_m-1}s(i)\), \par \(x_k=\sum_{i=0}^{k-1}, k=0,1,\dots,F_m-1\), \par \(\widetilde{\mathcal F}_m=\{(x_k/L,x_{\pi(k)}/L),k=0,1,\dots,F_m-1\}\). \par\noindent In this case, the authors prove \par \(\lim_{m\to\infty}|\widetilde{\mathcal F}_m|\text{disp }(\widetilde{\mathcal F}_m) =1+\frac{2}{\sqrt{5}}\).
For proof they use, e.g., \(F^2_{m-2}+F^2_m-3F_mF_{m-2}=(-1)^m\). Using the theory of uniform distribution of sequences it is also conjectured that the Fibonacci lattice \(\mathcal F_m\) have a smallest possible \(L_2\) discrepancy, since for \(m\le 7\) this is known. Note that in uniform distribution theory the \(s\)-dimensional dispersion is defined as follows: If \(\mathbf x_1,\mathbf x_2,\dots,\mathbf x_N\) belong to \([0,1]^s\) then the dispersion \(d_N\) of \(\mathbf x_n\) is defined by \(d_N(\mathbf x_1,\dots,\mathbf x_N)=\sup_{\mathbf x\in[0,1]^s} \min_{1\le n\le N}|\mathbf x-\mathbf x_n|\), where \(|\mathbf x-\mathbf x_n|\) is the Euclidean distance. Basic results of such dispersion are in [\textit{H. Niederreiter}, Random number generation and quasi-Monte Carlo methods. Philadelphia, PA: SIAM (1992; Zbl 0761.65002)].Small Gál sums and applicationshttps://zbmath.org/1472.112242021-11-25T18:46:10.358925Z"de la Bretèche, Régis"https://zbmath.org/authors/?q=ai:de-la-breteche.regis"Munsch, Marc"https://zbmath.org/authors/?q=ai:munsch.marc"Tenenbaum, Gérald"https://zbmath.org/authors/?q=ai:tenenbaum.geraldLet \(\alpha\in(0,1]\). Given \(\mathcal{M}\subseteq\mathbb{N}\) the so-called traditional Gál sum is defined by
\[
S_\alpha(\mathcal{M})=\sum_{m,n\in\mathcal{M}}\left(\frac{\gcd(m,n)}{\mathrm{lcm}(m,n)}\right)^\alpha.
\]
Let \(\mathcal{M}_N=\{1,2,\dots,N\}\). For the two-variable arithmetic function \(f:\mathbb{N}^2\to\mathbb{C}\) and \(\mathbf{c}=(c_n)_{1\leq n\leq N}\in\mathbb{R}^N\) let
\[
G(\alpha, f, \mathbf{c}; N):=\sum_{m,n\in \mathcal{M}_N}\frac{\left(\gcd(m,n)\right)^\alpha}{f(m,n)}\,c_mc_n,
\]
and \(G_N(\alpha, f, \mathbf{c}):=N\inf G(\alpha, f, \mathbf{c}; N)\), where infimum runs over \(\mathbf{c}\in\left(\mathbb{R}^+\right)^N\) with \(\|\mathbf{c}\|_1=1\). Letting \(f_1(m,n)=m+n\) and \(f_2(m,n)=\sqrt{mn}\), in the paper under review the authors consider \(\mathcal{V}_N:=G_N(1, f_1, \mathbf{c})\) and \(\mathcal{T}_N:=G_N(1, f_2, \mathbf{c})\), and prove that for \(N\geq 3\) one has the approximations
\[
\left(\log N\right)^\eta\ll\mathcal{V}_N\leq\frac{1}{2}\,\mathcal{T}_N\ll\left(\log N\right)^\eta\left(\log\log N\right)^3,
\]
where \(\eta\approxeq 0.16656<1/6\) determined explicitly. They provide some applications, where this minimization question arises naturally. As the first application the authors give a logarithmic improvement on Burgess' bound on multiplicative character sums \(S(M,N;\chi):=\sum_{M<n\leq M+N}\chi(n)\) where \(\chi\) is a Dirichlet character to the modulus \(p\). The second application is related with nonvanishing of theta functions \(\vartheta(x;\chi)=\sum_{n\geq 1}\chi(n)\mathrm{e}^{-\pi n^2 x/p}\), for which it turns out that a related minimization problem yields better results. As an application of this latter estimate, they obtain lower bounds for low moments of character sums. This minimization problem might also have applications in metric Diophantine approximationErdősian functions and an identity of Gausshttps://zbmath.org/1472.112392021-11-25T18:46:10.358925Z"Chatterjee, Tapas"https://zbmath.org/authors/?q=ai:chatterjee.tapas"Khurana, Suraj Singh"https://zbmath.org/authors/?q=ai:khurana.suraj-singhSummary: A famous identity of Gauss gives a closed form expression for the values of the digamma function \(\psi(x)\) at rational arguments \(x\) in terms of elementary functions. Linear combinations of such values are intimately connected with a conjecture of Erdős which asserts non vanishing of an infinite series associated to a certain class of periodic arithmetic functions. In this note we give a different proof for the identity of Gauss using an orthogonality like relation satisfied by these functions. As a by product we are able to give a new interpretation for \(n\)th Catalan number in terms of these functions.Lower bound for the least common multiplehttps://zbmath.org/1472.112462021-11-25T18:46:10.358925Z"Sury, B."https://zbmath.org/authors/?q=ai:sury.balasubramanianSummary: It is well known that the prime number theorem can be phrased as the statement that the least common multiple (lcm) of the first \(n\) natural numbers is asymptotic to the exponential of \(n\). Suitable weaker bounds of this lcm already suffice to deduce certain striking properties of primes such as the existence of a prime between \(n\) and \(2n\) for sufficiently large \(n\). In this note we prove in an elementary manner that the lcm of the first \(n\) natural numbers is bigger than \(2^n\) when \(n\) is bigger than 6.A note on extensions of multilinear maps defined on multilinear varietieshttps://zbmath.org/1472.112732021-11-25T18:46:10.358925Z"Gowers, W. T."https://zbmath.org/authors/?q=ai:gowers.william-timothy"Milićević, L."https://zbmath.org/authors/?q=ai:milicevic.lukaThe motivation and most of the background material for the paper under review come from the authors' paper [``A quantitative inverse theorem for the \(U^{4}\) norm over finite fields'', Preprint, \url{arXiv:1712.00241}] and the second author's article [Geom. Funct. Anal. 29, No. 5, 1503--1530 (2019, Zbl 1442.11027)].
The summary by the authors is a good description of this article: ``Let \(G_{1},\ldots,G_{k}\) be finite-dimensional vector spaces over a prime field
\(F_{p}\). \ A multilinear variety of codimension at most \(d\) is a subset of \(G_{1}\times\ldots\times G_{k}\) defined as the zero set of \(d\) forms, each of which is multilinear on some subset of the coordinates. A map \(\phi\) defined on a multilinear variety \(B\) is multilinear if for each coordinate \(c\) and all choices of \(x_{i}\in G_{i}\), \(i\neq c\), the restriction map \(y\longmapsto\phi(x_{1},\ldots,x_{c-1},y,x_{c+1},\ldots,x_{k})\) is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension at most \(d\) coincides on a multilinear variety of codimension \(O_{k}(d^{O_{k}(1)})\) with a multilinear map defined on the whole of \(G_{1}\times\ldots\times G_{k}\). \ Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results''.Degree of orthomorphism polynomials over finite fieldshttps://zbmath.org/1472.112932021-11-25T18:46:10.358925Z"Allsop, Jack"https://zbmath.org/authors/?q=ai:allsop.jack"Wanless, Ian M."https://zbmath.org/authors/?q=ai:wanless.ian-mSummary: An \textit{orthomorphism} over a finite field \(\mathbb{F}_q\) is a permutation \(\theta:\mathbb{F}_q\to\mathbb{F}_q\) such that the map \(x\mapsto\theta(x)-x\) is also a permutation of \(\mathbb{F}_q\). The \textit{degree} of an orthomorphism of \(\mathbb{F}_q\), that is, the degree of the associated reduced permutation polynomial, is known to be at most \(q-3\). We show that this upper bound is achieved for all prime powers \(q\notin\{2,3,5,8\}\). We do this by finding two orthomorphisms in each field that differ on only three elements of their domain. Such orthomorphisms can be used to construct 3-homogeneous Latin bitrades.Generalized \(F\)-signatures of Hibi ringshttps://zbmath.org/1472.130102021-11-25T18:46:10.358925Z"Higashitani, Akihiro"https://zbmath.org/authors/?q=ai:higashitani.akihiro"Nakajima, Yusuke"https://zbmath.org/authors/?q=ai:nakajima.yusukeLet \(R\) be a \(d\)-dimensional Noetherian ring of characteristic \(p>0\). \(R\) is said to have FFRT (finite \(F\)-representation type) if there is a finite set of isomorphism classes of finitely generated indecomposable modules \(\{M_0, \ldots, M_n\}\) such that for any \(e \in \mathbb{N}\) there are \(c_{i,e} \geq 0\), such that \[R^{1/p^e} \cong M_0^{\oplus c_{0,e}}\oplus M_1^{\oplus c_{1,e}}\oplus \cdots \oplus M_n^{\oplus c_{n,e}}.\] The generalized \(F\)-signature of \(M_i\) with respect to \(R\) is \(s(M_i,R):=\underset{e \rightarrow \infty}\lim\displaystyle\frac{c_{i,e}}{p^{ed}}\).
A Hibi ring is a special type of toric ring defined via a poset. For toric rings \(R\) of characteristic \(p\), it is known that \(R\) has FFRT and the indecomposable modules of \(R\) are the conical divisors of \(R\). The goal of this nice paper is to determine the generalized \(F\)-signatures for the conical divisors of a Hibi ring.
The main theorem determines the generalized \(F\)-signature for a conical divisor of a Segre product of polynomial rings of dimension \(d\), which is a Hibi ring, in terms of the number of elements of the symmetric group on a set of \(d\) elements which certain descent properties. The authors claim that the methods used to prove this result can also be used to determine the generalized \(F\)-signature for a conical divisor for other Hibi rings; their running example of a Hibi ring which is not a Segre product provides an illustration of this claim.Minimal cellular resolutions of the edge ideals of forestshttps://zbmath.org/1472.130122021-11-25T18:46:10.358925Z"Barile, Margherita"https://zbmath.org/authors/?q=ai:barile.margherita"Macchia, Antonio"https://zbmath.org/authors/?q=ai:macchia.antonioThe authors consider edge ideals of forests and give an explicit construction of a cellular resolution that is minimal. Their minimal resolution is based on the Lyubeznik resolution and on the discrete Morse theory. They present a procedure for selecting the admissible symbols which generate the free modules of the resolution. In the end of the paper, they compute the graded Betti numbers and the projective dimension.On perfect co-annihilating-ideal graph of a commutative Artinian ringhttps://zbmath.org/1472.130132021-11-25T18:46:10.358925Z"Mirghadim, S. M. Saadat"https://zbmath.org/authors/?q=ai:mirghadim.s-m-saadat"Nikmehr, M. J."https://zbmath.org/authors/?q=ai:nikmehr.mohammad-javad"Nikandish, R."https://zbmath.org/authors/?q=ai:nikandish.rezaLet \(R\) be a commutative ring with identity \(1\neq 0.\) The co-annihilating-ideal graph of \(R,\) denoted by \(A_R,\) is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of \(R\) and two distinct vertices \(I\) and \(J\) are adjacent whenever \(Ann(I) \cap Ann(J) = (0).\) In this paper, authors characterize all Artinian rings for which both of the graphs \(A_R\) and its complement \(\overline{A}_R\) are chordal. Moreover authors obtained all Artinian commutative rings \(R\) for which \(A_R\) and its complement \(\overline{A}_R\) are perfect. In particular, it is proved that, for a commutative Artinian ring \(R,\) the graph \(\overline{A}_R\) is a perfect graph if and only if \(|Max(R)|\leq 4.\)On distance Laplacian spectrum of zero divisor graphs of the ring \(\mathbb{Z}_n \)https://zbmath.org/1472.130142021-11-25T18:46:10.358925Z"Pirzada, S."https://zbmath.org/authors/?q=ai:pirzada.shariefuddin"Rather, B. A."https://zbmath.org/authors/?q=ai:rather.bilal-a"Chishti, T. A."https://zbmath.org/authors/?q=ai:chishti.tariq-aLet \(R\) be a finite commutative ring with identity \(1\neq 0.\) The zero-divisor graph \(\Gamma(R)\) is the simple undirected graph with the set of all non-zero zero-divisors \(Z^*(R)\) as the vertex set and two distinct vertices \(x\) and \(y\) are adjacent if \(x.y=0\) in \(R.\) Zero-divisor graphs of commutative rings are well studied in several aspects of graph theory for the past three decades. For a graph \(G,\) let \(A(G)\) be the adjacency matrix \(G.\) Let \(Deg(G)\) be the diagonal matrix of vertex degrees of vertices in \(G.\) The matrices \(L(G) = Deg(G) - A(G)\) and \(Q(G) = Deg(G) + A(G)\) are respectively the Laplacian and the signless Laplacian matrices and these matrices are real symmetric and positive semi-definite. It is assumed that \(0=\lambda_n\leq \lambda_{n-1}\leq\cdots\leq \lambda_1\) are the Laplacian eigenvalues of \(L(G).\) In this paper, authors obtained the distance Laplacian spectrum of the zero divisor graphs \(\Gamma(\mathbb{Z}_n\)) for different values of \(n\in \{pq, p^2q, (pq)^2, p^z~\text{for some}~ z\geq 2\}\) where \(p\) and \(q(p < q)\) are distinct primes. Further it is proved that the zero-divisor graph \(\Gamma(\mathbb{Z}_n)\) is distance Laplacian integral if and only if \(n\) is prime power or product of two distinct primes.Algebraic properties of edge ideals of some vertex-weighted oriented cyclic graphshttps://zbmath.org/1472.130192021-11-25T18:46:10.358925Z"Wang, Hong"https://zbmath.org/authors/?q=ai:wang.hong.7|wang.hong|wang.hong.2|wang.hong.5|wang.hong.1|wang.hong.3|wang.hong.4"Zhu, Guangjun"https://zbmath.org/authors/?q=ai:zhu.guangjun"Xu, Li"https://zbmath.org/authors/?q=ai:xu.li"Zhang, Jiaqi"https://zbmath.org/authors/?q=ai:zhang.jiaqiLet \(\mathcal{D}=(V(\mathcal{D}),E(\mathcal{D}),\omega)\) be a vertex-weighted oriented graph, i.e., a digraph with no loop, multiple or bidirected edge, which is equipped with a weight function \(\omega\) defined from \(V(\mathcal{D})=\{1,\ldots,n\}\) to the positive integers. Let \(\mathbb{K}\) be a field and \(S=\mathbb{K}[x_1,\ldots,x_n]\) the polynomial ring over \(\mathbb{K}\). Then, the edge ideal associated with \(\mathcal{D}\) is the monomial ideal in \(S\) generated by all the monomials of the form \(x_{i}x_{j}^{\omega(j)}\), where \((i,j)\in E(\mathcal{D})\) is a directed edge of \(\mathcal{D}\). \par This class of ideals was introduced in [\textit{C. Paulsen} et al., J. Algebra Appl. 12, No. 5, 24p (2013; Zbl 1266.05048)], as a generalization of the well-studied edge ideal of simple graphs initiated by \textit{A. Simis} et al. [J. Algebra. 167, No. 2, 389--416 (1994; Zbl 0816.13003)]. \par The authors of the paper under review aim to study some homological invariants of \(I(\mathcal{D})\). More precisely, by using methods of Betti splitting and polarization of monomial ideals, they present some characteristic-free formulas for the regularity and the projective dimension of \(I(\mathcal{D})\) for some special classes of vertex-weighted oriented graphs \(\mathcal{D}\).On the minimal free resolution of symbolic powers of cover ideals of graphshttps://zbmath.org/1472.130232021-11-25T18:46:10.358925Z"Fakhari, S. A. Seyed"https://zbmath.org/authors/?q=ai:seyed-fakhari.seyed-aminAuthor's abstract: For any graph \(G\), assume that \(J(G)\) is the cover ideal of \(G\). Let \(J(G)^{(k)}\) denote the \(k\)th symbolic power of \(J(G)\). We characterize all graphs \(G\) with the property that \(J(G)^{(k)}\) has a linear resolution for some (equivalently, for all) integer \(k\geq 2\). Moreover, it is shown that for any graph \(G\), the sequence
\((reg(J(G)^{(k)})_{k=1}^{\infty}\) is nondecreasing. Furthermore, we compute the largest degree of minimal generators of \(J(G)^{(k)}\) when \(G\) is either an unmixed of a claw-free graph.Explicit Pieri inclusionshttps://zbmath.org/1472.130252021-11-25T18:46:10.358925Z"Hunziker, Markus"https://zbmath.org/authors/?q=ai:hunziker.markus"Miller, John A."https://zbmath.org/authors/?q=ai:miller.john-a"Sepanski, Mark"https://zbmath.org/authors/?q=ai:sepanski.mark-rSummary: By the Pieri rule, the tensor product of an exterior power and a finite-dimensional irreducible representation of a general linear group has a multiplicity-free decomposition. The embeddings of the constituents are called Pieri inclusions and were first studied by \textit{J. Weyman} [Schur functors and resolutions of minors. Brandeis University, Waltham USA (PhD Thesis) (1980)] and described explicitly by \textit{P. J. Olver} [``Differential hyperforms I'', University of Minnesota, Mathematics Report, 82--101 (1980), \url{https://www-users.cse.umn.edu/~olver/a_/hyper.pdf}]. More recently, these maps have appeared in the work of \textit{D. Eisenbud} et al. [Ann. Inst. Fourier 61, No. 3, 905--926 (2011; Zbl 1239.13023)] and of \textit{S. V. Sam} [J. Softw. Algebra Geom. 1, 5--10 (2009; Zbl 1311.13039)] and Weyman to compute pure free resolutions for classical groups.
In this paper, we give a new closed form, non-recursive description of Pieri inclusions. For partitions with a bounded number of distinct parts, the resulting algorithm has polynomial time complexity whereas the previously known algorithm has exponential time complexity.Virtual resolutions of monomial ideals on toric varietieshttps://zbmath.org/1472.130272021-11-25T18:46:10.358925Z"Yang, Jay"https://zbmath.org/authors/?q=ai:yang.jayGiven a smooth toric variety \(X=X(\Sigma)\) and a \(\mathrm{Pic}(X)\)-graded module \(M\), then a free complex \(F\) of graded \(\mathrm{k}[\Sigma]\)-modules is a virtual resolution of \(M\) if the corresponding complex \(\widetilde{F}\) of vector bundles on \(X\) is a resolution of \(\widetilde{M}\). In the paper under review, the author uses cellular resolutions of monomial ideals to prove an analog of Hilbert's syzygy theorem for virtual resolutions of monomial ideals on smooth toric varieties.Gotzmann monomials in four variableshttps://zbmath.org/1472.130312021-11-25T18:46:10.358925Z"Bonanzinga, Vittoria"https://zbmath.org/authors/?q=ai:bonanzinga.vittoria"Eliahou, Shalom"https://zbmath.org/authors/?q=ai:eliahou.shalomA monomial ideal, \(J\subset K[x_1,\dots,x_n]\), is called Borel-stable if, for every monomial \(v\in J\) and for every index \(j\), \[ x_j\mid v \implies x_i({v}/{x_j}) \in J,\text{ for all }1\leq i \leq j. \] If \(u\) is a monomial, the authors denote by \(B(u)\) the smallest Borel stable monomial ideal that contains \(u\) and call \(u\) a Gotzmann monomial if the Hilbert function of \(B(u)\) attains Macaulay's lower bound for a certain degree and, hence, on [\textit{G. Gotzmann}, Math. Z. 158, 61--70 (1978; Zbl 0352.13009) ]. Their goal is to classify all Gotzmann monomials in \textit{four} variables. All monomials in one or two variables are Gotzmann monomials; the case of three variables is the contained in Proposition 8 of [\textit{S. Murai}, Ill. J. Math. 51, No. 3, 843--852 (2007; Zbl 1155.13012) ].
The main result of the article (Theorem 7.7) states that a monomial in four variables \(x_1^ax_2^bx_3^cx_4^t\) is a Gotzmann monomial if and only if \[ t\geq \binom{\binom{b}{2}}{2} + \frac{b+4}{3}\binom{b}{2} + (b+1)\binom{c+1}{2}+\binom{c+1}{3}-c. \]On quasi-equigenerated and Freiman cover ideals of graphshttps://zbmath.org/1472.130362021-11-25T18:46:10.358925Z"Drabkin, Benjamin"https://zbmath.org/authors/?q=ai:drabkin.benjamin"Guerrieri, Lorenzo"https://zbmath.org/authors/?q=ai:guerrieri.lorenzoThe present paper addresses the problem of computing the number of generators of the powers of a homogeneous ideal \(I\) in a polynomial ring \(R=\mathbf{k}[x_1,\dots,x_n]\). If all the generators of \(I\) have the same degree with respect to a standard (resp. non-standard) \(\mathbf{N}\)-grading then we say that \(I\) is equigenerated (resp. quasi-equigenerated). \textit{J. Herzog} et al. show in [Int. J. Algebra Comput. 29, No. 5, 827--847 (2019; Zbl 1423.13105)] that the number of generators of \(I^2\) is bigger than or equal to \(l(I)\mu(I)-\binom{l(I)}{2}\), where \(\mu(I)\) is the number of generators of \(I\) and \(l(I)\) is the analytic spread of \(I\). Other bounds for any powers of \(I\) are given by \textit{J. Herzog} and \textit{G. Zhu} [Commun. Algebra 47, No. 1, 407--423 (2019; Zbl 1410.13007)]. These bounds are consequence of a theorem by Freiman in additive number theory and therefore in the case that the bound for \(I^2\) is met, \(I\) is called a Freiman ideal, i.e. \(I\) is an equigenerated monomial ideal such that \(\mu(I^2)=l(I)\mu(I)-\binom{l(I)}{2}\).
The authors study Freiman ideals among cover ideals of graphs, and show that in general, graphs that are close enough to be complete have Freiman cover ideal. He also characterize Freiman cover ideals among pairs of complete graphs sharing a vertex, circulant graphs and whiskered graphs.On the depth and Stanley depth of the integral closure of powers of monomial idealshttps://zbmath.org/1472.130372021-11-25T18:46:10.358925Z"Seyed Fakhari, S. A."https://zbmath.org/authors/?q=ai:seyed-fakhari.seyed-aminLet $M$ be a finitely generated $\mathbb{Z}^n$-graded $S$-module, where $S=\mathbb{K}[x_1,\dots,x_n]$ and $\mathbb{K}$ is a field. For any homogenuous element $u\in M$, the $\mathbb{K}$-subspace $u\mathbb{K}[Z]$ is called a \textit{Stanley space} of dimension $|Z|$, if it is a free $\mathbb{K}[Z]$-module. A decomposition $\mathcal{D}$ of $M$ as a finite direct sum of Stanley spaces is called a \textit{Stanley decomposition} of $M$. The minimum dimension of a Stanley space in $\mathcal{D}$ is called the Stanley depth of $\mathcal{D}$ and is denoted by $\mathrm{sdepth}(\mathcal{D})$. The \textit{Stanley depth} of $M$ is defined as: $$\mathrm{sdepth}(M):=\max \{\mathrm{sdepth}(\mathcal{D}) |\mathcal{D}\text{ is a Stanley decomposition of }M\}.$$ This depth is defined to be $\infty$ for the zero module. The inequality $\mathrm{depth}(M)\leq\mathrm{sdepth}(M)$ is known as \textit{Stanley's inequality}. Let $G$ be a graph with edge ideal $I(G)$, $k\gg 0$ and $p$ be the number of its bipartite connected components. In the present paper it is shown that:
\begin{enumerate}
\item[(i)] $\mathrm{sdepth}(S/I(G))\geq p$; In particular, $S/\overline{I(G)^k}$ satisfies Stanley inequality.
\item[(ii)] $\mathrm{sdepth}(\overline{I(G)^k})\geq p+1$. In particular, $\overline{I(G)^k}$ satisfies Stanley inequality.
\item[(iii)] If $G$ is a connected bipartite graph whose girth is $g$ and $k\leq g/2+1$, then $\mathrm{sdepth}(\overline{I(G)^k})\geq 2$.
\end{enumerate}
Let $I\subset S$ be a nonzero monomial ideal with the analytic spread $l(I)$. Among other results, it is proved that the sequence $\{\mathrm{depth}(I^k/I^{k+1})\}^\infty_{k=0}$ converges to $n-l(I)$. Finally, it is shown that if $I$ is an integrally closed monomial ideal, then $\mathrm{depth}(S/I^m)\leq\mathrm{depth}(S/I)$ and $\mathrm{Ass}(S/I)\subseteq\mathrm{Ass}(S/I^m)$, for every integer $m\geq 1$.Projective dimension and regularity of edge ideals of some weighted oriented graphshttps://zbmath.org/1472.130382021-11-25T18:46:10.358925Z"Zhu, Guangjun"https://zbmath.org/authors/?q=ai:zhu.guangjun"Xu, Li"https://zbmath.org/authors/?q=ai:xu.li"Wang, Hong"https://zbmath.org/authors/?q=ai:wang.hong.7|wang.hong|wang.hong.2"Tang, Zhongming"https://zbmath.org/authors/?q=ai:tang.zhongmingA vertex-weighted oriented graph is a triplet \(D=(V(D),E(D),w)\), where \(V(D)=\{x_1,\dots,x_n\}\) is the vertex set, \(E(D)\) is the edge set and \(w:V(D)\rightarrow\mathbb{N}\) is a weight function. The edge ideal of \(D\), denoted by \(I(D)\), is the ideal of the polynomial ring \(S=k[x_1,x_2,\dots,x_n]\) which is generated by monomials \(\{x_ix_j^{w_j}|x_ix_j\in E(D)\}\). In the present article, the authors consider edge ideals of vertex-weighted oriented graphs and they provide some formulas for the projective dimension and the regularity of edge ideals associated to special vertex weighted rooted graphs. In fact, it is shown that if \(D\) satisfies one of the conditions
\begin{enumerate}
\item[(i)] \(D\) is a weighted oriented star graph;
\item[(ii)] \(D\) is a weighted oriented rooted forest such that \(w(x)\geq 2\) if \(d(x)\neq 1\);
\item[(iii)] \(D\) is a weighted oriented rooted cycle such that \(w(x)\geq 2\) for every vertex \(x\in V(D)\);
\end{enumerate}
then the projective dimension and regularity of the edge ideal is calculated by the following formulas: \(\mathrm{pd}(I(D)=|E(D)|-1;\ \ \mathrm{reg}(I(D))=\sum_{x\in V(D)}w(x)-|E(D)|+1\).Building maximal green sequences via component preserving mutationshttps://zbmath.org/1472.130402021-11-25T18:46:10.358925Z"Bucher, Eric"https://zbmath.org/authors/?q=ai:bucher.eric"Machacek, John"https://zbmath.org/authors/?q=ai:machacek.john-m"Runburg, Evan"https://zbmath.org/authors/?q=ai:runburg.evan"Yeck, Abe"https://zbmath.org/authors/?q=ai:yeck.abe"Zwede, Ethan"https://zbmath.org/authors/?q=ai:zwede.ethanSummary: We introduce a new method for producing both maximal green and reddening sequences of quivers. The method, called component preserving mutations, generalizes the notion of direct sums of quivers and can be used as a tool to both recover known reddening sequences as well as find reddening sequences that were previously unknown. We use the method to produce and recover maximal green sequences for many bipartite recurrent quivers that show up in the study of periodicity of \(T\)-systems and \(Y\)-systems. Additionally, we show how our method relates to the dominance phenomenon recently considered by Reading. Given a maximal green sequence produced by our method, this relation to dominance gives a maximal green sequence for infinitely many other quivers. Other applications of this new methodology are explored including computing of quantum dilogarithm identities and determining minimal length maximal green sequences.Secant planes of a general curve via degenerationshttps://zbmath.org/1472.140152021-11-25T18:46:10.358925Z"Cotterill, Ethan"https://zbmath.org/authors/?q=ai:cotterill.ethan"He, Xiang"https://zbmath.org/authors/?q=ai:he.xiang"Zhang, Naizhen"https://zbmath.org/authors/?q=ai:zhang.naizhenIn this paper, Osserman's theory of limit linear series, as a generalization of that of Eisenbud-Harris for compact type curves, is reviewed. Built on that, two constructions of a moduli space of inclusions of limit linear series are provided. Both spaces agree set-theoretically, but the second one, described as an intersection of determinantal loci of vector bundles helps proving a smoothing theorem for inclusion of limit linear series in special cases. Based on this, explicit formulas for counting inclusion of limit linear series, and equivalently for the number of secant planes to the image of a curve under a map are calculated. The paper ends with an example showing that the moduli of included limit linear series may have a component of unexpectedly large dimension.
Reviewer's remark: In Definition 4.1, the divisor \(D\) seems undefined -- perhaps \(Y\) may have been intended?Hurwitz theory of elliptic orbifolds. Ihttps://zbmath.org/1472.140342021-11-25T18:46:10.358925Z"Engel, Philip"https://zbmath.org/authors/?q=ai:engel.philip-miltonSummary: An \textit{elliptic orbifold} is the quotient of an elliptic curve by a finite group. In [Invent. Math. 145, No. 1, 59--103 (2001; Zbl 1019.32014)], \textit{A. Eskin} and \textit{A. Okounkov} proved that generating functions for the number of branched covers of an elliptic curve with specified ramification are quasimodular forms for \(\operatorname{SL}2_(\mathbb{Z})\). In [Prog. Math. 253, 1--25 (2006; Zbl 1136.14039)], they generalized this theorem to branched covers of the quotient of an elliptic curve by \(\pm 1\), proving quasimodularity for \(\Gamma_0(2)\). We generalize their work to the quotient of an elliptic curve by \(\langle\zeta N\rangle\) for \(N=3, 4, 6\), proving quasimodularity for \(\Gamma(N)\), and extend their work in the case \(N=2\).
It follows that certain generating functions of hexagon, square and triangle tilings of compact surfaces are quasimodular forms. These tilings enumerate lattice points in moduli spaces of flat surfaces. We analyze the asymptotics as the number of tiles goes to infinity, providing an algorithm to compute the Masur-Veech volumes of strata of cubic, quartic, and sextic differentials. We conclude a generalization of the Kontsevich-Zorich conjecture: these volumes are polynomial in \(\pi\).The determinant inner product and the Heisenberg product of \(\mathrm{Sym}(2)\)https://zbmath.org/1472.150052021-11-25T18:46:10.358925Z"Crasmareanu, Mircea"https://zbmath.org/authors/?q=ai:crasmareanu.mirceaLet \(A\) be a finite subset of a field and denote by \(D^{n(A)}\) the set of all possible determinants of matrices with entries in \(A\). In this paper, the following problem, typical in additive combinatorics, is investigated: how big is the image set of the determinant function compared to the set \(A\)? Interesting results are obtained, that remain also true also for the set of permanents.Rank one perturbations of matrix pencilshttps://zbmath.org/1472.150182021-11-25T18:46:10.358925Z"Dodig, Marija"https://zbmath.org/authors/?q=ai:dodig.marija"Stošić, Marko"https://zbmath.org/authors/?q=ai:stosic.markoUsing some of their earlier results (see for example [the first author, Linear Algebra Appl. 438, No. 8, 3155--3173 (2013; Zbl 1269.15029)]), the authors completely resolve the open problem of describing all possible Kronecker invariants of an arbitrary matrix pencil under rank one perturbations. Their solution is explicit and constructive, and is valid for arbitrary pencils. They combine results on one-row matrix pencil completions with combinatorial results on double generalized majorization, and develop some new techniques.Almost gentle algebras and their trivial extensionshttps://zbmath.org/1472.160132021-11-25T18:46:10.358925Z"Green, Edward L."https://zbmath.org/authors/?q=ai:green.edward-lee"Schroll, Sibylle"https://zbmath.org/authors/?q=ai:schroll.sibylleSummary: In this paper we define almost gentle algebras, which are monomial special multiserial algebras generalizing gentle algebras. We show that the trivial extensions of an almost gentle algebra by its minimal injective co-generator is a symmetric special multiserial algebra and hence a Brauer configuration algebra. Conversely, we show that any almost gentle algebra is an admissible cut of a unique Brauer configuration algebra and, as a consequence, we obtain that every Brauer configuration algebra with multiplicity function identically one is the trivial extension of an almost gentle algebra. We show that a hypergraph is associated with every almost gentle algebra \(A\), and that this hypergraph induces the Brauer configuration of the trivial extension of \(A\). Among other things, this gives a combinatorial criterion to decide when two almost gentle algebras have isomorphic trivial extensions.Rota-Baxter operators on a sum of fieldshttps://zbmath.org/1472.160442021-11-25T18:46:10.358925Z"Gubarev, V."https://zbmath.org/authors/?q=ai:gubarev.vyacheslav-f|gubarev.vsevolod-yurevich|gubarev.v-vOn the action of the Koszul map on the enveloping algebra of the general linear Lie algebrahttps://zbmath.org/1472.170252021-11-25T18:46:10.358925Z"Brini, Andrea"https://zbmath.org/authors/?q=ai:brini.andrea"Teolis, Antonio"https://zbmath.org/authors/?q=ai:teolis.antonio-g-bLet \(gl_n\) be the general linear Lie algebra (over the complex numbers \(\mathbf{C}\)), and let \(U(gl_n)\) be its universal enveloping algebra. Denote by \(\mathbf{C}(M_{n,n})\) the commutative polynomial algebra in the \(n^2\) variables \((i\mid j)\), \(1\le i,j\le n\) where \((i\mid j)\) are the corresponding entries of the generic \(n\times n\) matrix \(M_{n,n}\). Clearly the latter commutative algebra is isomorphic to Sym\((gl_n)\), and a natural isomorphism is given by the correspondence \((i\mid j)\mapsto e_{ij}\) where \(e_{ij}\) is the matrix with 1 at position \((i,j)\), and zeros elsewhere. The main result of the paper under review is the construction of an equivariant isomorphism \(K\colon U(gl_n)\to \mathbf{C}(M_{n,n})\). Furthermore the isomorphism \(K\) sends the Capelli bitableaux from \(U(gl_n)\), denoted by \([S\mid T]\) to the corresponding determinantal bitableaux \((S\mid T)\in \mathbf{C}(M_{n,n})\). Additionally, \(K\) sends the \(*\)-bitableaux in \(U(gl_n)\) to the corresponding permanental \(*\)-bitableaux in \(\mathbf{C}(M_{n,n})\). \par An interesting corollary of the construction of the isomorphism \(K\) isn deduced. Since \(gl_n\) acts on \(U(gl_n)\) by the adjoint representation, and also on \(\mathbf{C}(M_{n,n})\) (by polarization), one gets that \(Z\), the centre of \(U(gl_n)\) is in fact the algebra of invariants. Hence it maps via \(K\) to \(\mathbf{C}(M_{n,n}){(ad \, gl_n})\).Young-Capelli bitableaux, Capelli immanants in \(\mathbb{U}(\mathrm{gl}(n))\) and the Okounkov quantum immanantshttps://zbmath.org/1472.170262021-11-25T18:46:10.358925Z"Brini, A."https://zbmath.org/authors/?q=ai:brini.andrea"Teolis, A."https://zbmath.org/authors/?q=ai:teolis.antonio-g-bLet \(gl_n\) be the general linear Lie algebra over the complex numbers \(\mathbb{C}\). Its universal enveloping algebra \(U=U(gl_n)\) admits a basis consisting of the standard Capelli bitableaux and another basis consisting of the standard Young-Capelli bitableaux. The authors of the paper under review introduce another spanning set of \(U\), the so-called Capelli immanants. A similar notion already appeared in the study of the centre of \(U\), see fro example [\textit{A. Okounkov}, Transform. Groups 1, No. 1--2, 99--126 (1996; Zbl 0864.17014)]. Let \(\mathbb{C}[M_{n,n}]\) be the polynomial algebra in the \(n^2\) variables \(x_{ij}\), \(1\le i,j\le n\); one can view it as the polynomial algebra for one generic \(n\times n\) matrix. The latter algebra is, in fact, the symmetric algebra for \(gl_n\), and is closely related to \(U\). There is a natural isomorphism \(K\) of \(\mathbb{C}[M_{n,n}]\) and Sym\((gl_n\), called the Bitableax correspondence isomorphism or Koszul map (BCK for short). The authors transfer the action of the Capelli immanants to \(\mathbb{C}[M_{n,n}]\). They consider the inverse isomorphism \(K^{-1}\) and show that the images of the YoungCapelli bitableaux are exactly the standard right symmetrized bitableaux. They prove that the Capelli immanants are combinations of standard Young-Capelli bitableaux of the same shape. \par It turns out that the quantum immanants introduced by Okounkov in the above cited paper are simple combinations of diagonal Capelli immanants; the corresponding coefficients are given explicitly. A canonical presentations of quantum immanants in terms of double Young-Capelli bitableaux is given as well.FFLV-type monomial bases for type Bhttps://zbmath.org/1472.170332021-11-25T18:46:10.358925Z"Makhlin, Igor"https://zbmath.org/authors/?q=ai:makhlin.igorSummary: We present a combinatorial monomial basis (or, more precisely, a family of monomial bases) in every finite-dimensional irreducible $\mathfrak{so}_{2n+1}$-module. These bases are in many ways similar to the FFLV bases for types $A$ and $C$. They are also defined combinatorially via sums over Dyck paths in certain triangular grids. Our sums, however, involve weights depending on the length of the corresponding root. Accordingly, our bases also induce bases in certain degenerations of the modules but these degenerations are obtained not from the filtration by PBW degree but by a weighted version thereof.A relationship between Gelfand-Tsetlin bases and Chari-Loktev bases for irreducible finite dimensional representations of special linear Lie algebrashttps://zbmath.org/1472.170362021-11-25T18:46:10.358925Z"Raghavan, K. N."https://zbmath.org/authors/?q=ai:raghavan.komaranapuram-n"Ravinder, B."https://zbmath.org/authors/?q=ai:ravinder.bhimarthi"Viswanath, Sankaran"https://zbmath.org/authors/?q=ai:viswanath.sankaranSummary: We consider two bases for an arbitrary finite dimensional irreducible representation of a complex special linear Lie algebra: the classical Gelfand-Tsetlin basis and the relatively new Chari-Loktev basis. Both are parametrized by the set of (integral Gelfand-Tsetlin) patterns with a fixed bounding sequence determined by the highest weight of the representation. We define the \textit{row-wise dominance} partial order on this set of patterns, and prove that the transition matrix between the two bases is triangular with respect to this partial order. We write down explicit expressions for the diagonal elements of the transition matrix.
For the entire collection see [Zbl 1428.13001].Endowing evolution algebras with properties of discrete structureshttps://zbmath.org/1472.170992021-11-25T18:46:10.358925Z"González-López, Rafael"https://zbmath.org/authors/?q=ai:gonzalez-lopez.rafael"Núñez, Juan"https://zbmath.org/authors/?q=ai:nunez-valdes.juanIn this paper, the authors delve into the basic properties characterizing the directed graph that is uniquely associated to an evolution algebra, and which was introduced in [\textit{J. P. Tian}, Evolution algebras and their applications. Berlin: Springer (2008; Zbl 1136.17001)]. These properties are conveniently translated to algebraic concepts and results concerning the evolution algebra under consideration. In particular, the authors focus on the adjacency of graphs, whose immediate translation into algebraic language enables them to introduce the concepts of adjacency, walk, trail, circuit, path and cycle of an evolution algebra. Also the notions of strongly and weakly connected evolution algebras are introduced as the algebraic equivalences of the same concepts in graph theory. It enables the authors to introduce the notions of distance, girth, circumference, eccentricity, center, radio, diameter and geodesic of an evolution algebra, together with the concepts of Eulerian and Hamiltonian evolution algebras. Some basic results on these topics are then described. The relationship among all of these notions and their analogous in graph theory are visually illustrated throughout the paper.On automorphism groups of deleted wreath productshttps://zbmath.org/1472.200012021-11-25T18:46:10.358925Z"Dobson, Ted"https://zbmath.org/authors/?q=ai:dobson.ted"Miklavič, Štefko"https://zbmath.org/authors/?q=ai:miklavic.stefko"Šparl, Primož"https://zbmath.org/authors/?q=ai:sparl.primozSummary: Let \(\Gamma_1\) and \(\Gamma_2\) be digraphs. The \textit{deleted wreath product} of \(\Gamma_1\) \textit{and} \(\Gamma_2\), denoted \(\Gamma_1\wr_d\Gamma_2\), is the digraph with vertex set \(V(\Gamma_1) \times V(\Gamma_2)\) and arc set \(\{((x_1,y_1),(x_2,y_2)):(x_1,x_2)\in A(\Gamma_1) \text{ and } y_1 \neq y_2 \text{ or } x_1 = x_2 \text{ and } (y_1,y_2)\in A(\Gamma_2)\}\). We study the automorphism group of \(\Gamma_1 \wr_d \Gamma_2\), which always contains a natural subgroup isomorphic to \({\text{Aut}}(\Gamma_1)\times{\text{Aut}}(\Gamma_2)\). In particular, we focus on the characterization of digraph pairs \(\Gamma_1, \Gamma_2\) such that \(\Gamma_1 \wr_d \Gamma_2\) admits automorphisms not contained in this natural subgroup of \(\text{Aut}(\Gamma_1 \wr_d \Gamma_2)\). We provide methods to construct such pairs of digraphs, and also give several sufficient conditions under which no such additional automorphisms exist. As a corollary of our results, we provide a method for constructing new half-arc-transitive graphs from known ones using deleted wreath products.On the one dimensional representations of Ariki-Koike algebras at roots of unityhttps://zbmath.org/1472.200052021-11-25T18:46:10.358925Z"Jacon, Nicolas"https://zbmath.org/authors/?q=ai:jacon.nicolasSummary: We study the natural labeling of the one dimensional representations for Ariki-Koike algebras at roots of unity. For Hecke algebras of types \(A\) and \(B\), some of these representations can be identified with the socle of the Steinberg representation of a finite reductive group. We here give closed formulas for them. This uses, in particular, several results concerning crystal isomorphisms and the Mullineux involution.Determinants of representations of Coxeter groupshttps://zbmath.org/1472.200082021-11-25T18:46:10.358925Z"Ghosh, Debarun"https://zbmath.org/authors/?q=ai:ghosh.debarun"Spallone, Steven"https://zbmath.org/authors/?q=ai:spallone.stevenSummary: In [\textit{A. Ayyer} et al., J. Comb. Theory, Ser. A 150, 208--232 (2017; Zbl 1362.05012)], the authors characterize the partitions of \(n\) whose corresponding representations of \(S_n\) have nontrivial determinant. The present paper extends this work to all irreducible finite Coxeter groups \(W\). Namely, given a nontrivial multiplicative character \(\omega \) of \(W\), we give a closed formula for the number of irreducible representations of \(W\) with determinant \(\omega \). For Coxeter groups of type \(B_n\) and \(D_n\), this is accomplished by characterizing the bipartitions associated to such representations.Mathieu groups and its degree prime-power graphshttps://zbmath.org/1472.200132021-11-25T18:46:10.358925Z"Qin, Chao"https://zbmath.org/authors/?q=ai:qin.chao.1"Yan, Yanxiong"https://zbmath.org/authors/?q=ai:yan.yanxiong"Shum, Karping"https://zbmath.org/authors/?q=ai:shum.kar-ping"Chen, Guiyun"https://zbmath.org/authors/?q=ai:chen.guiyunSummary: Let \(\mathrm{cd}(G)\) be the set of irreducible complex character degrees of a finite group \(G\). The degree graph of \(G\) related to \(\mathrm{cd}(G)\) was defined. It was proved many finite simple groups (but not all Mathieu groups) are uniquely determined by their orders and degree graphs. We hope to define a new graph related to \(\mathrm{cd}(G)\) such that more simple groups can be uniquely determined by their orders and this newly defined graphs. Here a degree prime-power graph is defined and it is proved that all Mathieu groups can be determined uniquely by their orders and degree prime-power graphs.Wedge-direct sums of table algebras and applications to association schemes. Ihttps://zbmath.org/1472.200152021-11-25T18:46:10.358925Z"Xu, Bangteng"https://zbmath.org/authors/?q=ai:xu.bangtengSummary: In the remarkable paper [J. Algebra 396, 220--271 (2013; Zbl 1333.16021)] on fusion rings, \textit{H. I. Blau} introduced a very important operation on two table algebras (the Blau-construction). It is very difficult to describe the structure of table algebras obtained by recursively applying the Blau-construction. In this paper, we define the wedge-direct sum for a sequence of table algebras. Using the wedge-direct sum of a sequence of table algebras, we are able to not only give a clear description of the structure of table algebras obtained by recursively applying the Blau-construction but also obtain the characterization and classification of a class of \(p\)-table algebras. The structure of the Bose-Mesner algebras of the wreath products and wedge products of association schemes can also be clearly described in terms of the wedge-direct sum. As an application, we get some new and known results about the wreath products and wedge products of association schemes.Wedge-direct sums of table algebras and applications to association schemes. IIhttps://zbmath.org/1472.200162021-11-25T18:46:10.358925Z"Xu, Bangteng"https://zbmath.org/authors/?q=ai:xu.bangtengSummary: Extensions of association schemes and table algebras have been studied in many papers in the last two decades. A remarkable operation (the Blau-construction) on two table algebras in [J. Algebra 396, 220--271 (2013; Zbl 1333.16021] provides an important method to construct the extension of table algebras. The wedge-direct sum of a sequence of table algebras introduced in Part I [the author, Commun. Algebra 47, No. 8, 3309--3328 (2019; Zbl 1472.20015)] is a useful tool for the study of structures and characterizations of table algebras. Using the wedge-direct sum of a sequence of table algebras, in Part I we gave a new perspective on the Blau-construction, and obtained a clear description of the structure of table algebras constructed by recursively applying the Blau-construction. \(p\)-table algebras \((p\)-schemes) are an important class of table algebras (association schemes). In this part we will give the characterization and classification of a class of \(p\)-table algebras, by showing that they are isomorphic to the wedge-direct sum of a sequence of group algebras of finite \(p\)-groups. In order to determine whether two distinct sequences of table algebras yield the isomorphic wedge-direct sums or not, we will need to discuss the redundancy of the wedge-direct sum, and prove a Krull-Schmidt type theorem. Applications to association schemes will also be discussed.Maximal cocliques in \(\mathrm{PSL}_2(q)\)https://zbmath.org/1472.200252021-11-25T18:46:10.358925Z"Saunders, Jack"https://zbmath.org/authors/?q=ai:saunders.jackSummary: The generating graph of a finite group is a structure which can be used to encode certain information about the group. It was introduced by \textit{M. W. Liebeck} and \textit{A. Shalev} [J. Algebra 184, No. 1, 31--57 (1996; Zbl 0870.20014)] and has been further investigated by \textit{A. Lucchini} et al. [J. Aust. Math. Soc. 103, No. 1, 91--103 (2017; Zbl 1425.20010)], and others. We investigate maximal cocliques (totally disconnected induced subgraphs of the generating graph) in \(\mathrm{PSL}_2(q)\) for \(q\) a prime power and provide a classification of the ``large'' cocliques when \(q\) is prime. We then provide an interesting geometric example which contradicts this result when \(q\) is not prime and illustrate why the methods used for the prime case do not immediately extend to the prime-power case with the same result.The solubility graph associated with a finite grouphttps://zbmath.org/1472.200262021-11-25T18:46:10.358925Z"Akbari, B."https://zbmath.org/authors/?q=ai:akbari.banafsheh|akbari.behzad"Lewis, Mark L."https://zbmath.org/authors/?q=ai:lewis.mark-l"Mirzajani, J."https://zbmath.org/authors/?q=ai:mirzajani.javad"Moghaddamfar, A. R."https://zbmath.org/authors/?q=ai:moghaddamfar.ali-rezaThe solubility graph \(\Gamma_{\mathcal S}(G)\) associated with a finite group \(G\) is a simple graph whose vertices are the elements of \(G,\) and there is an edge between two distinct elements \(x\) and \(y\) if and only if \(\langle x, y \rangle\) is a soluble subgroup of \(G.\) The authors prove that \(\Gamma_{\mathcal S}(G)\) is connected, with diameter at most 11 (but they believe that the correct bound is much smaller than 11). Most of the paper, is focused on the sets of neighbors of vertices in \(\Gamma_{\mathcal S}(G).\) In particular, setting \(\mathrm{Sol}_G(x)=\{g\in G\mid \langle x, g\rangle \text { is soluble}\},\) they investigate how restrictions on the structure of this set influence the structure of \(G.\) For example, they prove that \(G\) is soluble if and only if \(\mathrm{Sol}_G(x)\) is a subgroup of \(G\) for all \(x \in G.\)A note on abelian partitionable groupshttps://zbmath.org/1472.200522021-11-25T18:46:10.358925Z"Foguel, Tuval"https://zbmath.org/authors/?q=ai:foguel.tuval-s"Hiller, Josh"https://zbmath.org/authors/?q=ai:hiller.joshSummary: A group \(G\) has an abelian partition if it has a set theoretic partition into disjoint commutative subsets \(A_0, A_1,\ldots, A_n\) where \(\vert A_i\vert > 1\) for all \(i\) and the identity is in \(A_0\), such a group is called an abelian partitionable group (AP-group). The problem of classifying AP-groups was recently taken up by \textit{A. Mahmoudifar} et al. [Int. Electron. J. Algebra 25, 224--231 (2019; Zbl 1425.20016)] who classified all groups with \(n = 2, 3\). The motivation for this problem can be found in graph theory where partitions of graphs into induced complete subgraphs is of great importance. We achieve a partial classification of AP-groups and introduce a family of groups with no abelian partition.On the existence of non-negative bases in subgroups of free groups of Schreier varietieshttps://zbmath.org/1472.200562021-11-25T18:46:10.358925Z"Kruglov, I. A."https://zbmath.org/authors/?q=ai:kruglov.igor-aleksandrovich"Cherednik, I. V."https://zbmath.org/authors/?q=ai:cherednik.ivan-vSummary: We show that a subgroup \(H\) of a free group \(F(X)\) has a non-negative (with respect to \(X\)) basis if and only if \(H\) is generated by the set of all its non-negative (with respect to \(X\)) elements. A similar result is proved for subgroups of free Abelian groups.Varieties of profinite graphshttps://zbmath.org/1472.200602021-11-25T18:46:10.358925Z"Acharyya, Amrita"https://zbmath.org/authors/?q=ai:acharyya.amrita"Corson, Jon M."https://zbmath.org/authors/?q=ai:corson.jon-michael"Das, Bikash"https://zbmath.org/authors/?q=ai:das.bikash-ranjanSummary: We consider pro-\(\mathcal{C}\) graphs for certain categories of finite graphs which we call pseudovarieties. After exploring some of the general theory, we specialize to a particular pseudovariety, denoted by \(\mathcal{E}\), that arises naturally in constructing end point compactifications of connected abstract graphs. The structure of pro-\(\mathcal{E}\) graphs and their fundamental profinite groups are shown to be analogous in certain ways to that of abstract graphs and their fundamental groups.The generalized lifting property of Bruhat intervalshttps://zbmath.org/1472.200782021-11-25T18:46:10.358925Z"Caselli, Fabrizio"https://zbmath.org/authors/?q=ai:caselli.fabrizio"Sentinelli, Paolo"https://zbmath.org/authors/?q=ai:sentinelli.paoloSummary: In [\textit{E. Tsukerman} and \textit{L. Williams}, Adv. Math. 285, 766--810 (2015; Zbl 1323.05010)], it is shown that every Bruhat interval of the symmetric group satisfies the so-called generalized lifting property. In this paper, we show that a Coxeter group satisfies this property if and only if it is finite and simply-laced.Asphericity of groups defined by graphshttps://zbmath.org/1472.200802021-11-25T18:46:10.358925Z"Bereznyuk, V. Yu."https://zbmath.org/authors/?q=ai:bereznyuk.v-yuSummary: A graph \(\Gamma\) labeled by a set \(S\) defines a group \(G(\Gamma)\) whose set of generators is the set \(S\) of labels and whose relations are all words which can be read on closed paths of this graph. We introduce the notion of an aspherical graph and prove that such a graph defines an aspherical group presentation. This result generalizes a theorem of \textit{D. Gruber} [Trans. Am. Math. Soc. 367, No. 3, 2051--2078 (2015; Zbl 1368.20030)] on graphs satisfying the graphical \(C(6)\)-condition and makes it possible to obtain new graphical conditions of asphericity similar to some classical conditions.Connectedness of spheres in Cayley graphshttps://zbmath.org/1472.200812021-11-25T18:46:10.358925Z"Brieussel, Jérémie"https://zbmath.org/authors/?q=ai:brieussel.jeremie"Gournay, Antoine"https://zbmath.org/authors/?q=ai:gournay.antoineSummary: We introduce the notion of connection thickness of spheres in a Cayley graph, related to dead-ends and their retreat depth. It was well-known that connection thickness is bounded for finitely presented one-ended groups. We compute that for natural generating sets of lamplighter groups on a line or on a tree, connection thickness is linear or logarithmic respectively. We show that it depends strongly on the generating set. We give an example where the metric induced at the (finite) thickness of connection gives diameter of order \(n^2\) to the sphere of radius \(n\). We also discuss the rarity of dead-ends and the relationships of connection thickness with cut sets in percolation theory and with almost-convexity. Finally, we present a list of open questions about spheres in Cayley graphs.A reciprocity law and the skew Pieri rule for the symplectic grouphttps://zbmath.org/1472.200972021-11-25T18:46:10.358925Z"Howe, Roger"https://zbmath.org/authors/?q=ai:howe.roger-e"Lávička, Roman"https://zbmath.org/authors/?q=ai:lavicka.roman"Lee, Soo Teck"https://zbmath.org/authors/?q=ai:lee.soo-teck"Souček, Vladimír"https://zbmath.org/authors/?q=ai:soucek.vladimirSummary: We use the theory of skew duality to show that decomposing the tensor product of \(k\) irreducible representations of the symplectic group \(\operatorname{Sp}_{2 m} = \operatorname{Sp}_{2 m}(\mathbb{C})\) is equivalent to branching from \(\operatorname{Sp}_{2n}\; \text{to}\; \operatorname{Sp}_{2 n_1} \times \cdots \times \operatorname{Sp}_{2 n_k}\), where \(n, n_1, \ldots, n_k\) are positive integers such that \(n = n_1 + \cdots + n_k\) and the \(n_j\)s depend on \(m\) as well as the representations in the tensor product. Using this result and a work of Lepowsky, we obtain a \textit{skew Pieri rule} for \(\operatorname{Sp}_{2m}\), i.e., a description of the irreducible decomposition of the tensor product of an irreducible representation of the symplectic group \(\operatorname{Sp}_{2m}\) with a fundamental representation.
\copyright 2017 American Institute of PhysicsOn the structures of hive algebras and tensor product algebras for general linear groups of low rankhttps://zbmath.org/1472.200992021-11-25T18:46:10.358925Z"Kim, Donggyun"https://zbmath.org/authors/?q=ai:kim.donggyun"Kim, Sangjib"https://zbmath.org/authors/?q=ai:kim.sangjib"Park, Euisung"https://zbmath.org/authors/?q=ai:park.euisungPolynomials associated with the fragments of coset diagramshttps://zbmath.org/1472.201132021-11-25T18:46:10.358925Z"Razaq, Abdul"https://zbmath.org/authors/?q=ai:razaq.abdul"Mushtaq, Qaiser"https://zbmath.org/authors/?q=ai:mushtaq.qaiser"Yousaf, Awais"https://zbmath.org/authors/?q=ai:yousaf.awaisSummary: The coset diagrams for \(\mathrm{PSL}(2, \mathbb{Z})\) are composed of fragments, and the fragments are further composed of circuits. Mushtaq has found that, the condition for the existence of a fragment in coset diagram is a polynomial \(f\) in \(\mathbb{Z}[z]\). Higman has conjectured that, the polynomials related to the fragments are monic and for a fixed degree, there are finite number of such polynomials. In this paper, we consider a family \(\Omega\) of fragments such that each fragment in \(\Omega\) contains one vertex fixed by a pair of words \((xy)^{q_1}(xy^{-1})^{q_2}, (xy^{-1})^{s_1}(xy)^{s_2}\), where \(s_1, s_2, q_1, q_2 \in \mathbb{Z}^+\), and prove Higman's conjecture for the polynomials obtained from \(\Omega\). At the end, we answer the question; for a fixed degree n, how many polynomials are evolved from \(\Omega\).The idempotent-generated subsemigroup of the Kauffman monoidhttps://zbmath.org/1472.201362021-11-25T18:46:10.358925Z"Dolinka, Igor"https://zbmath.org/authors/?q=ai:dolinka.igor"East, James"https://zbmath.org/authors/?q=ai:east.jamesSummary: We characterise the elements of the (maximum) idempotent-generated subsemigroup of the Kauffman monoid in terms of combinatorial data associated with certain normal forms. We also calculate the smallest size of a generating set and idempotent generating set.On elements of finite semigroups of order-preserving and decreasing transformationshttps://zbmath.org/1472.201402021-11-25T18:46:10.358925Z"Yağcı, Melek"https://zbmath.org/authors/?q=ai:yagci.melek"Korkmaz, Emrah"https://zbmath.org/authors/?q=ai:korkmaz.emrahFor \(n\in\mathbb{N}\), let \(\mathcal{C}_n\) be the semigroup of all order-preserving and decreasing transformations on \(X_n=\{1,\ldots,n\}\), under its natural order, and let \(N(\mathcal{C}_n)\) be the set of all nilpotent elements of \(\mathcal{C}_n\) and let Fix\((\alpha)=\{x\in X_n:xa=x\}\) for any transformation \(\alpha\). An element \(a\) of a finite semigroup is called \(m\)-potent (\(m\)-nilpotent) element if \(a^{m+1}=a^m\) \((a^m=0)\) and \(a,a^2,\ldots, a^m\) are distinct. The aim of this paper is to find a formulae for the number of \(m\)-nilpotent elements, In Section 2, the authors obtain a formulae for the number of \(m\)-nilpotent elements in \(N(\mathcal{C}_n)\) for \(2\leq m \leq n-1\). In Section 3, the authors obtain a formulae for the number of \(m\)-potent elements of \(C_{n,Y} = \{\alpha\in \mathcal{C}_n :\) Fix\((\alpha) = Y\}\) where \(Y\) is a subset of \(X_n\).Projective indecomposable modules and quivers for monoid algebrashttps://zbmath.org/1472.201422021-11-25T18:46:10.358925Z"Margolis, Stuart"https://zbmath.org/authors/?q=ai:margolis.stuart-w"Steinberg, Benjamin"https://zbmath.org/authors/?q=ai:steinberg.benjaminSummary: We give a construction of the projective indecomposable modules and a description of the quiver for a large class of monoid algebras including the algebra of any finite monoid whose principal right ideals have at most one idempotent generator. Our results include essentially all families of finite monoids for which this has been done previously, for example, left regular bands, \(\mathcal{J}\)-trivial and \(\mathcal{R}\)-trivial monoids and left regular bands of groups.The intersection graph of a finite Moufang loophttps://zbmath.org/1472.201442021-11-25T18:46:10.358925Z"Hasanzadeh Bashir, H."https://zbmath.org/authors/?q=ai:hasanzadeh-bashir.h"Iranmanesh, A."https://zbmath.org/authors/?q=ai:iranmanesh.aliIn this paper, the authors define the intersection graph of a loop analogously as it is defined for a group. They show some basic results about such graphs: they characterise Moufang loops of odd orders with complete intersection graphs, they characerise finite Moufang loops having forests as intersection graphs or they construct examples of non-isomorphic Moufang loops with isomorphic intersection graphs.On the enumeration and asymptotic growth of free quasigroup wordshttps://zbmath.org/1472.201452021-11-25T18:46:10.358925Z"Smith, Jonathan D. H."https://zbmath.org/authors/?q=ai:smith.jonathan-d-h"Wang, Stefanie G."https://zbmath.org/authors/?q=ai:wang.stefanie-gThe paper counts the number of reduced quasigroup words of given length and in a given number of variables. In other terms, it counts the growth of finitely generated free quasigroups with respect to the word length.
The main result is a recursive formula counting reduced words. There is an interesting discussion of its asymptotic growth, and of the relationship with the Catalan numbers, resulting in several conjectures supported by numerical experiments.Towards a classification of networks with asymmetric inputshttps://zbmath.org/1472.340692021-11-25T18:46:10.358925Z"Aguiar, Manuela"https://zbmath.org/authors/?q=ai:aguiar.manuela-a-d"Dias, Ana"https://zbmath.org/authors/?q=ai:dias.ana-paula-s"Soares, Pedro"https://zbmath.org/authors/?q=ai:soares.pedro-a-junThe heat equation on the finite Poincaré upper half-planehttps://zbmath.org/1472.354172021-11-25T18:46:10.358925Z"Dedeo, M. R."https://zbmath.org/authors/?q=ai:dedeo.michelle-r"Velasquez, Elinor"https://zbmath.org/authors/?q=ai:velasquez.elinorSummary: A differential-difference operator is used to model the heat equation on a finite graph analogue of Poincaré's upper half-plane. Finite analogues of the classical theta functions are shown to be solutions to the heat equation in this setting.Decomposition of random walk measures on the one-dimensional torushttps://zbmath.org/1472.370072021-11-25T18:46:10.358925Z"Gilat, Tom"https://zbmath.org/authors/?q=ai:gilat.tomSummary: The main result of this paper is a decomposition theorem for a measure on the one-dimensional torus. Given a sufficiently large subset \(S\) of the positive integers, an arbitrary measure on the torus is decomposed as the sum of two measures. The first one \(\mu_1\) has the property that the random walk with initial distribution \(\mu_1\) evolved by the action of \(S\) equidistributes very fast. The second measure \(\mu_2\) in the decomposition is concentrated on very small neighborhoods of a small number of points.A symmetric quantum calculushttps://zbmath.org/1472.390122021-11-25T18:46:10.358925Z"Brito Da Cruz, Artur M. C."https://zbmath.org/authors/?q=ai:brito-da-cruz.artur-m-c"Martins, Natália"https://zbmath.org/authors/?q=ai:martins.natalia-f"Torres, Delfim F. M."https://zbmath.org/authors/?q=ai:torres.delfim-f-mSummary: We introduce the \(\alpha\), \(\beta\)-symmetric difference derivative and the \(\alpha\), \(\beta\)-symmetric Nörlund sum. The associated symmetric quantum calculus is developed, which can be seen as a generalization of the forward and backward \(h\)-calculus.
For the entire collection see [Zbl 1277.00035].Equi-distributed property and spectral set conjecture on \(\mathbb{Z}_{p^2} \times \mathbb{Z}_p\)https://zbmath.org/1472.430072021-11-25T18:46:10.358925Z"Shi, Ruxi"https://zbmath.org/authors/?q=ai:shi.ruxiSummary: In this paper, we show an equi-distributed property in 2-dimensional finite abelian groups \(\mathbb{Z}_{p^n} \times \mathbb{Z}_{p^m}\), where \(p\) is a prime number. By using this equi-distributed property, we prove that Fuglede's spectral set conjecture holds on groups \(\mathbb{Z}_{p^2} \times \mathbb{Z}_p\), namely, a set in \(\mathbb{Z}_{p^2} \times \mathbb{Z}_p\) is a spectral set if and only if it is a translational tile.An operad of non-commutative independences defined by treeshttps://zbmath.org/1472.460682021-11-25T18:46:10.358925Z"Jekel, David"https://zbmath.org/authors/?q=ai:jekel.david"Liu, Weihua"https://zbmath.org/authors/?q=ai:liu.weihuaSummary: %\DeclareMathSymbol{\boxright}{3}{mathb}{'151}
We study certain notions of \(N\)-ary non-commutative independence, which generalize free, Boolean, and monotone independence. For every rooted subtree \(\mathcal{T}\) of an \(N\)-regular rooted tree, we define the \(\mathcal{T}\)-free product of \(N\) non-commutative probability spaces and the \(\mathcal{T}\)-free additive convolution of \(N\) non-commutative laws.
These \(N\)-ary additive convolution operations form a topological symmetric operad which includes the free, Boolean, monotone, and anti-monotone convolutions, as well as the orthogonal and subordination convolutions. Using the operadic framework, the proof of convolution identities such as
%\(\mu\boxplus\nu=\mu\rhd ( \nu\boxempty\kern-9pt\vdash \mu ) \)
%\(\mu\boxplus\nu=\mu\rhd(\nu \boxright \mu)\)
\(\mu\boxplus\nu = \mu \vartriangleright (\nu\,\square\!\!\!\!\!\vdash \!\mu) \)
can be reduced to combinatorial manipulations of trees. In particular, we obtain a decomposition of the \(\mathcal{T} \)-free convolution into iterated Boolean and orthogonal convolutions, which generalizes work of \textit{R. Lenczewski} [J. Funct. Anal. 246, No. 2, 330--365 (2007; Zbl 1129.46055)].
We also develop a theory of \(\mathcal{T} \)-free independence that closely parallels the free, Boolean, and monotone cases, provided that the root vertex has more than one neighbor. This includes combinatorial moment formulas, cumulants, a central limit theorem, and classification of distributions that are infinitely divisible with bounded support. In particular, we study the case where the root vertex of \(\mathcal{T}\) has \(n\) children and each other vertex has \(d\) children, and we relate the \(\mathcal{T} \)-free convolution powers to free and Boolean convolution powers and the Belinschi-Nica semigroup.An infinite family of \(m\)-ovoids of \(Q(4,q)\)https://zbmath.org/1472.510022021-11-25T18:46:10.358925Z"Feng, Tao"https://zbmath.org/authors/?q=ai:feng.tao"Tao, Ran"https://zbmath.org/authors/?q=ai:tao.ranLet \(Q(4,q)\) denote the point line geometry associated to a non-singular quadratic form on the \(5\)-dimensional vector space over the finite field of order \(q\), \(q=p^h\), \(p\) prime, \(h \geq 1\). It is well known that the Witt index of such a form equals \(2\). The singular subspaces of dimension \(1\) and \(2\), are considered as points and lines, respectively, in the projective space \(\mathrm{PG}(4,q)\), and it is well known that this point line geometry is an example of a generalized quadrangle, or order \(q\), embedded in \(\mathrm{PG}(4,q)\). This GQ is one of the so-called finite classical generalized quadrangles.
Let \(\mathcal{S}\) be a GQ. An {\em ovoid} of \(\mathcal{S}\) is a set \(\mathcal{O}\) of points such that every line of \(\mathcal{S}\) contains exactly one point of \(\mathcal{O}\). Let \(m \geq 1\). An {\em \(m\)-ovoid} of \(\mathcal{S}\) is a set \(\mathcal{O}\) of points such that every line contains exactly \(m\) points of \(\mathcal{O}\). The concept of an \(m\)-ovoid might look as a straightforward generalization of an ovoid. However, as ovoids of GQs are relatively rare, \(m\)-ovoids are quite exceptional. Some non-existence results are available, but construction results are, so far, quite rare. The nice paper under review provides a construction of \(\frac{q-1}{2}\)-ovoids of \(Q(4,q)\), \(q \equiv 1 \pmod{4}\), \(q > 5\), admitting \(C_{\frac{q^2-1}{2}} \rtimes C_2\) as automorphism group.Further consequences of the colorful Helly hypothesishttps://zbmath.org/1472.520112021-11-25T18:46:10.358925Z"Martínez-Sandoval, Leonardo"https://zbmath.org/authors/?q=ai:martinez-sandoval.leonardo"Roldán-Pensado, Edgardo"https://zbmath.org/authors/?q=ai:roldan-pensado.edgardo"Rubin, Natan"https://zbmath.org/authors/?q=ai:rubin.natanSummary: Let \(\mathcal{F}\) be a family of convex sets in \(\mathbb{R}^d,\) which are colored with \(d + 1\) colors. We say that \(\mathcal{F}\) satisfies the Colorful Helly Property if every rainbow selection of \(d + 1\) sets, one set from each color class, has a non-empty common intersection. The Colorful Helly Theorem of Lovász states that for any such colorful family \(\mathcal{F}\) there is a color class \(\mathcal{F}_i \subset \mathcal{F},\) for \(1 \le i \le d + 1,\) whose sets have a non-empty intersection. We establish further consequences of the Colorful Helly hypothesis. In particular, we show that for each dimension \(d \ge 2\) there exist numbers \(f(d)\) and \(g(d)\) with the following property: either one can find an additional color class whose sets can be pierced by \(f(d)\) points, or all the sets in \(\mathcal{F}\) can be crossed by \(g(d)\) lines.The general formula for the Ehrhart polynomial of polytopes with applicationshttps://zbmath.org/1472.520142021-11-25T18:46:10.358925Z"Sadiq, Fatema A."https://zbmath.org/authors/?q=ai:sadiq.fatema-a"Salman, Shatha A."https://zbmath.org/authors/?q=ai:salman.shatha-a"Sabri, Raghad I."https://zbmath.org/authors/?q=ai:sabri.raghad-iSummary: Recently, polytopes have shown wide applications in a lot of situations. For example, a cyclic polytope is very important in different areas of science like solutions to extremum problems (the Upper Bound Conjecture). Polytopes serve as bases for diverse constructions (from triangulations to bimatrix games). In addition, we give the general form for the product of simplex polytopes and an algorithm for these computations.The equivariant Ehrhart theory of the permutahedronhttps://zbmath.org/1472.520162021-11-25T18:46:10.358925Z"Ardila, Federico"https://zbmath.org/authors/?q=ai:ardila.federico"Supina, Mariel"https://zbmath.org/authors/?q=ai:supina.mariel"Vindas-Meléndez, Andrés R."https://zbmath.org/authors/?q=ai:vindas-melendez.andres-rIn [\textit{A. Stapledon}, Adv. Math. 226, No. 4, 3622--3654 (2011; zbl 1218.52014)] Stapledon introduced \textit{equivariant Ehrhart theory}, a variant of Ehrhart theory that takes group actions into account. For a lattice polytope \(P\) whose vertices lie in the lattice \(M\) and a group \(G\) acting on \(M\), one can define the \textit{equivariant \(H^*\)-series} \(H^*[z]\) which can be written as \(\sum_{i\geq 0} H_i^*z^i\) for appropriate virtual characters \(H_i^*\). Stapledon asks whether or not this series is effective, i.e, whether all the \(H_i^*\) are characters of representations of \(G\), and proposes the \textit{effectiveness conjecture} which states that the effectiveness of the equivariant \(H^*\)-series is equivalent to two other properties, namely
\begin{itemize}
\item[(i)] the toric variety of \(P\) admits a \(G\)-invariant non-degenerate hypersurface,
\item[(ii)] the equivariant \(H^*\)-series is a polynomial.
\end{itemize}
It is already known that (i) is a sufficient and (ii) is a necessary condition.
The present paper proves the effectiveness conjecture and three minor conjectures in the case of permutahedra under the action of the symmetric group.Multi-splits and tropical linear spaces from nested matroidshttps://zbmath.org/1472.520192021-11-25T18:46:10.358925Z"Schröter, Benjamin"https://zbmath.org/authors/?q=ai:schroter.benjaminSummary: We present an explicit combinatorial description of a special class of facets of the secondary polytopes of hypersimplices. These facets correspond to polytopal subdivisions called multi-splits. We show that the maximal cells in a multi-split of a hypersimplex are matroid polytopes of nested matroids. Moreover, we derive a description of all multi-splits of a product of simplices. Additionally, we present a computational result to derive explicit lower bounds on the number of facets of secondary polytopes of hypersimplices.On the density of the thinnest covering of \(\mathbb{R}^n\)https://zbmath.org/1472.520242021-11-25T18:46:10.358925Z"Jung, Soon-Mo"https://zbmath.org/authors/?q=ai:jung.soon-mo"Nam, Doyun"https://zbmath.org/authors/?q=ai:nam.doyunLet \(K\) be a convex body in \(\mathbb{R}^n\). Let \(\mathcal{U}(K)\) denote the family of all coverings \(\{K + x_i\}_{i\geq 1}\) of the entire space \(\mathbb{R}^n\), where \(x_i\in \mathbb{R}^n\), \(i\geq 1\). Given \(s>0\), let \(Q_s = \prod _{i\leq n} [-s, s)\) and \(R_s = \prod _{i\leq n} (-s, s)\). The covering density of \(K\) is defined as \[ \theta (K) = \inf_{\{K + x_i\}_{i}\in \mathcal{U}(K)} \liminf _{s\to \infty} \frac{1}{\mbox{vol} (Q_s)} \sum_{K+x_i \subset Q_s } \mbox{vol}(K+x_i). \] It is known that \[ \theta (K) = \inf_{\{K + x_i\}_{i}\in \mathcal{U}(K)} \liminf _{s\to \infty} \frac{1}{\mbox{vol} (Q_s)} \sum_{(K+x_i)\cap Q_s = \emptyset } \mbox{vol} (K+x_i). \] It follows from results of \textit{H.~Groemer} [Math. Z. 81, 260--278 (1963; Zbl 0123.39104)] that for any fixed \(t>0\),
\begin{align*}
\theta (K) &= \lim_{s\to \infty }\inf_{\{K + x_i\}_{i}\in \mathcal{U}(K)} \frac{1}{\mbox{vol} (Q_s)} \sum_{(K+x_i)\cap Q_s = \emptyset } \mbox{vol} (K+x_i) \\
&= \lim_{\lambda \to 0^+ }\inf_{\{\lambda K + x_i\}_{i}\in \mathcal{U}(\lambda K)} \frac{1}{\mbox{vol} (R_t)} \sum_{(\lambda K+x_i)\cap R_t = \emptyset } \mbox{vol}(\lambda K+x_i) .
\end{align*}
The authors provide another proof of this result. They also mention that yet another proof given in [\textit{S.-M. Jung}, Commun. Korean Math. Soc. 10, No. 3, 621--632 (1995, Zbl 0943.52002)] is based on an unverified assertion and contains some ambiguities.Tilings with congruent edge coronaehttps://zbmath.org/1472.520262021-11-25T18:46:10.358925Z"Tomenes, Mark D."https://zbmath.org/authors/?q=ai:tomenes.mark-d"De Las Peñas, Ma. Louise Antonette N."https://zbmath.org/authors/?q=ai:de-las-penas.ma-louise-antonette-nA normal tiling of the Euclidean plane is a cover of the plane by non-overlapping closed topological discs such that their diameters are bounded from above and their inradii are bounded from below by positive constants, and such that the intersection of any two tiles is either empty or a singleton or an arc. Such arcs are called edges of the tiling. A centred edge corona is composed of the centre of an edge and of all tiles having a non-empty intersection with that edge. A tiling is called edge-transitive or isotoxal if its symmetry group acts transitively on the set of all its edges. The authors show that every normal tiling with pairwise congruent centred edge coronae is isotoxal, and they classify such tilings.
For the entire collection see [Zbl 1467.52001].Exact hyperplane covers for subsets of the hypercubehttps://zbmath.org/1472.520352021-11-25T18:46:10.358925Z"Aaronson, James"https://zbmath.org/authors/?q=ai:aaronson.james"Groenland, Carla"https://zbmath.org/authors/?q=ai:groenland.carla"Grzesik, Andrzej"https://zbmath.org/authors/?q=ai:grzesik.andrzej"Johnston, Tom"https://zbmath.org/authors/?q=ai:johnston.tom"Kielak, Bartłomiej"https://zbmath.org/authors/?q=ai:kielak.bartlomiejThe exact cover of \(B\subseteq \{0, 1\}^n\), denoted by \(\mathrm{ec}(B)\), is a set of hyperplanes in \(\mathbb{R}^n\) whose union intersects \(\{0, 1\}^n\) exactly at \(B\) -- that is, the points in \(\{0,1\}^n\setminus B\) are not covered. If just one point, say \(0\), is removed, then \(\mathrm{ec}(\{0,1\}^n\setminus\{0\})=n\) from work of \textit{N. Alon} and \textit{Z. Füredi} [Eur. J. Comb. 14, No. 2, 79--83 (1993; Zbl 0773.52011)]. In this article, the authors generalize this to up to \(4\) points removed. They show that \(\mathrm{ec}(\{0,1\}^n\setminus S) = n-1\) if \(|S|\in\{2,3\}\), and if \(|S|=4\), then \(\mathrm{ec}(\{0,1\}^n\setminus S)\) is \(n-1\) if the four points of \(S\) are not coplanar, and \(n-2\) otherwise. The authors prove asymptotic bounds concerning the following two numbers: for \(n,k\in\mathbb{N}\),
\begin{align*}
\mathrm{ec}(n, k) &= \max\{\mathrm{ec}(\{0,1\}^n\setminus S) : S\subseteq \{0,1\}^n, \; |S|=k \}, \\
\mathrm{ec}(n) &= \max\{\mathrm{ec}(B) : B\subseteq \{0,1\}^n \}.
\end{align*}
They close by posing two problems that would, if answered affirmatively, tighten the bounds they determined.Worpitzky-compatible subarrangements of braid arrangements and cocomparability graphshttps://zbmath.org/1472.520372021-11-25T18:46:10.358925Z"Tran, Tan Nhat"https://zbmath.org/authors/?q=ai:tran.tan-nhat"Tsuchiya, Akiyoshi"https://zbmath.org/authors/?q=ai:tsuchiya.akiyoshiSummary: The class of Worpitzky-compatible subarrangements of a Weyl arrangement together with an associated Eulerian polynomial was recently introduced by \textit{A. U. Ashraf} et al. [Adv. Appl. Math. 120, Article ID 102064, 24 p. (2020; Zbl 1447.52026)], which brings the characteristic and Ehrhart quasi-polynomials into one formula. The subarrangements of the braid arrangement, the Weyl arrangement of type \(A\), are known as the graphic arrangements. We prove that the Worpitzky-compatible graphic arrangements are characterized by cocomparability graphs. This can be regarded as a counterpart of the characterization by Stanley and Edelman-Reiner of free and supersolvable graphic arrangements in terms of chordal graphs. Our main result yields new formulas for the chromatic and graphic Eulerian polynomials of cocomparability graphs.Smoothness filtration of the magnitude complexhttps://zbmath.org/1472.550062021-11-25T18:46:10.358925Z"Gomi, Kiyonori"https://zbmath.org/authors/?q=ai:gomi.kiyonoriLet \((X, d)\) be a metric space. A sequence of points \(\langle x_0, x_1, \dots, x_n\rangle\) (\(x_i\in X\)) is said to be a proper \(n\)-chain of length \(\ell\) if \(x_{i-1}\neq x_i\) (\(i=1, \dots, n\)) and \(d(x_0, x_1)+d(x_1, x_2)+\dots +d(x_{n-1}, x_n)=\ell\). Let \(C_n^\ell(X)\) be the free abelian group generated by proper \(n\)-chains of length \(\ell\). One can naturally define the boundary map \(\partial : C_n^\ell(X)\longrightarrow C_{n-1}^\ell(X)\) whose homology group \(H_n^\ell(X):=H_n(C_*^\ell(X))\) is called the \textit{magnitude homology} of \((X, d)\).
The notion of magnitude homology was first introduced by \textit{R. Hepworth} and \textit{S. Willerton} [Homology Homotopy Appl. 19, No. 2, 31--60 (2017; Zbl 1377.05088)] for a finite metric space defined by a graph. Later, it was generalized to a metric space (furthermore, enriched category) by \textit{T. Leinster} and \textit{M. Shulman} [``Magnitude homology of enriched categories and metric spaces'', Preprint, \url{arXiv:1711.00802}]. The computation of magnitude homology is, in general, difficult. In particular, if there exists a \textit{\(4\)-cut}, that is a chain \(\langle x_0, x_1, x_2, x_3\rangle\) satisfying
\[
\begin{split} d(x_0, x_3) &< d(x_0, x_1) + d(x_1, x_2) + d(x_2, x_3)\\
&= d(x_0, x_2) + d(x_2, x_3) = d(x_0, x_1) + d(x_1, x_3), \end{split}
\]
then the computation becomes complicated. Indeed, a previous work by \textit{R. Kaneta} and \textit{M. Yoshinaga} [Bull. Lond. Math. Soc. 53, No. 3, 893--905 (2021; Zbl 1472.55007)] showed that if the metric space \((X, d)\) does not have \(4\)-cuts, then the computation of the magnitude homology is reduced to that of the order complexes for posets.
The present paper extends and refines the previous works by using spectral sequences. In a proper chain \(\langle x_0, x_1, \dots, x_n\rangle\), a point \(x_i\) is said to be a \textit{smooth point} if \(d(x_{i-1}, x_{i})+d(x_{i}, x_{i+1})=d(x_{i-1}, x_{i+1})\) (otherwise, it is called a \textit{singular point}). Denote the number of smooth points in \(x\) by \(\sigma(x)\) and the submodule of \(C_n^\ell(X)\) generated by proper chains with \(\sigma(x)\leq p\) by \(F_pC_n^\ell(X)\). Then, \(F_pC_*^\ell(X)\) defines a filtration on the magnitude chain complex \(C_*^\ell(X)\). The associated spectral sequence is the main object of the present paper. The author completely describes at which page the spectral sequence degenerates. Precise results are as follows.
\begin{itemize}
\item[(a)] The spectral sequence always degenerates at \(E^4\).
\item[(b)] The spectral sequence degenerates at \(E^2\) if and only if the metric space \((X, d)\) does not contain \(4\)-cuts.
\item[(c)] The spectral sequence degenerates at \(E^3\) if and only if there does not exist a chain \(x=\langle x_0, x_1, x_2, x_3, x_4\rangle\) such that both \(\langle x_0, x_1, x_2, x_3\rangle\) and \(\langle x_1, x_2, x_3, x_4\rangle\) are \(4\)-cuts.
\end{itemize}
The author also applies the spectral sequence to a number of concrete examples.Geometry of uniform spanning forest components in high dimensionshttps://zbmath.org/1472.600182021-11-25T18:46:10.358925Z"Barlow, Martin T."https://zbmath.org/authors/?q=ai:barlow.martin-t"Járai, Antal A."https://zbmath.org/authors/?q=ai:jarai.antal-aSummary: We study the geometry of the component of the origin in the uniform spanning forest of \(\mathbb{Z}^d\) and give bounds on the size of balls in the intrinsic metric.Concentration inequalities for bounded functionals via log-Sobolev-type inequalitieshttps://zbmath.org/1472.600362021-11-25T18:46:10.358925Z"Götze, Friedrich"https://zbmath.org/authors/?q=ai:gotze.friedrich-w"Sambale, Holger"https://zbmath.org/authors/?q=ai:sambale.holger"Sinulis, Arthur"https://zbmath.org/authors/?q=ai:sinulis.arthurSummary: In this paper, we prove multilevel concentration inequalities for bounded functionals \(f = f(X_1, \dots , X_n)\) of random variables \(X_1, \dots , X_n\) that are either independent or satisfy certain logarithmic Sobolev inequalities. The constants in the tail estimates depend on the operator norms of \(k\)-tensors of higher order differences of \(f\). We provide applications for both dependent and independent random variables. This includes deviation inequalities for empirical processes \(f(X) = \sup_{g \in{\mathcal{F}}} {|g(X)|}\) and suprema of homogeneous chaos in bounded random variables in the Banach space case \(f(X) = \sup_t{\Vert \sum_{i_1 \ne \dots \ne i_d} t_{i_1 \dots i_d} X_{i_1} \cdots X_{i_d}\Vert }_{{\mathcal{B}}} \). The latter application is comparable to earlier results of Boucheron, Bousquet, Lugosi, and Massart and provides the upper tail bounds of Talagrand. In the case of Rademacher random variables, we give an interpretation of the results in terms of quantities familiar in Boolean analysis. Further applications are concentration inequalities for \(U\)-statistics with bounded kernels \(h\) and for the number of triangles in an exponential random graph model.Central limit theorems from the roots of probability generating functionshttps://zbmath.org/1472.600462021-11-25T18:46:10.358925Z"Michelen, Marcus"https://zbmath.org/authors/?q=ai:michelen.marcus"Sahasrabudhe, Julian"https://zbmath.org/authors/?q=ai:sahasrabudhe.julianSummary: For each \(n\), let \(X_n \in \{0, \ldots, n \}\) be a random variable with mean \(\mu_n\), standard deviation \(\sigma_n\), and let \[P_n(z) = \sum_{k = 0}^n \mathbb{P}(X_n = k) z^k,\] be its probability generating function. We show that if none of the complex zeros of the polynomials \(\{P_n(z) \}\) is contained in a neighborhood of \(1 \in \mathbb{C}\) and \(\sigma_n > n^\varepsilon\) for some \(\varepsilon > 0\), then \(X_n^\ast = (X_n - \mu_n) \sigma_n^{- 1}\) is asymptotically normal as \(n \to \infty\): that is, it tends in distribution to a random variable \(Z \sim \mathcal{N}(0, 1)\). On the other hand, we show that there exist sequences of random variables \(\{X_n \}\) with \(\sigma_n > C \log n\) for which \(P_n(z)\) has no roots near 1 and \(X_n^\ast\) is not asymptotically normal. These results disprove a conjecture of Pemantle and improve upon various results in the literature. We go on to prove several other results connecting the location of the zeros of \(P_n(z)\) and the distribution of the random variable \(X_n\).A threshold for cutoff in two-community random graphshttps://zbmath.org/1472.601172021-11-25T18:46:10.358925Z"Ben-Hamou, Anna"https://zbmath.org/authors/?q=ai:ben-hamou.annaSummary: In this paper, we are interested in the impact of communities on the mixing behavior of the nonbacktracking random walk. We consider sequences of sparse random graphs of size \(N\) generated according to a variant of the classical configuration model which incorporates a two-community structure. The strength of the bottleneck is measured by a parameter \(\alpha\) which roughly corresponds to the fraction of edges that go from one community to the other. We show that if \(\alpha \gg \frac{1}{\log N}\), then the nonbacktracking random walk exhibits cutoff at the same time as in the one-community case, but with a larger cutoff window, and that the distance profile inside this window converges to the Gaussian tail function. On the other hand, if \(\alpha \ll \frac{1}{\log N}\) or \(\alpha \asymp \frac{1}{\log N}\), then the mixing time is of order \(1/\alpha\) and there is no cutoff.Mixing time and eigenvalues of the abelian sandpile Markov chainhttps://zbmath.org/1472.601192021-11-25T18:46:10.358925Z"Jerison, Daniel C."https://zbmath.org/authors/?q=ai:jerison.daniel-c"Levine, Lionel"https://zbmath.org/authors/?q=ai:levine.lionel"Pike, John"https://zbmath.org/authors/?q=ai:pike.johnSummary: The abelian sandpile model defines a Markov chain whose states are integer-valued functions on the vertices of a simple connected graph \(G\). By viewing this chain as a (nonreversible) random walk on an abelian group, we give a formula for its eigenvalues and eigenvectors in terms of ``multiplicative harmonic functions'' on the vertices of \(G\). We show that the spectral gap of the sandpile chain is within a constant factor of the length of the shortest noninteger vector in the dual Laplacian lattice, while the mixing time is at most a constant times the smoothing parameter of the Laplacian lattice. We find a surprising inverse relationship between the spectral gap of the sandpile chain and that of simple random walk on \(G\): If the latter has a sufficiently large spectral gap, then the former has a small gap! In the case where \(G\) is the complete graph on \(n\) vertices, we show that the sandpile chain exhibits cutoff at time \(\frac{1}{4\pi ^2}n^3\log n\).Significance-based community detection in weighted networkshttps://zbmath.org/1472.621232021-11-25T18:46:10.358925Z"Palowitch, John"https://zbmath.org/authors/?q=ai:palowitch.john"Bhamidi, Shankar"https://zbmath.org/authors/?q=ai:bhamidi.shankar"Nobel, Andrew B."https://zbmath.org/authors/?q=ai:nobel.andrew-bSummary: Community detection is the process of grouping strongly connected nodes in a network. Many community detection methods for un-weighted networks have a theoretical basis in a null model. Communities discovered by these methods therefore have interpretations in terms of statistical significance. In this paper, we introduce a null for weighted networks called the continuous configuration model. First, we propose a community extraction algorithm for weighted networks which incorporates iterative hypothesis testing under the null. We prove a central limit theorem for edge-weight sums and asymptotic consistency of the algorithm under a weighted stochastic block model. We then incorporate the algorithm in a community detection method called CCME. To benchmark the method, we provide a simulation framework involving the null to plant ``background'' nodes in weighted networks with communities. We show that the empirical performance of CCME on these simulations is competitive with existing methods,
particularly
when overlapping communities and background nodes are present. To further validate the method, we present two real-world networks with potential background nodes and analyze them with CCME, yielding results that reveal macro-features of the corresponding systems.Maximin distance designs based on densest packingshttps://zbmath.org/1472.621252021-11-25T18:46:10.358925Z"Yang, Liuqing"https://zbmath.org/authors/?q=ai:yang.liuqing"Zhou, Yongdao"https://zbmath.org/authors/?q=ai:zhou.yongdao"Liu, Min-Qian"https://zbmath.org/authors/?q=ai:liu.min-qianSummary: Computer experiments play a crucial role when physical experiments are expensive or difficult to be carried out. As a kind of designs for computer experiments, maximin distance designs have been widely studied. Many existing methods for obtaining maximin distance designs are based on stochastic algorithms, and these methods will be infeasible when the run size or number of factors is large. In this paper, we propose some deterministic construction methods for maximin \(L_2\)-distance designs in two to five dimensions based on densest packings. The resulting designs have large \(L_2\)-distances and are mirror-symmetric. Some of them have the same \(L_2\)-distances as the existing optimal maximin distance designs, and some of the others are completely new. Especially, the resulting 2-dimensional designs possess a good projection property.Using symbolic networks to analyse dynamical properties of disease outbreakshttps://zbmath.org/1472.621522021-11-25T18:46:10.358925Z"Herrera-Diestra, José L."https://zbmath.org/authors/?q=ai:herrera-diestra.jose-l"Buldú, Javier M."https://zbmath.org/authors/?q=ai:buldu.javier-m"Chavez, Mario"https://zbmath.org/authors/?q=ai:chavez.mario"Martínez, Johann H."https://zbmath.org/authors/?q=ai:martinez.johann-hSummary: We introduce a new methodology, which is based on the construction of epidemic networks, to analyse the evolution of epidemic time series. First, we translate the time series into ordinal patterns containing information about local fluctuations in disease prevalence. Each pattern is associated with a node of a network, whose (directed) connections arise from consecutive appearances in the series. The analysis of the network structure and the role of each pattern allows them to be classified according to the enhancement of entropy/complexity along the series, giving a different point of view about the evolution of a given disease.A framework for second-order eigenvector centralities and clustering coefficientshttps://zbmath.org/1472.650432021-11-25T18:46:10.358925Z"Arrigo, Francesca"https://zbmath.org/authors/?q=ai:arrigo.francesca"Higham, Desmond J."https://zbmath.org/authors/?q=ai:higham.desmond-j"Tudisco, Francesco"https://zbmath.org/authors/?q=ai:tudisco.francescoSummary: We propose and analyse a general tensor-based framework for incorporating second-order features into network measures. This approach allows us to combine traditional pairwise links with information that records whether triples of nodes are involved in wedges or triangles. Our treatment covers classical spectral methods and recently proposed cases from the literature, but we also identify many interesting extensions. In particular, we define a mutually reinforcing (spectral) version of the classical clustering coefficient. The underlying object of study is a constrained nonlinear eigenvalue problem associated with a cubic tensor. Using recent results from nonlinear Perron-Frobenius theory, we establish existence and uniqueness under appropriate conditions, and show that the new spectral measures can be computed efficiently with a nonlinear power method. To illustrate the added value of the new formulation, we analyse the measures on a class of synthetic networks. We also give computational results on centrality and link prediction for real-world networks.Performance and stability of direct methods for computing generalized inverses of the graph Laplacianhttps://zbmath.org/1472.650522021-11-25T18:46:10.358925Z"Benzi, Michele"https://zbmath.org/authors/?q=ai:benzi.michele"Fika, Paraskevi"https://zbmath.org/authors/?q=ai:fika.paraskevi"Mitrouli, Marilena"https://zbmath.org/authors/?q=ai:mitrouli.marilenaThe authors present and study two direct algorithms for computing particular generalized inverses of graph Laplacians: the group inverse and the absorption inverse. The group generalized inverse of a square matrix \(A\) is the unique \(A^\sharp\) such that \(AA^\sharp A=A\), \(A^\sharp AA^\sharp=A^\sharp\) and \(AA^\sharp=A^\sharp A\), while absorption inverse further generalizes group inverse for graphs with absorption. The authors first improve the direct method they previously developed in [Linear Algebra Appl. 574, 123--158 (2019; Zbl. 1433.65065)], and show that this improvement is forward stable with high relative component-wise accuracy. The second proposed algorithm is based on bottleneck matrices, which represents inverses of the principal \((n-1)\times(n-1)\)-minors of the Laplacian, with lower numerical stability than the first method. Experimental results show that both methods presented here provide faster algorithms for handling dense problems.Numerical-solution-for-nonlinear-Klein-Gordon equation via operational-matrix by clique polynomial of complete graphshttps://zbmath.org/1472.651282021-11-25T18:46:10.358925Z"Kumbinarasaiah, S."https://zbmath.org/authors/?q=ai:kumbinarasaiah.s"Ramane, H. S."https://zbmath.org/authors/?q=ai:ramane.harishchandra-s"Pise, K. S."https://zbmath.org/authors/?q=ai:pise.kartik-s"Hariharan, G."https://zbmath.org/authors/?q=ai:hariharan.govindSummary: This study introduced a generalized operational matrix using Clique polynomials of a complete graph and proposed the latest approach to solve the non-linear Klein-Gordon (KG) equation. KG equations describe many real physical phenomena in fluid dynamics, electrical engineering, biogenetics, tribology. By using the properties of the operational-matrix, we transform-the non-linear KG equation into a system-of algebraic-equations. Unknown coefficients to be determined by Newton's method. The present-technique is applied-to four problems, and the obtained-results are-compared with-another-method in the literature. Also, we discussed some theorems on convergence analysis and continuous property.Backjumping is exception handlinghttps://zbmath.org/1472.680282021-11-25T18:46:10.358925Z"Robbins, Ed"https://zbmath.org/authors/?q=ai:robbins.edward-henry|robbins.edward-l"King, Andy"https://zbmath.org/authors/?q=ai:king.andy"Howe, Jacob M."https://zbmath.org/authors/?q=ai:howe.jacob-mSummary: ISO Prolog provides catch and throw to realize the control flow of exception handling. This pearl demonstrates that catch and throw are inconspicuously amenable to the implementation of backjumping. In fact, they have precisely the semantics required: rewinding the search to a specific point and carrying of a preserved term to that point. The utility of these properties is demonstrated through an implementation of graph coloring with backjumping and a backjumping SAT solver that applies conflict-driven clause learning.General caching is hard: even with small pageshttps://zbmath.org/1472.680622021-11-25T18:46:10.358925Z"Folwarczný, Lukáš"https://zbmath.org/authors/?q=ai:folwarczny.lukas"Sgall, Jiří"https://zbmath.org/authors/?q=ai:sgall.jiriSummary: \textit{Caching} (also known as \textit{paging}) is a classical problem concerning page replacement policies in two-level memory systems. \textit{General caching} is the variant with pages of different sizes and fault costs. The strong NP-hardness of its two important cases, the \textit{fault model} (each page has unit fault cost) and the \textit{bit model} (each page has the same fault cost as size) has been established. We prove that this already holds when page sizes are bounded by a small constant: The bit and fault models are strongly NP-complete even when page sizes are limited to \(\{1, 2, 3\}\).
Considering only the decision versions of the problems, general caching is equivalent to the \textit{unsplittable flow on a path problem} and therefore our results also improve the hardness results about this problem.
For the entire collection see [Zbl 1326.68015].Effectiveness of structural restrictions for hybrid CSPshttps://zbmath.org/1472.680692021-11-25T18:46:10.358925Z"Kolmogorov, Vladimir"https://zbmath.org/authors/?q=ai:kolmogorov.vladimir"Rolínek, Michal"https://zbmath.org/authors/?q=ai:rolinek.michal"Takhanov, Rustem"https://zbmath.org/authors/?q=ai:takhanov.rustemSummary: Constraint Satisfaction Problem (CSP) is a fundamental algorithmic problem that appears in many areas of Computer Science. It can be equivalently stated as computing a homomorphism \({\mathbf {R}\rightarrow \boldsymbol{\Gamma}}\) between two relational structures, e.g. between two directed graphs. Analyzing its complexity has been a prominent research direction, especially for the \textit{fixed template CSPs} where the right side \(\boldsymbol{\Gamma}\) is fixed and the left side \(\mathbf {R}\) is unconstrained.
Far fewer results are known for the \textit{hybrid} setting that restricts both sides simultaneously. It assumes that \(\mathbf {R}\) belongs to a certain class of relational structures (called a \textit{structural restriction} in this paper). We study which structural restrictions are \textit{effective}, i.e. there exists a fixed template \(\boldsymbol{\Gamma}\) (from a certain class of languages) for which the problem is tractable when \(\mathbf {R}\) is restricted, and NP-hard otherwise. We provide
a characterization for structural restrictions that are \textit{closed under inverse homomorphisms}. The criterion is based on the \textit{chromatic number} of a relational structure defined in this paper; it generalizes the standard chromatic number of a graph.
As our main tool, we use the algebraic machinery developed for fixed template CSPs. To apply it to our case, we introduce a new construction called a ``lifted language''. We also give a characterization for structural restrictions corresponding to minor-closed families of graphs, extend results to certain Valued CSPs (namely conservative valued languages), and state implications for (valued) CSPs with ordered variables and for the maximum weight independent set problem on some restricted families of graphs.
For the entire collection see [Zbl 1326.68015].On structural parameterizations of the matching cut problemhttps://zbmath.org/1472.680712021-11-25T18:46:10.358925Z"Aravind, N. R."https://zbmath.org/authors/?q=ai:aravind.n-r"Kalyanasundaram, Subrahmanyam"https://zbmath.org/authors/?q=ai:kalyanasundaram.subrahmanyam"Kare, Anjeneya Swami"https://zbmath.org/authors/?q=ai:kare.anjeneya-swamiSummary: In an undirected graph, a matching cut is a partition of vertices into two sets such that the edges across the sets induce a matching. The Matching Cut problem is the problem of deciding whether a given graph has a matching cut. The Matching Cut problem can be expressed using a monadic second-order logic (MSOL) formula and hence is solvable in linear time for graphs with bounded tree-width. However, this approach leads to a running time of \(f(\phi,t)n^{O(1)}\), where \(\phi\) is the length of the MSOL formula, \(t\) is the tree-width of the graph and \(n\) is the number of vertices of the graph.
In [Theor. Comput. Sci. 609, Part 2, 328--335 (2016; Zbl 1331.68109)], \textit{D. Kratsch} and \textit{V. B. Le}
asked to give a single exponential algorithm for the Matching Cut problem with tree-width alone as the parameter. We answer this question by giving a \(2^{O(t)}n^{O(1)}\) time algorithm. We also show the tractability of the Matching Cut problem when parameterized by neighborhood diversity and other structural parameters.
For the entire collection see [Zbl 1378.68013].Minimum degree up to local complementation: bounds, parameterized complexity, and exact algorithmshttps://zbmath.org/1472.681102021-11-25T18:46:10.358925Z"Cattanéo, David"https://zbmath.org/authors/?q=ai:cattaneo.david"Perdrix, Simon"https://zbmath.org/authors/?q=ai:perdrix.simonSummary: The local minimum degree of a graph is the minimum degree that can be reached by means of local complementation. For any \(n\), there exist graphs of order \(n\) which have a local minimum degree at least \(0.189n\), or at least \(0.110n\) when restricted to bipartite graphs. Regarding the upper bound, we show that the local minimum degree is at most \(\frac{3}{8}n+o(n)\) for general graphs and \(\frac{n}{4}+o(n)\) for bipartite graphs, improving the known \(\frac{n}{2}\) upper bound. We also prove that the local minimum degree is smaller than half of the vertex cover number (up to a logarithmic term). The local minimum degree problem is NP-Complete and hard to approximate. We show that this problem, even when restricted to bipartite graphs, is in W[2] and FPT-equivalent to the \textsc{EvenSet} problem, whose W[1]-hardness is a long standing open question. Finally, we show that the local minimum degree is computed by a \(\mathcal O^*(1.938^n)\)-algorithm, and a \(\mathcal O^*(1.466^n)\)-algorithm for the
bipartite graphs.
For the entire collection see [Zbl 1326.68015].Sliding token on bipartite permutation graphshttps://zbmath.org/1472.681122021-11-25T18:46:10.358925Z"Fox-Epstein, Eli"https://zbmath.org/authors/?q=ai:fox-epstein.eli"Hoang, Duc A."https://zbmath.org/authors/?q=ai:hoang.duc-a"Otachi, Yota"https://zbmath.org/authors/?q=ai:otachi.yota"Uehara, Ryuhei"https://zbmath.org/authors/?q=ai:uehara.ryuheiSummary: \textsc{Sliding Token} is a natural reconfiguration problem in which vertices of independent sets are iteratively replaced by neighbors. We develop techniques that may be useful in answering the conjecture that \textsc{Sliding Token} is polynomial-time decidable on bipartite graphs. Along the way, we give efficient algorithms for \textsc{Sliding Token} on bipartite permutation and bipartite distance-hereditary graphs.
For the entire collection see [Zbl 1326.68015].On hardness of the joint crossing numberhttps://zbmath.org/1472.681152021-11-25T18:46:10.358925Z"Hliněný, Petr"https://zbmath.org/authors/?q=ai:hlineny.petr"Salazar, Gelasio"https://zbmath.org/authors/?q=ai:salazar.gelasioSummary: The Joint Crossing Number problem asks for a simultaneous embedding of two disjoint graphs into one surface such that the number of edge crossings (between the two graphs) is minimized. It was introduced by
\textit{S. Negami} in [J. Graph Theory 36, No. 1, 8--23 (2001; Zbl 0971.05037)]
in connection with diagonal flips in triangulations of surfaces, and subsequently investigated in a general form for small-genus surfaces. We prove that all of the commonly considered variants of this problem are NP-hard already in the orientable surface of genus 6, by a reduction from a special variant of the anchored crossing number problem of
\textit{S. Cabello} and \textit{B. Mohar} [SIAM J. Comput. 42, No. 5, 1803--1829 (2013; Zbl 1282.05033)].
For the entire collection see [Zbl 1326.68015].Coloring temporal graphshttps://zbmath.org/1472.681172021-11-25T18:46:10.358925Z"Marino, Andrea"https://zbmath.org/authors/?q=ai:marino.andrea"Silva, Ana"https://zbmath.org/authors/?q=ai:silva.ana-carolina|silva.ana-maria-f|silva.ana-t-c|silva.ana-lSummary: A \textit{temporal graph} is a pair \((G,\lambda)\) where \(G\) is a simple graph and \(\lambda\) is a function assigning to each edge time labels telling at which snapshots each edge is active. As recently defined by
\textit{G. B. Mertzios} et al. [J. Comput. Syst. Sci. 120, 97--115 (2021; Zbl 07365382)],
a \textit{temporal coloring} of \((G,\lambda)\) is a sequence of colorings of the vertices of the snapshots such that each edge is properly colored at least once. We first focus on \textit{t-persistent} and \textit{t-occurrent} temporal graphs. The former (resp. the latter) are temporal graphs where each edge of \(G\) stays active for at least \(t\) consecutive (resp. not-necessarily consecutive) snapshots. We study which values of \(t\) make the problem polynomial-time solvable. We also investigate the complexity of the problem when restricted to temporal graphs \((G,\lambda)\) such that \(G\) has bounded treewidth.Retracted article: ``A distance vector similarity metric for complex networks''https://zbmath.org/1472.681182021-11-25T18:46:10.358925Z"Meghanathan, Natarajan"https://zbmath.org/authors/?q=ai:meghanathan.natarajanFrom the text: The editor-in-chief and publisher have retracted this article in agreement with the author. The article was simultaneously submitted to and published online in Computing and in
[J. King Saud Univ. -- Comput. Inf. Sci., \url{doi:10.1016/j.jksuci.2017.06.007}].
The online version of this article contains the full text of the retracted article as electronic supplementary material.The network-untangling problem: from interactions to activity timelineshttps://zbmath.org/1472.681202021-11-25T18:46:10.358925Z"Rozenshtein, Polina"https://zbmath.org/authors/?q=ai:rozenshtein.polina"Tatti, Nikolaj"https://zbmath.org/authors/?q=ai:tatti.nikolaj"Gionis, Aristides"https://zbmath.org/authors/?q=ai:gionis.aristidesSummary: In this paper we study a problem of determining when entities are active based on their interactions with each other. We consider a set of entities \(V\) and a sequence of time-stamped edges \(E\) among the entities. Each edge \((u,v,t)\in E\) denotes an interaction between entities \(u\) and \(v\) at time \(t\). We assume an activity model where each entity is active during at most \(k\) time intervals. An interaction \((u, v, t)\) can be \textit{explained} if at least one of \(u\) or \(v\) are active at time \(t\). Our goal is to reconstruct the \textit{activity intervals} for all entities in the network, so as to explain the observed interactions. This problem, the \textit{network-untangling problem}, can be applied to discover event timelines from complex entity interactions. We provide two formulations of the network-untangling problem: (i) minimizing the total interval length over all entities (sum version), and (ii) minimizing the maximum interval length (max version). We study separately the two problems for \(k=1\) and \(k>1\) activity intervals per entity. For the case \(k=1\), we show that the sum problem is \textbf{NP}-hard, while the max problem can be solved optimally in linear time. For the sum problem we provide efficient algorithms motivated by realistic assumptions. For the case of \(k>1\), we show that both formulations are inapproximable. However, we propose efficient algorithms based on alternative optimization. We complement our study with an evaluation on synthetic and real-world datasets, which demonstrates the validity of our concepts and the good performance of our algorithms.Waypoint routing on bounded treewidth graphshttps://zbmath.org/1472.681212021-11-25T18:46:10.358925Z"Schierreich, Šimon"https://zbmath.org/authors/?q=ai:schierreich.simon"Suchý, Ondřej"https://zbmath.org/authors/?q=ai:suchy.ondrejSummary: In the \textsc{Waypoint Routing Problem} one is given an undirected capacitated and weighted graph \(G\), a source-destination pair \(s,t\in V(G)\) and a set \(W\subseteq V(G)\), of \textit{waypoints}. The task is to find a walk which starts at the source vertex \(s\), visits, in any order, all waypoints, ends at the destination vertex \(t\), respects edge capacities, that is, traverses each edge at most as many times as is its capacity, and minimizes the cost computed as the sum of costs of traversed edges with multiplicities. We study the problem for graphs of bounded treewidth and present a new algorithm for the problem working in \(2^{\mathcal{O}(\mathrm{tw})}\cdot n\) time, significantly improving upon the previously known algorithms. We also show that this running time is optimal for the problem under Exponential Time Hypothesis.Generating random hyperbolic graphs in subquadratic timehttps://zbmath.org/1472.681232021-11-25T18:46:10.358925Z"von Looz, Moritz"https://zbmath.org/authors/?q=ai:von-looz.moritz"Meyerhenke, Henning"https://zbmath.org/authors/?q=ai:meyerhenke.henning"Prutkin, Roman"https://zbmath.org/authors/?q=ai:prutkin.romanSummary: Complex networks have become increasingly popular for modeling various real-world phenomena. Realistic generative network models are important in this context as they simplify complex network research regarding data sharing, reproducibility, and scalability studies. \textit{Random hyperbolic graphs} are a very promising family of geometric graphs with unit-disk neighborhood in the hyperbolic plane. Previous work provided empirical and theoretical evidence that this generative graph model creates networks with many realistic features.
In this work, we provide the first generation algorithm for random hyperbolic graphs with subquadratic running time. We prove a time complexity of \(O((n^{3/2}+m) \log n)\) with high probability for the generation process. This running time is confirmed by experimental data with our implementation. The acceleration stems primarily from the reduction of pairwise distance computations through a polar quadtree, which we adapt to hyperbolic space for this purpose and
which
can be of independent interest. In practice we improve the running time of a previous implementation (which allows more general neighborhoods than the unit disk) by at least two orders of magnitude this way. Networks with billions of edges can now be generated in a few minutes.
For the entire collection see [Zbl 1326.68015].End vertices of graph searches on bipartite graphshttps://zbmath.org/1472.681252021-11-25T18:46:10.358925Z"Zou, Meibiao"https://zbmath.org/authors/?q=ai:zou.meibiao"Wang, Zhifeng"https://zbmath.org/authors/?q=ai:wang.zhifeng"Wang, Jianxin"https://zbmath.org/authors/?q=ai:wang.jianxin"Cao, Yixin"https://zbmath.org/authors/?q=ai:cao.yixinSummary: For a graph search algorithm, the end vertex problem is concerned with which vertices of a graph can be the last visited by this algorithm. We show that for both lexicographic depth-first search and maximum cardinality search, the end vertex problem is NP-complete on bipartite graphs, even if the maximum degree of the graph is bounded.Mining explainable local and global subgraph patterns with surprising densitieshttps://zbmath.org/1472.681332021-11-25T18:46:10.358925Z"Deng, Junning"https://zbmath.org/authors/?q=ai:deng.junning"Kang, Bo"https://zbmath.org/authors/?q=ai:kang.bo"Lijffijt, Jefrey"https://zbmath.org/authors/?q=ai:lijffijt.jefrey"De Bie, Tijl"https://zbmath.org/authors/?q=ai:de-bie.tijlSummary: The connectivity structure of graphs is typically related to the attributes of the vertices. In social networks for example, the probability of a friendship between any pair of people depends on a range of attributes, such as their age, residence location, workplace, and hobbies. The high-level structure of a graph can thus possibly be described well by means of patterns of the form `the subgroup of all individuals with certain properties X are often (or rarely) friends with individuals in another subgroup defined by properties Y', ideally relative to their expected connectivity. Such rules present potentially actionable and generalizable insight into the graph. Prior work has already considered the search for dense subgraphs (`communities') with homogeneous attributes. The first contribution in this paper is to generalize this type of pattern to densities between a \textit{pair of subgroups}, as well as between \textit{all pairs from a set of subgroups that partition the vertices}. Second, we develop a novel information-theoretic approach for quantifying the subjective interestingness of such patterns, by contrasting them with prior information an analyst may have about the graph's connectivity. We demonstrate empirically that in the special case of dense subgraphs, this approach yields results that are superior to the state-of-the-art. Finally, we propose algorithms for efficiently finding interesting patterns of these different types.A survey of community detection methods in multilayer networkshttps://zbmath.org/1472.681432021-11-25T18:46:10.358925Z"Huang, Xinyu"https://zbmath.org/authors/?q=ai:huang.xinyu"Chen, Dongming"https://zbmath.org/authors/?q=ai:chen.dongming"Ren, Tao"https://zbmath.org/authors/?q=ai:ren.tao"Wang, Dongqi"https://zbmath.org/authors/?q=ai:wang.dongqiSummary: Community detection is one of the most popular researches in a variety of complex systems, ranging from biology to sociology. In recent years, there's an increasing focus on the rapid development of more complicated networks, namely multilayer networks. Communities in a single-layer network are groups of nodes that are more strongly connected among themselves than the others, while in multilayer networks, a group of well-connected nodes are shared in multiple layers. Most traditional algorithms can rarely perform well on a multilayer network without modifications. Thus, in this paper, we offer overall comparisons of existing works and analyze several representative algorithms, providing a comprehensive understanding of community detection methods in multilayer networks. The comparison results indicate that the promoting of algorithm efficiency and the extending for general multilayer networks are also expected in the forthcoming studies.An optimal algorithm for tiling the plane with a translated polyominohttps://zbmath.org/1472.682082021-11-25T18:46:10.358925Z"Winslow, Andrew"https://zbmath.org/authors/?q=ai:winslow.andrewSummary: We give a \(O(n)\)-time algorithm for determining whether translations of a polyomino with \(n\) edges can tile the plane. The algorithm is also a \(O(n)\)-time algorithm for enumerating all regular tilings, and we prove that at most \(\varTheta (n)\) such tilings exist.
For the entire collection see [Zbl 1326.68015].A \(4+\epsilon\) approximation for \(k\)-connected subgraphshttps://zbmath.org/1472.682152021-11-25T18:46:10.358925Z"Nutov, Zeev"https://zbmath.org/authors/?q=ai:nutov.zeevSummary: We obtain approximation ratio \(4+\frac{2}{\ell}\approx 4+\frac{4\lg k}{\lg n-\lg k}\) for the (undirected) \(k\)-\textsc{Connected Subgraph} problem, where \(\ell=\lfloor\frac{\lg n-\lg k+1}{2\lg k+1} \rfloor\) is the largest integer such that \(2^{\ell-1}k^{2\ell+1}\leq n\). For large values of \(n\) this improves the ratio 6 of
\textit{J. Cheriyan} and \textit{L. A. Végh} [SIAM J. Comput. 43, No. 4, 1342--1362 (2014; Zbl 1303.05097)]
when \(n\geq k^3\) (the case \(\ell=1)\). Our result implies an fpt-approximation ratio \(4+\epsilon\) that matches (up to the ``\(+\epsilon\)'' term) the best known ratio 4 for \(k=6,7\) for both the general and the easier augmentation versions of the problem. Similar results are shown for the problem of covering an arbitrary symmetric crossing supermodular biset function.Quantitative classification of vortical flows based on topological features using graph matchinghttps://zbmath.org/1472.760362021-11-25T18:46:10.358925Z"Krueger, Paul S."https://zbmath.org/authors/?q=ai:krueger.paul-s"Hahsler, Michael"https://zbmath.org/authors/?q=ai:hahsler.michael"Olinick, Eli V."https://zbmath.org/authors/?q=ai:olinick.eli-v"Williams, Sheila H."https://zbmath.org/authors/?q=ai:williams.sheila-h"Zharfa, Mohammadreza"https://zbmath.org/authors/?q=ai:zharfa.mohammadrezaSummary: Vortical flow patterns generated by swimming animals or flow separation (e.g. behind bluff objects such as cylinders) provide important insight to global flow behaviour such as fluid dynamic drag or propulsive performance. The present work introduces a new method for quantitatively comparing and classifying flow fields using a novel graph-theoretic concept, called a weighted Gabriel graph, that employs critical points of the velocity vector field, which identify key flow features such as vortices, as graph vertices. The edges (connections between vertices) and edge weights of the weighted Gabriel graph encode local geometric structure. The resulting graph exhibits robustness to minor changes in the flow fields. Dissimilarity between flow fields is quantified by finding the best match (minimum difference) in weights of matched graph edges under relevant constraints on the properties of the edge vertices, and flows are classified using hierarchical clustering based on computed dissimilarity. Application of this approach to a set of artificially generated, periodic vortical flows demonstrates high classification accuracy, even for large perturbations, and insensitivity to scale variations and number of periods in the periodic flow pattern. The generality of the approach allows for comparison of flows generated by very different means (e.g. different animal species).Quantum state transfer on a class of circulant graphshttps://zbmath.org/1472.810432021-11-25T18:46:10.358925Z"Pal, Hiranmoy"https://zbmath.org/authors/?q=ai:pal.hiranmoySummary: We study the existence of quantum state transfer on non-integral circulant graphs. We find that continuous-time quantum walks on quantum networks based on certain circulant graphs with \(2^k (k \in \mathbb{Z})\) vertices exhibit pretty good state transfer when there is no external dynamic control over the system. We generalize a few previously known results on pretty good state transfer on circulant graphs, and this way we re-discover all integral circulant graphs on \(2^k\) vertices exhibiting perfect state transfer.On discrete spectrum of a model graph with loop and small edgeshttps://zbmath.org/1472.810692021-11-25T18:46:10.358925Z"Borisov, D. I."https://zbmath.org/authors/?q=ai:borisov.denis-i"Konyrkulzhaeva, M. N."https://zbmath.org/authors/?q=ai:konyrkulzhaeva.maral-nurlanovna"Mukhametrakhimova, A. I."https://zbmath.org/authors/?q=ai:mukhametrakhimova.a-iSummary: We consider a perturbed graph consisting of two infinite edges, a loop, and a glued arbitrary finite graph \(\gamma \epsilon\) with small edges, where \(\gamma \epsilon\) is obtained by \(\epsilon^{-1}\) times contraction of some fixed graph and \(\epsilon\) is a small parameter. On the perturbed graph, we consider the Schrödinger operator whose potential on small edges can singularly depend on \(\epsilon\) with the Kirchhoff condition at internal vertices and the Dirichlet or Neumann condition at the boundary vertices. For the perturbed eigenvalue and the corresponding eigenfunction we prove the holomorphy with respect to \(\epsilon\) and propose a recurrent algorithm for finding all coefficients of their Taylor series.Dependence of the density of states outer measure on the potential for deterministic Schrödinger operators on graphs with applications to ergodic and random modelshttps://zbmath.org/1472.811042021-11-25T18:46:10.358925Z"Hislop, Peter D."https://zbmath.org/authors/?q=ai:hislop.peter-d"Marx, Christoph A."https://zbmath.org/authors/?q=ai:marx.christoph-aSummary: We prove quantitative bounds on the dependence of the density of states on the potential function for discrete, deterministic Schrödinger operators on infinite graphs. While previous results were limited to random Schrödinger operators with independent, identically distributed potentials, this paper develops a deterministic framework, which is applicable to Schrödinger operators independent of the specific nature of the potential. Following ideas by Bourgain and Klein, we consider the density of states outer measure (DOSoM), which is well defined for all (deterministic) Schrödinger operators. We explicitly quantify the dependence of the DOSoM on the potential by proving a modulus of continuity in the \(\ell^\infty \)-norm. The specific modulus of continuity so obtained reflects the geometry of the underlying graph at infinity. For the special case of Schrödinger operators on \(\mathbb{Z}^d\), this implies the Lipschitz continuity of the DOSoM with respect to the potential. For Schrödinger operators on the Bethe lattice, we obtain log-Hölder dependence of the DOSoM on the potential. As an important consequence of our deterministic framework, we obtain a modulus of continuity for the density of states measure (DOSm) of ergodic Schrödinger operators in the underlying potential sampling function. Finally, we recover previous results for random Schrödinger operators on the dependence of the DOSm on the single-site probability measure by formulating this problem in the ergodic framework using the quantile function associated with the random potential.Glauber dynamics for Ising model on convergent dense graph sequenceshttps://zbmath.org/1472.820182021-11-25T18:46:10.358925Z"Acharyya, Rupam"https://zbmath.org/authors/?q=ai:acharyya.rupam"Stefankovic, Daniel"https://zbmath.org/authors/?q=ai:stefankovic.danielSummary: We study the Glauber dynamics for Ising model on (sequences of) dense graphs. We view the dense graphs through the lens of graphons [\textit{L. Lovász} and \textit{B. Szegedy}, J. Comb. Theory, Ser. B 96, No. 6, 933--957 (2006; Zbl 1113.05092)]. For the ferromagnetic Ising model with inverse temperature \(\beta\) on a convergent sequence of graphs \(\{G_n\}\) with limit graphon \(W\) we show fast mixing of the Glauber dynamics if \(\beta\lambda_1(W)<1\) and slow (torpid) mixing if \(\beta\lambda_1(W)>1\) (where \(\lambda_1(W)\) is the largest eigenvalue of the graphon). We also show that in the case \(\beta\lambda_1(W)=1\) there is insufficient information to determine the mixing time (it can be either fast or slow).
For the entire collection see [Zbl 1372.68012].Phase transition for the interchange and quantum Heisenberg models on the Hamming graphhttps://zbmath.org/1472.820212021-11-25T18:46:10.358925Z"Adamczak, Radosław"https://zbmath.org/authors/?q=ai:adamczak.radoslaw"Kotowski, Michał"https://zbmath.org/authors/?q=ai:kotowski.michal"Miłoś, Piotr"https://zbmath.org/authors/?q=ai:milos.piotrSummary: We study a family of random permutation models on the Hamming graph \(H(2,n)\) (i.e., the 2-fold Cartesian product of complete graphs), containing the interchange process and the cycle-weighted interchange process with parameter \(\theta>0\). This family contains the random walk representation of the quantum Heisenberg ferromagnet. We show that in these models the cycle structure of permutations undergoes a phase transition -- when the number of transpositions defining the permutation is \(\le cn^2\), for small enough \(c>0\), all cycles are microscopic, while for more than \(\geq Cn^2\) transpositions, for large enough \(C>0\), macroscopic cycles emerge with high probability.
We provide bounds on values \(C, c\) depending on the parameter \(\theta\) of the model, in particular for the interchange process we pinpoint exactly the critical time of the phase transition. Our results imply also the existence of a phase transition in the quantum Heisenberg ferromagnet on \(H(2,n)\), namely for low enough temperatures spontaneous magnetization occurs, while it is not the case for high temperatures.
At the core of our approach is a novel application of the cyclic random walk, which might be of independent interest. By analyzing explorations of the cyclic random walk, we show that sufficiently long cycles of a random permutation are uniformly spread on the graph, which makes it possible to compare our models to the mean-field case, i.e., the interchange process on the complete graph, extending the approach used earlier by Schramm.Complex systems: features, similarity and connectivityhttps://zbmath.org/1472.900202021-11-25T18:46:10.358925Z"Comin, Cesar H."https://zbmath.org/authors/?q=ai:comin.cesar-henrique"Peron, Thomas"https://zbmath.org/authors/?q=ai:peron.thomas-k-dm"Silva, Filipi N."https://zbmath.org/authors/?q=ai:silva.filipi-nascimento"Amancio, Diego R."https://zbmath.org/authors/?q=ai:amancio.diego-r"Rodrigues, Francisco A."https://zbmath.org/authors/?q=ai:rodrigues.francisco-aparecido"Costa, Luciano da F."https://zbmath.org/authors/?q=ai:costa.luciano-da-fontoura|da-f-costa.lucianoSummary: The increasing interest in complex networks research has been motivated by intrinsic features of this area, such as the generality of the approach to represent and model virtually any discrete system, and the incorporation of concepts and methods deriving from many areas, from statistical physics to sociology, which are often used in an independent way. Yet, for this same reason, it would be desirable to integrate these various aspects into a more coherent and organic framework, which would imply in several benefits normally allowed by the systematization in science, including the identification of new types of problems and the cross-fertilization between fields. More specifically, the identification of the main areas to which the concepts frequently used in complex networks can be applied paves the way to adopting and applying a larger set of concepts and methods deriving from those respective areas. Among the several areas that have been used in complex networks research, pattern recognition, optimization, linear algebra, and time series analysis seem to play a particularly basic and recurrent role. In the present manuscript, we propose a systematic way to integrate the concepts from these diverse areas regarding complex networks research. In order to do so, we start by grouping the multidisciplinary concepts into three main groups of representations, namely features, similarity, and network connectivity. Then we show that several of the analysis and modeling approaches to complex networks can be thought as a composition of maps between these three groups, with emphasis on nine main types of mappings, which are presented and illustrated. For instance, we argue that many models used to generate networks can be understood as a mapping from features to similarity, and then to network connectivity concepts. Such a systematization of principles and approaches also provides an opportunity to review some of the most closely related works in the literature, which is also developed in this article.Symmetry adapted Gram spectrahedrahttps://zbmath.org/1472.900832021-11-25T18:46:10.358925Z"Heaton, Alexander"https://zbmath.org/authors/?q=ai:heaton.alexander"Hoşten, Serkan"https://zbmath.org/authors/?q=ai:hosten.serkan"Shankar, Isabelle"https://zbmath.org/authors/?q=ai:shankar.isabelleA reduction heuristic for the all-colors shortest path problemhttps://zbmath.org/1472.901082021-11-25T18:46:10.358925Z"Carrabs, Francesco"https://zbmath.org/authors/?q=ai:carrabs.francesco"Cerulli, Raffaele"https://zbmath.org/authors/?q=ai:cerulli.raffaele"Raiconi, Andrea"https://zbmath.org/authors/?q=ai:raiconi.andreaSummary: The All-Colors Shortest Path (ACSP) is a recently introduced NP-Hard optimization problem, in which a color is assigned to each vertex of an edge weighted graph, and the aim is to find the shortest path spanning all colors. The solution path can be not simple, that is it is possible to visit multiple times the same vertices if it is a convenient choice. The starting vertex can be constrained (ACSP) or not (ACSP-UE). We propose a reduction heuristic based on the transformation of any ACSP-UE instance into an Equality Generalized Traveling Salesman Problem one. Computational results show the algorithm to outperform the best previously known one.A trust model for spreading gossip in social networks: a multi-type bootstrap percolation modelhttps://zbmath.org/1472.910242021-11-25T18:46:10.358925Z"Bhansali, Rinni"https://zbmath.org/authors/?q=ai:bhansali.rinni"Schaposnik, Laura P."https://zbmath.org/authors/?q=ai:schaposnik.laura-pSummary: We introduce here a multi-type bootstrap percolation model, which we call \textit{}\( \mathcal{T} \)-Bootstrap Percolation \(( \mathcal{T} \)-BP), and apply it to study information propagation in social networks. In this model, a social network is represented by a graph \(G\) whose vertices have different labels corresponding to the type of role the person plays in the network (e.g. a student, an educator etc.). Once an initial set of vertices of \(G\) is randomly selected to be carrying a gossip (e.g. to be infected), the gossip propagates to a new vertex provided it is transmitted by a minimum threshold of vertices with different labels. By considering random graphs, which have been shown to closely represent social networks, we study different properties of the \(\mathcal{T} \)-BP model through numerical simulations, and describe its implications when applied to rumour spread, fake news and marketing strategies.The transsortative structure of networkshttps://zbmath.org/1472.910302021-11-25T18:46:10.358925Z"Ngo, Shin-Chieng"https://zbmath.org/authors/?q=ai:ngo.shin-chieng"Percus, Allon G."https://zbmath.org/authors/?q=ai:percus.allon-g"Burghardt, Keith"https://zbmath.org/authors/?q=ai:burghardt.keith"Lerman, Kristina"https://zbmath.org/authors/?q=ai:lerman.kristinaSummary: Network topologies can be highly non-trivial, due to the complex underlying behaviours that form them. While past research has shown that some processes on networks may be characterized by local statistics describing nodes and their neighbours, such as degree assortativity, these quantities fail to capture important sources of variation in network structure. We define a property called transsortativity that describes correlations among a node's neighbours. Transsortativity can be systematically varied, independently of the network's degree distribution and assortativity. Moreover, it can significantly impact the spread of contagions as well as the perceptions of neighbours, known as the majority illusion. Our work improves our ability to create and analyse more realistic models of complex networks.Using free association networks to extract characteristic patterns of affect dynamicshttps://zbmath.org/1472.910352021-11-25T18:46:10.358925Z"Dover, Yaniv"https://zbmath.org/authors/?q=ai:dover.yaniv"Moore, Zohar"https://zbmath.org/authors/?q=ai:moore.zoharSummary: The dynamics of human affect in day-to-day life are an intrinsic part of human behaviour. Yet, it is difficult to observe and objectively measure how affect evolves over time with sufficient resolution. Here, we suggest an approach that combines free association networks with affect mapping, to gain insight into basic patterns of affect dynamics. This approach exploits the established connection in the literature between association networks and behaviour. Using extant rich data, we find consistent patterns of the dynamics of the valence and arousal dimensions of affect. First, we find that the individuals represented by the data tend to feel a constant pull towards an affect-neutral global equilibrium point in the valence-arousal space. The farther the affect is from that point, the stronger the pull. We find that the drift of affect exhibits high inertia, i.e. is slow-changing, but with occasional discontinuous jumps of valence. We further find that, under certain conditions, another metastable equilibrium point emerges on the network, but one which represents a much more negative and agitated state of affect. Finally, we demonstrate how the affect-coded association network can be used to identify useful or harmful trajectories of associative thoughts that otherwise are hard to extract.Architecture and evolution of semantic networks in mathematics textshttps://zbmath.org/1472.910372021-11-25T18:46:10.358925Z"Christianson, Nicolas H."https://zbmath.org/authors/?q=ai:christianson.nicolas-h"Blevins, Ann Sizemore"https://zbmath.org/authors/?q=ai:blevins.ann-sizemore"Bassett, Danielle S."https://zbmath.org/authors/?q=ai:bassett.danielle-sSummary: Knowledge is a network of interconnected concepts. Yet, precisely how the topological structure of knowledge constrains its acquisition remains unknown, hampering the development of learning enhancement strategies. Here, we study the topological structure of semantic networks reflecting mathematical concepts and their relations in college-level linear algebra texts. We hypothesize that these networks will exhibit structural order, reflecting the logical sequence of topics that ensures accessibility. We find that the networks exhibit strong core-periphery architecture, where a dense core of concepts presented early is complemented with a sparse periphery presented evenly throughout the exposition; the latter is composed of many small modules each reflecting more narrow domains. Using tools from applied topology, we find that the expositional evolution of the semantic networks produces and subsequently fills knowledge gaps, and that the density of these gaps tracks negatively with community ratings of each textbook. Broadly, our study lays the groundwork for future efforts developing optimal design principles for textbook exposition and teaching in a classroom setting.Combinatorics of polymer-based models of early metabolismhttps://zbmath.org/1472.921082021-11-25T18:46:10.358925Z"Weller-Davies, Oliver"https://zbmath.org/authors/?q=ai:weller-davies.oliver"Steel, Mike"https://zbmath.org/authors/?q=ai:steel.michael-anthony"Hein, Jotun"https://zbmath.org/authors/?q=ai:hein.jotun-jSymmetric group of the genetic-code cubes. Effect of the genetic-code architecture on the evolutionary processhttps://zbmath.org/1472.921512021-11-25T18:46:10.358925Z"Sanchez, Robersy"https://zbmath.org/authors/?q=ai:sanchez.robersySummary: The current evidence supports that the genetic code architecture is optimized to minimize the transcriptional and translational errors and to preserve amino-acid hydrophobicity during mutational events. The genetic code is mathematically equivalent to a cube inserted in the ordinary three-dimensional (3D) space, which leads to consistent phylogenetic analyses of DNA protein-coding regions. Herein, the symmetric group \((GC,\circ)\) of the genetic-code cubes is formally developed. Next, it is shown that principal component (PC) scales of amino-acid derived from subsets of the genetic-code cubes are highly correlated with hydrophobicity and other physicochemical amino-acid properties. The effect of this architecture on the evolutionary process was modelled by a Weibull probability distribution to fit the evolutionary mutational cost estimated using amino acid PC-scales optimized on a set of homologous proteins. The application of Weibull model permits the identification of mutational events with high and low probabilities of fixation in gene populations. It is illustrated how this approach conveys a valuable information for \textit{de novo} vaccine design.Designing q-unique DNA sequences with integer linear programs and Euler tours in de Bruijn graphshttps://zbmath.org/1472.921622021-11-25T18:46:10.358925Z"D'Addario, Marianna"https://zbmath.org/authors/?q=ai:daddario.marianna"Kriege, Nils"https://zbmath.org/authors/?q=ai:kriege.nils-m"Rahmann, Sven"https://zbmath.org/authors/?q=ai:rahmann.svenSummary: DNA nanoarchitechtures require carefully designed oligonucleotides with certain non-hybridization guarantees, which can be formalized as the q-uniqueness property on the sequence level. We study the optimization problem of finding a longest q-unique DNA sequence. We first present a convenient formulation as an integer linear program on the underlying De Bruijn graph that allows to flexibly incorporate a variety of constraints; solution times for practically relevant values of q are short. We then provide additional insights into the problem structure using the quotient graph of the De Bruijn graph with respect to the equivalence relation induced by reverse complementarity. Specifically, for odd q the quotient graph is Eulerian, so finding a longest q-unique sequence is equivalent to finding an Euler tour and solved in linear time with respect to the output string length. For even q, self-complementary edges complicate the problem, and the graph has to be Eulerized by deleting a minimum number of edges. Two sub-cases arise, for one of which we present a complete solution, while the other one remains open.
For the entire collection see [Zbl 1472.92001].Computation and visualization of protein topology graphs including ligand informationhttps://zbmath.org/1472.921682021-11-25T18:46:10.358925Z"Schäfer, Tim"https://zbmath.org/authors/?q=ai:schafer.tim"May, Patrick"https://zbmath.org/authors/?q=ai:may.patrick-j-c"Koch, Ina"https://zbmath.org/authors/?q=ai:koch.inaSummary: Motivation: Ligand information is of great interest to understand protein function. Protein structure topology can be modeled as a graph with secondary structure elements as vertices and spatial contacts between them as edges. Meaningful representations of such graphs in 2D are required for the visual inspection, comparison and analysis of protein folds, but their automatic visualization is still challenging. We present an approach which solves this task, supports different graph types and can optionally include ligand contacts. Results: Our method extends the field of protein structure description and visualization by including ligand information. It generates a mathematically unique representation and high-quality 2D plots of the secondary structure of a protein based on a protein-ligand graph. This graph is computed from 3D atom coordinates in PDB files and the corresponding SSE assignments of the DSSP algorithm. The related software supports different notations and allows a rapid visualization of protein structures. It can also export graphs in various standard file formats so they can be used with other software. Our approach visualizes ligands in relationship to protein structure topology and thus represents a useful tool for exploring protein structures. Availability: The software is released under an open source license and available at \url{http://www.bioinformatik.uni-frankfurt.de} in the Software section under Visualization of Protein Ligand Graphs.
For the entire collection see [Zbl 1472.92001].Reconstructing ecological networks with noisy dynamicshttps://zbmath.org/1472.921752021-11-25T18:46:10.358925Z"Freilich, Mara A."https://zbmath.org/authors/?q=ai:freilich.mara-a"Rebolledo, Rolando"https://zbmath.org/authors/?q=ai:rebolledo.rolando"Corcoran, Derek"https://zbmath.org/authors/?q=ai:corcoran.derek"Marquet, Pablo A."https://zbmath.org/authors/?q=ai:marquet.pablo-aSummary: Ecosystems functioning is based on an intricate web of interactions among living entities. Most of these interactions are difficult to observe, especially when the diversity of interacting entities is large and they are of small size and abundance. To sidestep this limitation, it has become common to infer the network structure of ecosystems from time series of species abundance, but it is not clear how well can networks be reconstructed, especially in the presence of stochasticity that propagates through ecological networks. We evaluate the effects of intrinsic noise and network topology on the performance of different methods of inferring network structure from time-series data. Analysis of seven different four-species motifs using a stochastic model demonstrates that star-shaped motifs are differentially detected by these methods while rings are differentially constructed. The ability to reconstruct the network is unaffected by the magnitude of stochasticity in the population dynamics. Instead, interaction between the stochastic and deterministic parts of the system determines the path that the whole system takes to equilibrium and shapes the species covariance. We highlight the effects of long transients on the path to equilibrium and suggest a path forward for developing more ecologically sound statistical techniques.A short proof for graph energy is at least twice of minimum degreehttps://zbmath.org/1472.922652021-11-25T18:46:10.358925Z"Akbari, Saieed"https://zbmath.org/authors/?q=ai:akbari.saieed"Hosseinzadeh, Mohammad Ali"https://zbmath.org/authors/?q=ai:hosseinzadeh.mohammad-aliSummary: The energy \(\mathcal{E}(G)\) of a graph \(G\) is the sum of the absolute values of all eigenvalues of \textit{B. Zhou} [MATCH Commun. Math. Comput. Chem. 55, No. 1, 91--94 (2006; Zbl 1088.05055)] studied
the problem of bounding the graph energy in terms of the minimum degree together with other parameters. He proved his result for quadrangle-free graphs. Recently, \textit{X. Ma} [MATCH Commun. Math. Comput. Chem. 81, No. 2, 393--404 (2019; Zbl 1471.92451)] it is shown that for every graph \(G\), \(\mathcal{E}(G)\ge 2\delta(G)\), where \(\delta(G)\) is the minimum degree of \(G\), and the equality holds if and only if \(G\) is a complete multipartite graph with equal size of parts. Here, we provide a short proof for this result. Also, we give an affirmative answer to a problem proposed in X. Ma [loc. cit.].On the symmetric division deg index of molecular graphshttps://zbmath.org/1472.922662021-11-25T18:46:10.358925Z"Ali, Akbar"https://zbmath.org/authors/?q=ai:ali.akbar"Elumalai, Suresh"https://zbmath.org/authors/?q=ai:elumalai.suresh"Mansour, Toufik"https://zbmath.org/authors/?q=ai:mansour.toufikSummary: The symmetric division deg (SDD) index is one the 148 discrete Adriatic indices, introduced several years ago. The SDD index has already been proved a valuable index in the QSPR/QSAR (quantitative structure-property/activity relationships) studies. In the present paper, we firstly correct an upper bound on the SDD index of molecular trees, reported in the recent paper [MATCH Commun. Math. Comput. Chem. 82, 43--55 (2019)], by giving the best possible upper bound on the SDD index of any molecular \((n,m)\)-graph (a molecular graph with order \(n\) and size \(m)\). We then establish a lower bound on the SDD index of any molecular \((n,m)\)-graph. Finally, by extending a theorem of the aforementioned paper, we characterize the graphs with fifth to ninth minimum SDD indices from the class of all molecular trees having a fixed, but sufficiently large, order.On the complementary equienergetic graphshttps://zbmath.org/1472.922672021-11-25T18:46:10.358925Z"Ali, Akbar"https://zbmath.org/authors/?q=ai:ali.akbar"Elumalai, Suresh"https://zbmath.org/authors/?q=ai:elumalai.suresh"Mansour, Toufik"https://zbmath.org/authors/?q=ai:mansour.toufik"Rostami, Mohammad Ali"https://zbmath.org/authors/?q=ai:rostami.mohammad-aliSummary: Energy of a simple graph \(G\), denoted by \(\mathcal{E}(G)\), is the sum of the absolute values of the eigenvalues of \(G\). Two graphs with the same order and energy are called equienergetic graphs. A graph \(G\) with the property \(G\cong\overline{G}\) is called self-complementary graph, where \(\overline{G}\) denotes the complement of \(G\). Two non-self-complementary equienergetic graphs \(G_1\) and \(G_2\) satisfying the property \(G_1\cong\overline{G_2}\) are called complementary equienergetic graphs. Recently, [\textit{H. S. Ramane} et al., ``[Graphs
equienergetic with their complements'', MATCH Commun. Math. Comput. Chem. 82, 471--480 (2019)] initiated the study of the complementary equienergetic regular graphs and they asked to study the complementary equienergetic non-regular graphs. In this paper, by developing some computer codes and by making use of some software like Nauty, Maple and GraphTea, all the complementary equienergetic graphs with at most 10 vertices as well as all the members of the graph class \(\Omega=\{G: \mathcal{E}(L(G))=\mathcal{E}(\overline{L(G)})\), the order of \(G\) is at most 10\} are determined, where \(L(G)\) denotes the line graph of \(G\). In the cases where we could not find the closed forms of the eigenvalues and energies of the obtained graphs, we verify the graph energies using a high precision computing (2000 decimal places) of Maple. A result about a pair of complementary equienergetic graphs is also given at the end of this paper.Randic index and energyhttps://zbmath.org/1472.922682021-11-25T18:46:10.358925Z"Allem, Luiz Emilio"https://zbmath.org/authors/?q=ai:allem.luiz-emilio"Braga, Rodrigo O."https://zbmath.org/authors/?q=ai:braga.rodrigo-o"Pastine, Adriàn"https://zbmath.org/authors/?q=ai:pastine.adrianSummary: In this paper, we construct families of graphs that satisfy the conjecture for the Randić energy \(RE(G)\) proposed by \textit{I. Gutman} et al. [Linear Algebra Appl. 442, 50--57 (2014; Zbl 1282.05118)] based on the Randić index \(R_{-1}(G)\). More specifically, we provide upper bounds for the energy and we show how to add edges to \(TB\)-graphs that maintain the energy bounded.Short note on Randić energyhttps://zbmath.org/1472.922692021-11-25T18:46:10.358925Z"Allem, Luiz Emilio"https://zbmath.org/authors/?q=ai:allem.luiz-emilio"Molina, Gonzalo"https://zbmath.org/authors/?q=ai:molina.gonzalo"Pastine, Adrián"https://zbmath.org/authors/?q=ai:pastine.adrianSummary: In this paper, we consider the Randić energy \(RE\) of simple connected graphs. We provide upper bounds for \(RE\) in terms of the number of vertices and the nullity of the graph. We present families of graphs that satisfy the Conjecture proposed by \textit{I. Gutman} et al. [Linear Algebra Appl. 442, 50--57 (2014; Zbl 1282.05118)] about the maximal \(RE\). For example, we show that starlikes of odd order satisfy the conjecture.Some statistical results on Randić energy of graphshttps://zbmath.org/1472.922702021-11-25T18:46:10.358925Z"Altındağ, Ilkay"https://zbmath.org/authors/?q=ai:altindag.ilkaySummary: Let \(\widehat{G}_{n,m}\) be the set of all simple graphs with \(n\) vertices and \(m\) edges. In this paper, we establish the average value of the difference between Randić energy of two graphs selected randomly from the set \(\widehat{G}_{n,m}\). By means of this result, we get a criteria for deciding when two graphs are almost Randić equienergetic.A lower bound for the energy of hypoenergetic and non hypoenergetic graphshttps://zbmath.org/1472.922712021-11-25T18:46:10.358925Z"Andrade, Enide"https://zbmath.org/authors/?q=ai:andrade.enide"Carmona, Juan R."https://zbmath.org/authors/?q=ai:carmona.juan-r"Infante, Geraldine"https://zbmath.org/authors/?q=ai:infante.geraldine-m"Robbiano, María"https://zbmath.org/authors/?q=ai:robbiano.mariaSummary: Let \(G\) be a simple undirected graph with n vertices and m edges. The energy of \(G\); \(\mathcal{E}(G)\) corresponds to the sum of its singular values. This work obtains lower bounds for \(\mathcal{E}(G)\) where one of them generalizes a lower bound obtained by [\textit{B. J. McClelland}, ``Properties of the latent roots of a matrix: the estimation of \(\pi\)-electron energies'', J. Chem. Phys. 54, 640--643 (1971)] to the case of graphs with given nullity. An extension to the bipartite case is given and, in this case, it is shown that the lower bound \(2\sqrt{m}\) is improved. The equality cases are characterized. Moreover, a simple lower bound that considers the number of edges and the diameter of \(G\) is derived. A simple lower bound, which improves the lower bound \(2\sqrt{n-1}\), for the energy of trees with \(n\) vertices and diameter \(d\) is also obtained.On the energy of singular and non singular graphshttps://zbmath.org/1472.922722021-11-25T18:46:10.358925Z"Andrade, Enide"https://zbmath.org/authors/?q=ai:andrade.enide"Carmona, Juan R."https://zbmath.org/authors/?q=ai:carmona.juan-r"Poveda, Alex"https://zbmath.org/authors/?q=ai:poveda.alex"Robbiano, María"https://zbmath.org/authors/?q=ai:robbiano.mariaSummary: Let \(G\) be a simple undirected graph with \(n\) vertices, \(m\) edges, adjacency matrix \(A\), largest eigenvalue \(\rho\) and nullity \(\kappa\). The energy of \(G\), \(\mathcal{E}(G)\) is the sum of its singular values. In this work lower bounds for \(\mathcal{E}(G)\) in terms of the coefficient of \(\mu^\kappa\) in the expansion of characteristic polynomial, \(p(\mu)=\det(\mu I-A)\) are obtained. In particular one of the bounds generalizes a lower bound obtained by \textit{K. Ch. Das} et al. [MATCH Commun. Math. Comput. Chem. 70, No. 2, 663--668 (2013; Zbl 1299.05213)] to the case of graphs with given nullity. The bipartite case is also studied obtaining in this case, a sufficient condition to improve the spectral lower bound \(2\rho\). Considering an increasing sequence convergent to \(\rho\) a convergent increasing sequence of lower bounds for the energy of \(G\) is constructed.Geometric-arithmetic index and minimum degree of connected graphshttps://zbmath.org/1472.922732021-11-25T18:46:10.358925Z"Aouchiche, Mustapha"https://zbmath.org/authors/?q=ai:aouchiche.mustapha"El Hallaoui, Issmail"https://zbmath.org/authors/?q=ai:elhallaoui.issmail"Hansen, Pierre"https://zbmath.org/authors/?q=ai:hansen.pierreSummary: In the present paper, we prove lower and upper bounds for each of the ratios \(GA/\delta\), as well as a lower bound on \(GA/\sqrt\delta\), in terms of the order \(n\), over the class of connected graphs on \(n\) vertices, where \(GA\) and \(\delta\) denote the geometric-arithmetic index and the minimum degree, respectively. We also characterize the extremal graphs corresponding to each of those bounds. In order to prove our results, we provide a modified statement of a well-known lower bound on the geometric-arithmetic index in terms of minimum degree.Graphs of maximal energy with fixed maximal degreehttps://zbmath.org/1472.922742021-11-25T18:46:10.358925Z"Arizmendi, Octavio"https://zbmath.org/authors/?q=ai:arizmendi.octavio"Hidalgo, Jorge Fernandez"https://zbmath.org/authors/?q=ai:fernandez-hidalgo.jorgeSummary: We give a bound for the graph energy with given maximal degree in terms of the second and fourth moments of a graph. In the case in which the graph is \(d\)-regular we obtain the bound that is given in \textit{E. R. van Dam} et al. [J. Comb. Theory, Ser. B 107, 123--131 (2014; Zbl 1298.05217)]. through elementary methods.On energy of trees with perfect matchinghttps://zbmath.org/1472.922752021-11-25T18:46:10.358925Z"Ashraf, Firouzeh"https://zbmath.org/authors/?q=ai:ashraf.firouzehSummary: The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. We prove that the energy of a tree on n vertices with a perfect matching and maximum degree at most 3 is greater than \(1:21n-3:23\). This improves some known bounds on the energy of trees.The star sequence and the general first Zagreb indexhttps://zbmath.org/1472.922772021-11-25T18:46:10.358925Z"Bedratyuk, Leonid"https://zbmath.org/authors/?q=ai:bedratyuk.leonid"Savenko, Oleg"https://zbmath.org/authors/?q=ai:savenko.olegSummary: For a simple graph, we introduce a notion of the star sequence and prove that the star sequence and the frequently sequence of a graph are inverses of each other from a combinatorial point of view. As a consequence, we express the general first Zagreb index in terms of the star sequence. Also, we calculate the ordinary generating function and find a linear recurrence relation for the sequence of the general first Zagreb indexes.Exhaustive and metaheuristic exploration of two new structural irregularity measureshttps://zbmath.org/1472.922782021-11-25T18:46:10.358925Z"Boaventura-Netto, Paulo"https://zbmath.org/authors/?q=ai:boaventura-netto.paulo-oswaldo"de Lima, Leonardo"https://zbmath.org/authors/?q=ai:de-lima.leonardo-s"Caporossi, Gilles"https://zbmath.org/authors/?q=ai:caporossi.gillesSummary: Let \(G(V,E)\) be a graph with vertex set \(V\), \(|V|=n\), and edge set \(E\). In this paper, we introduce two new polynomial irregularity measures: \[IRR_m(G)=\frac{(\xi-1)}{n}+\sum_{(i,j)\in E}|(d_i\mu(d_i)-d_j\mu(d_j))|\] and \[IRR_d(G)=\frac{(\xi-1)}{n}+\sum_{(i,j)\in E}\left|\frac{d_i}{\mu(d_i)}-\frac{d_j}{\mu(d_j)}\right|,\] where \(d_i\) is the degree of the vertex \(v_i\in V\), \(\mu(d_i)\) is the degree multiplicity of \(v_i\) in the degree sequence and \(\xi\) is the number of (different) degree values of \(G\). The results of two explorations: one, exhaustive, of the graph sets from 4 to 10 vertices, and other, using AGX-III program on graphs from 11 to 30 vertices, both looking for extremal graphs of two new polynomial irregularity measures are presented. Some discussion on the obtained values and structures is presented. The use of AGX-III allowed us to identify typical structures for the extremal graphs associated with these measures. Some improvements were obtained through the variation of a parameter, with the aid of manual graph building by using an optimal strategy. These structures we built were of the type indicated by the heuristic. For the second measure, AGX-III showed extremal graphs based on unigraphic sequences which generate threshold graphs.On extremal graphs of weighted Szeged indexhttps://zbmath.org/1472.922792021-11-25T18:46:10.358925Z"Bok, Jan"https://zbmath.org/authors/?q=ai:bok.jan"Furtula, Boris"https://zbmath.org/authors/?q=ai:furtula.boris"Jedličkova, Nikola"https://zbmath.org/authors/?q=ai:jedlickova.nikola"Škrekovski, Rist"https://zbmath.org/authors/?q=ai:skrekovski.ristSummary: An extension of the well-known Szeged index was introduced recently, named as weighted Szeged index \((wSz(G))\). This paper is devoted to characterizing the extremal trees and graphs of this new topological invariant. In particular, we proved that the star is a tree having the maximal \(wSz(G)\). Finding a tree with the minimal \(wSz(G)\) is not an easy task to be done. Here, we present the minimal trees up to 25 vertices obtained by computer and describe the regularities which retain in them. Our preliminary computer tests suggest that a tree with the minimal \(wSz(G)\) is, at the same time, the connected graph of the given order that attains the minimal weighted Szeged index. Additionally, it is proven that among the bipartite connected graphs the complete balanced bipartite graph \(K_{\lfloor n/2\rfloor\lceil n/2\rceil}\) attains the maximal \(wSz(G)\). We believe that the \(K_{\lfloor n/2\rfloor\lceil n/2 \rceil}\) is a connected graph of given order that attains the maximum \(wSz(G)\).On the maximal RRR index of trees with many leaveshttps://zbmath.org/1472.922802021-11-25T18:46:10.358925Z"Božović, Vladimir"https://zbmath.org/authors/?q=ai:bozovic.vladimir"Kovijanić Vukićević, Žana"https://zbmath.org/authors/?q=ai:kovijanic-vukicevic.zana"Popivoda, Goran"https://zbmath.org/authors/?q=ai:popivoda.goran"Škrekovski, Riste"https://zbmath.org/authors/?q=ai:skrekovski.riste"Tepeh, Aleksandra"https://zbmath.org/authors/?q=ai:tepeh.aleksandraSummary: The reduced reciprocal Randić (RRR) index of the graph \(G=(V,E)\) is defined as RRR\((G)=\sum_{uv\in E}\sqrt{(d_u-1)(d_v-1)}\), where \(d_u\) and \(d_v\) denote the degrees of vertices \(u\) and \(v\), respectively. We characterize the trees of order \(n\) with \(p\) pendant vertices that maximize RRR index for every \(p\ge\lfloor n/2\rfloor\), which has been identified as an open problem by \textit{X. Ren} et al. [MATCH Commun. Math. Comput. Chem. 76, No. 1, 171--184 (2016; Zbl 1461.05069)]. The main observations which leads to the characterization is that the extremal tree is of height 2.New methods for calculating the degree distance and the Gutman indexhttps://zbmath.org/1472.922812021-11-25T18:46:10.358925Z"Brezovnik, Simon"https://zbmath.org/authors/?q=ai:brezovnik.simon"Tratnik, Niko"https://zbmath.org/authors/?q=ai:tratnik.nikoSummary: In the paper we develop new methods for calculating the two well-known topological indices, the degree distance and the Gutman index. Firstly, we prove that the Wiener index of a double vertex-weighted graph can be computed from the Wiener indices of weighted quotient graphs with respect to a partition of the edge set that is coarser than \(\Theta^*\)-partition. This result immediately gives a method for computing the degree distance of any graph. Next, we express the degree distance and the Gutman index of an arbitrary phenylene by using its hexagonal squeeze and inner dual. In addition, it is shown how these two indices of a phenylene can be obtained from the four quotient trees. Furthermore, reduction theorems for the Wiener index of a double vertex-weighted graph are presented. Finally, a formula for computing the Gutman index of a partial Hamming graph is obtained.On the energy of blossomed starshttps://zbmath.org/1472.922832021-11-25T18:46:10.358925Z"Chen, Wuxian"https://zbmath.org/authors/?q=ai:chen.wuxian"Yan, Weigen"https://zbmath.org/authors/?q=ai:yan.weigenSummary: The energy of a graph is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Given \(p\) integers \(n_1\ge n_2 \ge\cdots\ge n_p\ge 0\), let \(SC(n_1,n_2,\dots,n_p)\) be a tree obtained from a star \(K_{1,p}\) with \(p\) vertices \(v_1,v_2,\dots,v_p\) of degree one by attaching \(n_i\) pendent edges to vertex \(v_i\) for \(1\le i\le p\), which is called a blossomed star. Let \(\mathcal{SC}(n;n_1,n_2,\dots, n_p)=\{SC(n_1,n_2,\dots,n_p)|\sum^p_{i=1}n_i=n-p-1\}\). In this paper, we show that, among all blossomed stars in \(\mathcal{SC} (n;n_1,n_2,\dots,n_p)\), \(SC(n-p-1,0,\dots,0)\) has minimal energy and \(SC\underbrace{(r+1,\dots,r+1}_t,\underbrace{r,\dots,r}_{p-t}\) has maximal energy, where \(n-p-1=pr+t\), \(0\le t\le r-1\).Solution to a problem on the complexity of connective eccentric index of graphshttps://zbmath.org/1472.922842021-11-25T18:46:10.358925Z"Chen, Xiaolin"https://zbmath.org/authors/?q=ai:chen.xiaolin"Lian, Huishu"https://zbmath.org/authors/?q=ai:lian.huishuSummary: The connective eccentricity index of a graph \(G\) is defined as \(\xi^{\text{ce}}(G)=\sum_{v\in V}\frac{\deg(v)}{\text{ecc}(v)}\), where \(\deg(v)\) and ecc\((v)\) are the degree and the eccentricity of the vertex \(v\), respectively. The complexity of the connective eccentricity index of a graph \(G\), denoted by \(C_{\xi^{\text{ce}}}(G)\), is the number of different \(\frac{\deg(v)}{\text{ecc}(v)}\) for any vertex \(v\). \textit{Y. Alizadeh} and \textit{S. Klavžar} [MATCH Commun. Math. Comput. Chem. 76, No. 3, 659--667 (2016; Zbl 1461.05034)] studied this invariant and left a problem ``construct infinite families of graphs \(\{G_n\}_{n\to\infty}\) such that \(C_{\xi^{\text{ce}}}(Gn)= |V(G_n)|=n\)''. In this paper, we solve this problem by giving a construction of \(G_n\) for all \(n\ge 7\).The path and the star as extremal values of vertex-degree-based topological indices among treeshttps://zbmath.org/1472.922852021-11-25T18:46:10.358925Z"Cruz, Roberto"https://zbmath.org/authors/?q=ai:cruz.roberto"Rada, Juan"https://zbmath.org/authors/?q=ai:rada.juanSummary: We denote by \(\mathcal{T}_n\) the set of trees with \(n\) vertices, \(P_n\in\mathcal{T}_n\) is the path tree and \(S_n\in\mathcal{T}_n\) is the star tree. Let \(\varphi\) be a vertex-degree-based topological index defined over \(\mathcal{T}_n\). For any tree \(T\in\mathcal{T}_n\) we find an expression of \(\varphi(T)\) in terms of \(\varphi(P_n)\) and a function \(f_\varphi\) associated to \(\varphi\), in such a way that the derivatives of \(f_\varphi\) over a compact set gives information on when the path \(P_n\) is an extremal value of \(\varphi\) over \(\mathcal{T}_n\), for \(n\ge 3\). Similarly, we present results which give information on when the star \(S_n\) is an extremal value of \(\varphi\) over \(\mathcal{T}_n\), for \(n\ge 3\). As an application, we determine extremal trees for exponential vertex-degree-based topological indices.Note on the girth of a borderenergetic graphhttps://zbmath.org/1472.922862021-11-25T18:46:10.358925Z"Deng, Bo"https://zbmath.org/authors/?q=ai:deng.bo"Huo, Bofeng"https://zbmath.org/authors/?q=ai:huo.bofeng"Li, Xueliang"https://zbmath.org/authors/?q=ai:li.xueliangSummary: The energy \(\mathcal{E}(G)\) of a graph \(G\) is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. If a graph \(G\) of order \(n\) has the same energy as the complete graph \(K_n\), i.e., if \(\mathcal{E}(G)=2(n-1)\), then \(G\) is said to be borderenergetic. In this note, we investigate the girth of a borderenergetic graph \(G\) in the case that \(G\) is a dense graph, and get the result that the girth is 3.On \(L\)-borderenergetic graphs with maximum degree at most 4https://zbmath.org/1472.922872021-11-25T18:46:10.358925Z"Deng, Bo"https://zbmath.org/authors/?q=ai:deng.bo"Li, Xueliang"https://zbmath.org/authors/?q=ai:li.xueliangSummary: If a graph \(G\) of order \(n\) has the same Laplacian energy as the complete graph \(K_n\) does, i.e., if \(\mathcal{LE}(G)=2(n-1)\), then \(G\) is said to be \(L\)-borderenergetic. In this paper, we first prove that there are no 2-connected \(L\)-borderenergetic graphs of order \(n\ge 5\) with maximum degree \(\Delta=3\), which improves the result in \textit{B. Deng} et al. [MATCH Commun. Math. Comput. Chem. 77, No. 3, 607--616 (2017; Zbl 1470.92405)]. Then by surveying the \(L\)-borderenergetic graphs with maximum degree \(\Delta=4\), we present two asymptotically tight bounds on their sizes.(Laplacian) borderenergetic graphs and bipartite graphshttps://zbmath.org/1472.922882021-11-25T18:46:10.358925Z"Deng, Bo"https://zbmath.org/authors/?q=ai:deng.bo"Li, Xueliang"https://zbmath.org/authors/?q=ai:li.xueliang"Zhao, Haixing"https://zbmath.org/authors/?q=ai:zhao.haixingSummary: A graph \(G\) of order \(n\) and size \(m\) is (Laplacian) borderenergetic if it has the same (Laplacian) energy as the complete graph \(K_n\) does. In this paper, we prove that when \(m<\frac{2(n-1)^2}{n}\), a borderenergetic graph is not bipartite. Moreover, for a borderenergetic bipartite graph, we present a lower bound of its largest eigenvalue and an upper bound of its middle eigenvalue, respectively. Analogously, Laplacian borderenergetic bipartite graphs is observed and some asymptotically tight bounds on their first Zagreb indices are shown.Maximum and second maximum of geometric-arithmetic index of tricyclic graphshttps://zbmath.org/1472.922892021-11-25T18:46:10.358925Z"Deng, Hanyuan"https://zbmath.org/authors/?q=ai:deng.hanyuan"Elumalai, Suresh"https://zbmath.org/authors/?q=ai:elumalai.suresh"Balachandran, Selvaraj"https://zbmath.org/authors/?q=ai:balachandran.selvarajSummary: Geometric-arithmetic index is defined as \(GA(G)=\sum_{uv\in E(G)}\frac{2\sqrt{d_ud_v}}{d_u+d_v}\), where \(d_u\) denotes the degree of a vertex \(u\) in \(G\). In this paper, we obtain the first and second maximum values of geometric-arithmetic index for all tricyclic graphs on \(n\) vertices and the corresponding extremal graphs.Efficient computation of trees with minimal atom-bond connectivity index revisitedhttps://zbmath.org/1472.922902021-11-25T18:46:10.358925Z"Dimitrov, Darko"https://zbmath.org/authors/?q=ai:dimitrov.darko"Milosavljevic, Nikola"https://zbmath.org/authors/?q=ai:milosavljevic.nikolaSummary: The atom-bond connectivity (ABC) index is a vertex-degree-based graph invariant that found applications in chemistry. For a graph \(G\), the ABC index is defined as \(\sum_{uv\in E(G)}\sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}}\), where \(d(u)\) is the degree of vertex \(u\) in \(G\) and \(E(G)\) is the set of edges of \(G\). Here, we show several new properties of the degree sequences of the trees with minimal ABC index. We exploit them and some recently proven results of the structure of the minimal-ABC trees to improve the algorithm based on the degree sequence \textit{D. Dimitrov} [Appl. Math. Comput. 224, 663--670 (2013; Zbl 1334.05163)] and \textit{W. Lin} et al. [MATCH Commun. Math. Comput. Chem. 72, No. 3, 699--708 (2014; Zbl 1464.05335)]. The evaluation of the new algorithm shows that it is significantly faster than the known algorithms for identifying the trees with minimal ABC index.The Szeged and Wiener indices of line graphshttps://zbmath.org/1472.922912021-11-25T18:46:10.358925Z"Dobrynin, Andrey A."https://zbmath.org/authors/?q=ai:dobrynin.andrey-aSummary: The Wiener index and Szeged indices are structural descriptors based on distances between vertices of a graph \(G\). The Szeged index appears as generalization of Wiener's formula for acyclic molecules. The concept of line graph, \(L(G)\), for a graph \(G\) has found various applications in chemical research. Some results for the Szeged index of line graphs are presented. In particular, we are interesting in finding of graphs \(G\) with property \(Sz(G)=Sz(L(G))\). The obtained results will be compared with the similar properties of the Wiener index.On correlation of hyperbolic volumes of fullerenes with their propertieshttps://zbmath.org/1472.922922021-11-25T18:46:10.358925Z"Egorov, A. A."https://zbmath.org/authors/?q=ai:egorov.aleksandr-anatolevich|egorov.andrey-aleksandrovich"Vesnin, A. Yu."https://zbmath.org/authors/?q=ai:vesnin.andrei-yu|vesnin.andrei-yurevichSummary: We observe that fullerene graphs are one-skeletons of polyhedra, which can be realized with all dihedral angles equal to \(\pi /2\) in a hyperbolic 3-dimensional space. One of the most important invariants of such a polyhedron is its volume. We are referring this volume as a hyperbolic volume of a fullerene. It is known that some topological indices of graphs of chemical compounds serve as strong descriptors and correlate with chemical properties. We demonstrate that hyperbolic volume of fullerenes correlates with few important topological indices and so, hyperbolic volume can serve as a chemical descriptor too. The correlation between hyperbolic volume of fullerene and its Wiener index suggested few conjectures on volumes of hyperbolic polyhedra. These conjectures are confirmed for the initial list of fullerenes.On extremal properties of general graph entropieshttps://zbmath.org/1472.922932021-11-25T18:46:10.358925Z"Eliasi, Mehdi"https://zbmath.org/authors/?q=ai:eliasi.mehdiSummary: Determining extremal values of graph entropies for some given classes of graphs is intricate, because there is a lack of analytical methods to tackle this particular problem. In this paper we apply the strong mixing variables method for this propose. We characterized the graphs which attain the minimum values of the graph entropy, based on an arbitrary increasing convex information functional, among certain classes of graphs, namely, trees, unicyclic graphs and bicyclic graphs.Correcting the number of \(L\)-borderenergetic graphs of order 9 and 10https://zbmath.org/1472.922942021-11-25T18:46:10.358925Z"Elumalai, Suresh"https://zbmath.org/authors/?q=ai:elumalai.suresh"Rostami, Mohammad Ali"https://zbmath.org/authors/?q=ai:rostami.mohammad-aliSummary: In [\textit{Q. Tao} and \textit{Y. Hou}, MATCH Commun. Math. Comput. Chem. 77, No. 3, 595--606 (2017; Zbl 1466.92285)] 120 non-isomorphic noncomplete \(L\)-borderenergetic graphs of order 10 are reported. By computer search, we find that their total number is 232. The missing graphs are now determined.Maximum Balaban index and sum-Balaban index of tricyclic graphshttps://zbmath.org/1472.922952021-11-25T18:46:10.358925Z"Fang, Wei"https://zbmath.org/authors/?q=ai:fang.wei"Yu, Hongjie"https://zbmath.org/authors/?q=ai:yu.hongjie.1"Gao, Yubin"https://zbmath.org/authors/?q=ai:gao.yubin"Li, Xiaoxin"https://zbmath.org/authors/?q=ai:li.xiaoxin"Jing, Guangming"https://zbmath.org/authors/?q=ai:jing.guangming"Li, Zhongshan"https://zbmath.org/authors/?q=ai:li.zhongshanSummary: Balaban index and Sum-Balaban index were used in various quantitative structure-property relationship and quantitative structure activity relationship studies. In this paper, we characterize the graphs with the maximum Balaban index and maximum Sum-Balaban index of tricyclic graphs.Concordant generation of mark tables and USCI-CF (unit subduced cycle indices with chirality fittingness) tables on the basis of combined-permutation representationshttps://zbmath.org/1472.922962021-11-25T18:46:10.358925Z"Fujita, Shinsaku"https://zbmath.org/authors/?q=ai:fujita.shinsakuSummary: Combined-permutation representations (CPRs) have been used in the GAP (Groups, Algorithms, Programming) system to cover Fujita's USCI (Unit-Subduced-Cycle-Index) approach for symmetry-itemized enumerations of 3D structures. New GAP functions for constructing USCI-CF tables and for constructing the concordant mark tables have been developed to support the practical usage of Fujita's USCI approach. The source code containing these newely-defined functions is attached as an appendix. Concordant generation of mark tables and USCI-CF tables is applied to a CPR (degree \(=4+2)\) based on a tetrahedral skeleton as well as to another CPR (degree \(=10+2)\) based on an adamantane skeleton. Although these CPRs are different in their degrees, they are capable of generating an identical set of a mark table and a USCI table for the point group \(T_d\). The USCI-CF table enables us to generate a list of subduced cycle indices with chirality fittingness (SCI-CFs), which is multiplied by an inverse mark table to give a list of partial cycle indices with chirality fittingness (PCI-CFs). Each element of the list of PCI-CFs gives the PCI-CF for each subgroup. Thereby, symmetry-itemized enumeration based on a tetrahedral skeleton of Td is conducted by means of the PCI method of Fujita's USCI approach. The results are summarized in a tabular form. The relationship between PCI-CFs and CI-CFs is discussed.Standardization of mark tables and USCI-CF (unit subduced cycle indices with chirality fittingness) tables derived from different \(O_h\)-skeletonshttps://zbmath.org/1472.922972021-11-25T18:46:10.358925Z"Fujita, Shinsaku"https://zbmath.org/authors/?q=ai:fujita.shinsakuSummary: Permutation groups (PGs) are derived by respective sets of generators based on various \(O_h\)-skeletons (octahedron, cube, cuboctahedron, truncated octahedron, truncated hexahedron, and rhombic dodecahedron). Each set of generators is selected from combined-permutation representations (CPRs) which stabilize the vertices of each \(O_h\)-skeleton and contain 2-cycles due to mirror-permutations. Thus, the CPR of degree \(8(=6+2)\) for octahedron, the CPR of degree \(10(=8+2)\) for cube, the CPR of degree \(14(=12+2)\) for cuboctahedron, the CPR of degree \(26(=24+2)\) for truncated octahedron, the CPR of degree \(26(=24+2)\) for truncated hexahedron, and the CPR of degree \(16(=14+2)\) for rhombic dodecahedron are regarded as PGs. These PGs are found to be isomorphic to the point group \(O_h\), which is, in turn, composed of symmetry operations based on symmetry elements.Mark tables (tables of marks) of these PGs are different from each other when they are produced by the GAP system. By sorting rows and columns according to the respective non-redundant sets of subgroups (SSGs), however, these mark tables can be standardized to give a standard mark table for the point group \(O_h\). Concordant construction of a standard mark table and a USCI-CF (unit-subduced-cycle-index-with-chirality-fittingness) table for \(O_h\) is discussed by starting from each of the \(O_h\)-skeletons. After a set of SCI-CFs (subduced cycle indices with chirality fittingness) and a set of PCI-CFs (partial cycle indices with chirality fittingness) are generated, symmetry-itemized enumeration based on the PCI method of Fujita's USCI approach (S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry,
Springer-Verlag, Berlin-Heidelberg, 1991) is conducted by starting from each of the \(O_h\)-skeletons.Regular representations and coset representations combined with a Mirror-permutation. Concordant construction of the Mark table and the USCI-CF table of the point group \(T_d\)https://zbmath.org/1472.922982021-11-25T18:46:10.358925Z"Fujita, Shinsaku"https://zbmath.org/authors/?q=ai:fujita.shinsakuSummary: A combined-permutation representation (CPR) of degree \(26(=24+2)\) for a regular representation (RR) of degree 24 is derived algebraically from a multiplication table of the point group \(T_d\), where reflections are explicitly considered in the form of a mirror-permutation of degree 2. Thereby, the standard mark table and the standard USCI-CF table (unit-subduced-cycle-index-with-chirality-fittingness table) are concordantly generated by using the GAP functions \textsf{MarkTableforUSCI} and \textsf{constructUSCITable}, which have been developed by Fujita for the purpose of systematizing the concordant construction. A CPR for each coset representation (CR) \((G_i\setminus) T_d\) is obtained algebraically by means of the GAP function \textsf{CosetRepCF} developed by Fujita (Appendix A). On the other hand, CPRs for CRs are obtained geometrically as permutation groups by considering appropriate skeletons, where the point group \(T_d\) acts on an orbit of \(|T_d|/|G_i|\) positions to be equivalent in a given skeleton so as to generate the CPR of degree \(|T_d|/|G_i|\). An RR as a CPR is obtained by considering a regular body (RB), the \(|T_d|\) positions of which are considered to be governed by RR \((C_1\setminus)T_d\). These geometrically derived CPRs as groups are compared with the corresponding CPRs obtained algebraically.Average distance, connected hub number and connected domination numberhttps://zbmath.org/1472.922992021-11-25T18:46:10.358925Z"Gao, Xiao-Lu"https://zbmath.org/authors/?q=ai:gao.xiaolu"Xu, Shou-Jun"https://zbmath.org/authors/?q=ai:xu.shoujunSummary: Let \(G\) be a connected graph of given order \(n\) and let \(\mu(G)\) denote the average of all the distances between any two distinct vertices in \(G\). The connected hub number \(h_c(G)\) (resp., the connected domination number \(\gamma_c(G))\) of \(G\) is the smallest order of a connected subgraph \(S\) of \(G\) such that each pair of nonadjacent vertices outside \(S\) are joined by a path with all internal vertices in \(S\) (resp., each vertex outside \(S\) is adjacent to one vertex of \(S)\). It is easy to see that \(h_c(G)\le\gamma_c(G)\le h_c(G)+1\). In view of the close relationship between the two invariants, we can partition connected graphs into two classes and according to this partition, give sharp upper bounds on \(\mu(G)\) of the two classes of \(G\) in terms of \(h_c(G)\), respectively, and further characterize the extremal graphs. As a corollary, we give sharp upper bounds on \(\mu(G)\) in terms of \(\gamma_c(G)\), and characterize the extremal graphs. Since these graphs are trees, we further address the problem about 2-connected graphs and give some initial properties and results.On forgotten coindex of chemical graphshttps://zbmath.org/1472.923002021-11-25T18:46:10.358925Z"Ghalavand, Ali"https://zbmath.org/authors/?q=ai:ghalavand.ali"Ashrafi, Ali Reza"https://zbmath.org/authors/?q=ai:ashrafi.ali-rezaSummary: The forgotten coindex of a graph \(G\) is defined as \(\overline{F}(G)=\sum_{uv\ni E(G)}[\deg(u)^2+\deg(v)^2]\), where \(\deg(u)\) is the degree of the vertex \(u\) of \(G\). The main aim of this paper is to present some bounds for the forgotten coindex. An ordering of chemical graphs with respect to the forgotten coindex are also given.Some inequalities between degree- and distance-based topological indices of graphshttps://zbmath.org/1472.923012021-11-25T18:46:10.358925Z"Ghalavand, Ali"https://zbmath.org/authors/?q=ai:ghalavand.ali"Ashrafi, Reza Ali"https://zbmath.org/authors/?q=ai:ashrafi.reza-aliSummary: The aim of this paper is to present some inequalities between degree distance and Gutman index with the Zagreb and reformulated Zagreb indices of graphs.On the inverse problem for the Graovac-Pisanski indexhttps://zbmath.org/1472.923022021-11-25T18:46:10.358925Z"Ghorbani, Modjtaba"https://zbmath.org/authors/?q=ai:ghorbani.modjtaba"Klavžar, Sandi"https://zbmath.org/authors/?q=ai:klavzar.sandi"Rahmani, Shaghayegh"https://zbmath.org/authors/?q=ai:rahmani.shaghayeghSummary: The Graovac-Pisanski index is a topological index that at the same time involves graph distances and symmetries of a given graph. In this note it is proved that the values 0, \(n/2\), \(n\), \(3n/2\), and \(2n\) are the only values of the Graovac-Pisanski index (alias modified Wiener index) from the interval \([0,2n]\) or real numbers that are realizable in the class of all graphs of order \(n\). In the class of trees of order \(n\), only the values \(0,n\), and \(2n\) are realizable.Steiner (revised) Szeged index of graphshttps://zbmath.org/1472.923032021-11-25T18:46:10.358925Z"Ghorbani, Modjtaba"https://zbmath.org/authors/?q=ai:ghorbani.modjtaba"Li, Xueliang"https://zbmath.org/authors/?q=ai:li.xueliang"Maimani, Hamid Reza"https://zbmath.org/authors/?q=ai:maimani.hamid-reza"Mao, Yaping"https://zbmath.org/authors/?q=ai:mao.yaping"Rahmani, Shaghayegh"https://zbmath.org/authors/?q=ai:rahmani.shaghayegh"Rajabi-Parsa, Mina"https://zbmath.org/authors/?q=ai:rajabi-parsa.minaSummary: The Steiner distance in a graph, introduced by \textit{G. Chartrand} et al. Čas. Pštování Mat. 114, No. 4, 399--410 (1989; Zbl 0688.05040)], is a natural generalization of the concept of classical graph distance. For a connected graph \(G\) of order at least 2 and \(S\subseteq V(G)\), the Steiner distance \(d_G(S)\) of the set \(S\) of vertices in \(G\) is the minimum size of a connected subgraph whose vertex set contains or connects \(S\). In this paper, we introduce the concept of the Steiner (revised) Szeged index \((rSz_k(G))Sz_k(G)\) of a graph \(G\), which is a natural generalization of the well-known (revised) Szeged index of chemical use. We determine the \(Sz_k(G)\) for trees in general. Then we give a formula for computing the Steiner Szeged index of a graph in terms of orbits of automorphism group action on the edge set of the graph. Finally, we give sharp upper and lower bounds of \((rSz_k(G))Sz_k(G)\) of a connected graph \(G\), and establish some of its properties. Formulas of \((rSz_k(G))Sz_k(G)\) for large \(k\) are also given in this paper.Ordering of connected bipartite unicyclic graphs with large energieshttps://zbmath.org/1472.923042021-11-25T18:46:10.358925Z"Guo, Ji-Ming"https://zbmath.org/authors/?q=ai:guo.jiming"Qian, Hua"https://zbmath.org/authors/?q=ai:qian.hua"Shan, Hai-Ying"https://zbmath.org/authors/?q=ai:shan.haiying"Wang, Zhi-Wen"https://zbmath.org/authors/?q=ai:wang.zhiwenSummary: The energy of a graph is the sum of the absolute value of the eigenvalues of its adjacency matrix. In this paper, the first \(\lfloor\frac{n-5}{2}\rfloor\) largest energies of connected bipartite unicyclic graphs on \(n\ge 78\) vertices are determined which generalize some known results.Arithmetic-geometric spectral radius and energy of graphshttps://zbmath.org/1472.923052021-11-25T18:46:10.358925Z"Guo, Xin"https://zbmath.org/authors/?q=ai:guo.xin"Gao, Yubin"https://zbmath.org/authors/?q=ai:gao.yubinSummary: Let \(G\) be a graph of order \(n\) with vertex set \(V(G)=\{v_1,v_2, \dots,v_n\}\), and let \(d_i\) be the degree of the vertex \(v_i\) of \(G\) for \(i=1,2,\dots,n\). The arithmetic-geometric adjacency matrix \(A_{ag}(G)\) of \(G\) is defined so that its \((i,j)\)-entry is equal to \(\frac{d_i+d_j} {2\sqrt{d_id_j}}\) if the vertices \(v_i\) and \(v_j\) are adjacent, and 0 otherwise. The arithmetic-geometric spectral radius and arithmetic-geometric energy of \(G\) are the radius and energy of its arithmetic-geometric adjacency matrix, respectively. In this paper, some sharp lower and upper bounds on arithmetic-geometric radius and arithmetic-geometric energy are obtained, and the respective extremal graphs are characterized.Inverse problem for sigma indexhttps://zbmath.org/1472.923062021-11-25T18:46:10.358925Z"Gutman, Ivan"https://zbmath.org/authors/?q=ai:gutman.ivan-m"Togan, Muge"https://zbmath.org/authors/?q=ai:togan.muge"Yurttas, Aysun"https://zbmath.org/authors/?q=ai:yurttas.aysun"Cevik, Ahmet Sinan"https://zbmath.org/authors/?q=ai:cevik.ahmet-sinan"Cangul, Ismail Naci"https://zbmath.org/authors/?q=ai:cangul.ismail-naciSummary: If \(G\) is a (molecular) graph and \(d_v\) the degree of its vertex \(u\), then its sigma index is defined as \(\sigma(G)=\sum(d_u-d_v)^2\), with summation going over all pairs of adjacent vertices. Some basic properties of \(\sigma(G)\) are established. The inverse problem for topological indices is about the existence of a graph having its index value equal to a given non-negative integer. We study the problem for the sigma index and first show that \(\sigma(G)\) is an even integer. Then we construct graph classes in which \(\sigma(G)\) covers all positive even integers. We also study the inverse problem for acyclic, unicyclic, and bicyclic graphs.Comparing multiplicative Wiener index with other graph invariantshttps://zbmath.org/1472.923082021-11-25T18:46:10.358925Z"Hua, Hongbo"https://zbmath.org/authors/?q=ai:hua.hongboSummary: The Wiener index (W) of a connected graph is the sum of distances over all vertex pairs in this graph. As a variant of Wiener index, the multiplicative Wiener index (MW) of a connected graph is the product of distances over all vertex pairs in this graph. The first multiplicative Zagreb index (MZ) of a graph is the product of squares of degree over all vertices in this graph. \textit{K. Ch. Das} and \textit{I. Gutman} [Discrete Appl. Math. 206, 9--14 (2016; Zbl 1335.05057)] proved that for any bipartite connected graph of order \(n\ge 5\), MW\(>\)W. In this paper, we first generalize Das and Gutman's result by proving that if \(G\) is a connected graph of order \(n\ge 5\) and size \(m\) such that \(m\le\lfloor\frac n2\rfloor\lceil\frac n2\rceil-1\), then MW\(>\)W. Second, we compare MW with MZ for trees, and prove that MW\(\ge\)MZ for any tree with at least five vertices. Finally, we compare MW with the independence number for connected graph, and prove that MW is greater than independence number, with only two exceptions.Outerplanar graph data structure: a new computational analysis model of genome rearrangementshttps://zbmath.org/1472.923092021-11-25T18:46:10.358925Z"Jafarzadeh, Nafiseh"https://zbmath.org/authors/?q=ai:jafarzadeh.nafiseh"Iranmanesh, Ali"https://zbmath.org/authors/?q=ai:iranmanesh.aliSummary: The computational study of genome rearrangements is one of the most important research area in computational biology and bioinformatics. In this paper, we define a novel graph data structure as a rearrangement model for whole genome alignment in large scales. This model is capable of realizing non-collinear changes as well as collinear changes. Also we apply our rearrangement graphical model to present a dynamic programing method for alignment of an arbitrary sequence to a pan-genome reference which is encoded as an outerplanar graph. In this method, a gapped alignment is considered where the gaps could be affine, linear or constant.Koolen-Moulton-type upper bounds on the energy of a graphhttps://zbmath.org/1472.923102021-11-25T18:46:10.358925Z"Jahanbani, Akbar"https://zbmath.org/authors/?q=ai:jahanbani.akbar"Rodriguez Zambrano, Jonnathan"https://zbmath.org/authors/?q=ai:rodriguez-zambrano.jonnathanSummary: The energy of a graph \(G\), denoted by \(E(G)\), is defined as the sum of the absolute values of all eigenvalues of \(G\). In this paper, using a Koolen and Moulton demonstration technique, new lower bounds are obtained for the energy of a graph \(G\), that depends only the number of vertices, the number of edges, degree sequence and the spread of adjacency matrix of a graph given.Largest Wiener index of unicyclic graphs with given bipartitionhttps://zbmath.org/1472.923112021-11-25T18:46:10.358925Z"Jiang, Hui"https://zbmath.org/authors/?q=ai:jiang.hui"Li, Wenjing"https://zbmath.org/authors/?q=ai:li.wenjing.1Summary: The Wiener index of a connected graph is the sum of distances between all unordered pairs of its vertices. In this paper, we first identify the graphs whose Wiener index is second largest among trees with given bipartition. Based on this result, the largest Wiener index of unicyclic graphs with given bipartition is determined and the corresponding extremal graphs are characterized.The lower bound of revised Szeged index with respect to tricyclic graphshttps://zbmath.org/1472.923122021-11-25T18:46:10.358925Z"Ji, Shengjin"https://zbmath.org/authors/?q=ai:ji.shengjin"Hong, Yanmei"https://zbmath.org/authors/?q=ai:hong.yanmei"Liu, Mengmeng"https://zbmath.org/authors/?q=ai:liu.mengmeng"Wang, Jianfeng"https://zbmath.org/authors/?q=ai:wang.jianfeng|wang.jianfeng.2|wang.jianfeng.1Summary: The revised Szeged index of a graph is defined as \(Sz^*(G)= \sum_{e=uv\in E}\left(n_u(e)+\frac{n_0(e)}{2}\right)\left(n_v(e)+ \frac{n_0(e)}{2}\right)\), where \(n_u(e)\) and \(n_v(e)\) are, respectively, the number of vertices of \(G\) lying closer to vertex \(u\) than to vertex \(v\) and the number of vertices of \(G\) lying closer to vertex \(v\) than to vertex \(u\), and \(n_0(e)\) is the number of vertices equidistant to \(u\) and \(v\). In the paper, we acquired the lower bound of revised Szeged index among all tricyclic graphs, and the extremal graphs that attain the lower bound are determined.Mathematical aspects of Balaban indexhttps://zbmath.org/1472.923132021-11-25T18:46:10.358925Z"Knor, Martin"https://zbmath.org/authors/?q=ai:knor.martin"Škrekovski, Riste"https://zbmath.org/authors/?q=ai:skrekovski.riste"Tepeh, Aleksandra"https://zbmath.org/authors/?q=ai:tepeh.aleksandraSummary: Balaban index is defined as \(J(G)=\frac{m}{m-n+2}\sum\frac{1}{\sqrt{w(u)\cdot w(v)}}\), where the sum is taken over all edges of a connected graph \(G,n\) and \(m\) are the cardinalities of the vertex and the edge set of \(G\), respectively, and \(w(u)\) (resp. \(w(v))\) denotes the sum of distances from \(u\) (resp. \(v)\) to all the other vertices of \(G\). In the paper we summarize known results, clarify some ambiguities in the literature, and expose problems and conjectures on this molecular descriptor with attractive properties. In parallel, we discuss a related sum-Balaban index.On the difference between Wiener index and Graovac-Pisanski indexhttps://zbmath.org/1472.923142021-11-25T18:46:10.358925Z"Knor, Martin"https://zbmath.org/authors/?q=ai:knor.martin"Škrekovski, Riste"https://zbmath.org/authors/?q=ai:skrekovski.riste"Tepeh, Aleksandra"https://zbmath.org/authors/?q=ai:tepeh.aleksandraSummary: Let \(G\) be a connected graph. The Wiener index of \(G\) is the sum of all distances in \(G\), that is, \(W(G)=\sum_{u,v\in V(G)}\text{dist} (u,v)\). On the other hand, the Graovac-Pisanski index of \(G\) is \(GP(G) =\frac{|V(G)|}{2|\Aut(G)|}\sum_{u\in V(G)}\sum_{\alpha\in\Aut(G)} \text{dist}(u,\alpha(u))\), where \(\Aut(G)\) is the group of automorphisms of \(G\). In this paper we study the difference \(\Delta_W (G)=W(G)-GP(G)\). We show that this difference is nonnegative for trees, but there are graphs \(G\) for which \(\Delta_W(G)\) is negative. We also find infinitely many graphs \(G\) which are not vertex-transitive and yet \(\Delta_W(G)=0\). For trees we completely determine the set of values of \(\Delta_W(G)\).Extremal Wiener index of trees with prescribed path factorshttps://zbmath.org/1472.923162021-11-25T18:46:10.358925Z"Lin, Hong"https://zbmath.org/authors/?q=ai:lin.hongSummary: The Wiener index of a connected graph is defined as the sum of distances between all unordered pairs of its vertices. A graph \(G\) is said to have a \(P_r\)-factor if \(G\) contains a spanning subgraph \(F\) of \(G\) such that every component of \(F\) is a path with \(r\) vertices. In this paper, we characterize the trees which minimize and maximize the Wiener index among all trees on \(kr\) vertices \((k\ge 2,r\ge 2)\) with a \(P_r\)-factor respectively. This generalizes an early result concerning the minimum Wiener index of tree with perfect matchings, which was independently obtained by \textit{Z. Du} and \textit{B. Zhou} [MATCH Commun. Math. Comput. Chem. 63, No. 1, 101--112 (2010; Zbl 1299.05083)] as well as \textit{W. Lin} et al. [MATCH Commun. Math. Comput. Chem. 67, No. 2, 337--345 (2012; Zbl 1289.05376)].Complete characterization of trees with maximal augmented Zagreb indexhttps://zbmath.org/1472.923172021-11-25T18:46:10.358925Z"Lin, Wenshui"https://zbmath.org/authors/?q=ai:lin.wenshui"Dimitrov, Darko"https://zbmath.org/authors/?q=ai:dimitrov.darko"Škrekovski, Riste"https://zbmath.org/authors/?q=ai:skrekovski.risteSummary: The augmented Zagreb index (AZI) of an \(n\)-vertex graph \(G=(V, E)\) is defined as \(AZI(G)=\sum_{v_iv_j\in E}[d_id_j/(d_i+d_j-2)]^3\), where \(V=\{v_0,v_1,\dots,v_{n-1}\}\), \(n\ge 3\), and \(d_i\) denotes the degree of vertex \(v_i\) of \(G\). As a variant of the well-known atom-bond connectivity index, the \(AZI\) was shown to have the best predicting ability for a variety of physicochemical properties among several tested vertex-degree-based topological indices. In \textit{B. Furtula} et al. [J. Math. Chem. 48, No. 2, 370--380 (2010; Zbl 1196.92050)] proposed the problem of characterizing \(n\)-vertex tree(s) with maximal \(AZI\). In the present paper we solve this problem by proving that the \(n\)-vertex balanced double star uniquely maximizes \(AZI\) if \(n\ge 19\).Ordering of bicyclic graphs by matching energyhttps://zbmath.org/1472.923182021-11-25T18:46:10.358925Z"Liu, Xiangxiang"https://zbmath.org/authors/?q=ai:liu.xiangxiang"Wang, Ligong"https://zbmath.org/authors/?q=ai:wang.ligong"Xiao, Peng"https://zbmath.org/authors/?q=ai:xiao.pengSummary: Let \(G\) be a simple graph of order \(n\) and \(\mu_1,\mu_2,\dots, \mu_n\) be the roots of its matching polynomial. The matching energy is defined as the sum \(\sum^n_{i=1}|\mu_i|\), which was introduced by \textit{I. Gutman} and \textit{S. Wagner} [Discrete Appl. Math. 160, No. 15, 2177--2187 (2012; Zbl 1252.05120)]. For bicyclic graphs of order \(n\), the graphs with the first five smallest matching energies are determined and the graph with the second greatest matching energy is also determined in this paper.A model for HOMO-LUMO gap and maximum-weight matchinghttps://zbmath.org/1472.923192021-11-25T18:46:10.358925Z"Li, Xueliang"https://zbmath.org/authors/?q=ai:li.xueliang"Weng, Yindi"https://zbmath.org/authors/?q=ai:weng.yindiSummary: Let \(G_1,G_2\) be simple, connected, invertible graphs. The bridged graph is constructed from \(G_1\) and \(G_2\) by connecting selected pairs of vertices from \(G_1\) and \(G_2\) via new edges. The HOMO-LUMO gap is the difference between the smallest positive and largest negative eigenvalue of its adjacency matrix. The Kekulé pattern coincides with ``perfect matching''. In view of the importance of these two indices and the generality of results, we consider the HOMO-LUMO gap on the edge-weighted bridged graph. And in order to control structure, we present the integer program of the maximum-weight matching with vertex weight. Then we give a model of HOMO-LUMO gap and maximum-weight matching with the weight coefficient \(w\). The numerical results are presented.New bounds on the normalized Laplacian (Randić) energyhttps://zbmath.org/1472.923212021-11-25T18:46:10.358925Z"Maden, A. Dilek"https://zbmath.org/authors/?q=ai:maden.ayse-dilek|maden.a-dilek-gungorSummary: In this paper, we consider the energy of a simple graph with respect to its normalized Laplacian eigenvalues (Randić eigenvalues), which we call the normalized Laplacian energy (also Randić energy). We provide improved upper and lower bounds on these energies for connected (bipartite) graphs.Steiner Gutman indexhttps://zbmath.org/1472.923222021-11-25T18:46:10.358925Z"Mao, Yaping"https://zbmath.org/authors/?q=ai:mao.yaping"Das, Kinkar Ch."https://zbmath.org/authors/?q=ai:das.kinkar-chandraSummary: The concept of Gutman index SGut\((G)\) of a connected graph \(G\) was introduced in [\textit{I. Gutman}, ``Selected properties of the Schultz molecular topological index'', J. Chem. Inf. Comput. Sci. 34, No. 5, 1087--1089 (1994; \url{doi:10.1021/ci00021a009})].
The Steiner distance in a graph, introduced by \textit{G. Chartrand} et al. [Čas. Pěstování Mat. 114, No. 4, 399--410 (1989; Zbl 0688.05040)], is a natural generalization of the concept of classical graph distance. In this paper, we generalize the concept of Gutman index by Steiner distance. The Steiner Gutman \(k\)-index SGut\(_k(G)\) of \(G\) is defined by SGut\(_k(G)=\sum_{\substack{S \subseteq V(G)\\ |S|=k}}\left[\prod_{v\in S}\deg_G(v)\right]d_G(S)\), where \(d_G(S)\) is the Steiner distance of \(S\) and \(\deg_G(v)\) is the degree of \(v\) in \(G\). Expressions for SGut\(_k\) for some special graphs are obtained. We also give sharp upper and lower bounds of SGut\(_k\) of a connected graph, and get the expression of SGut\(_k(G)\) for \(k=n\), \(n-1\). Finally, we compare between \(k\)-center Steiner degree distance \(SDD_k\) and SGut\(_k\) of graphs.A unified approach to bounds for topological indices on trees and applicationshttps://zbmath.org/1472.923232021-11-25T18:46:10.358925Z"Martínez-Pérez, Alvaro"https://zbmath.org/authors/?q=ai:martinez-perez.alvaro"Rodríguez, José M."https://zbmath.org/authors/?q=ai:rodriguez.jose-manuelSummary: The aim of this paper is to use a unified approach in order to obtain new inequalities for a large family of topological indices restricted to trees and to characterize the set of extremal trees with respect to them. Our main results provide upper and lower bounds for a large class of topological indices on trees, fixing or not the maximum degree or the number of pendant vertices. This class includes the variable first Zagreb, the multiplicative second Zagreb, the Narumi-Katayama and the sum lordeg indices. In particular, our results on the sum lordeg index partially solve an open problem on this index.New lower bounds for the geometric-arithmetic indexhttps://zbmath.org/1472.923242021-11-25T18:46:10.358925Z"Martınez-Pérez, Alvaro"https://zbmath.org/authors/?q=ai:martinez-perez.alvaro"Rodrıguez, José M."https://zbmath.org/authors/?q=ai:rodriguez.jose-manuelSummary: The aim of this paper is to obtain new inequalities involving the geometric-arithmetic index \(GA_1\) and characterize graphs extremal with respect to them. Our main results provide lower bounds on \(GA_1(G)\) involving just the minimum and the maximum degree of the graph \(G\).Some inequalities for general sum-connectivity indexhttps://zbmath.org/1472.923262021-11-25T18:46:10.358925Z"Milovanović, I. Z."https://zbmath.org/authors/?q=ai:milovanovic.igor-z"Milovanović, E. I."https://zbmath.org/authors/?q=ai:milovanovic.emina-i"Matejic, M."https://zbmath.org/authors/?q=ai:matejic.marjan-mSummary: Let \(G\) be a simple connected graph with \(n\) vertices and \(m\) edges. Denote by \(d_1\ge d_2\ge\cdots\ge d_n>0\) and \(d(e_1)\ge d(e_2)\ge\cdots\ge d(e_m)>0\) sequences of vertex and edge degrees, respectively. Adjacency of the vertices \(i\) and \(j\) is denoted by \(i\sim j\). A vertex-degree topological index, referred to as general sum-connectivity index, is defined as \(\chi_\alpha=\chi_\alpha(G)=\sum_{i\sim j} (d_i+d_j)^\alpha\), where \(\alpha\) is an arbitrary real number. Lower and upper bounds for \(\chi_\alpha\) are obtained. We also prove one generalization of discrete Kantorovich inequality.The structure of ABC-minimal trees with given number of leaveshttps://zbmath.org/1472.923272021-11-25T18:46:10.358925Z"Mohar, Bojan"https://zbmath.org/authors/?q=ai:mohar.bojanSummary: The atom-bond connectivity (ABC) index is a degree-based molecular descriptor with diverse chemical applications. Recent work of \textit{W. Lin} et al. [MATCH Commun. Math. Comput. Chem. 76, No. 1, 131--140 (2016; Zbl 1461.05063)] gave rise to a conjecture about the minimum possible ABC-index of trees with a fixed number \(t\) of leaves. We show that this conjecture is incorrect and we prove what the correct answer is. It is shown that the extremal tree \(T_t\) is unique for \(t\ge 1195\), it has order \(|T_t|=t+\left\lfloor\frac{t}{10}\right\rfloor+1\) (when \(t\bmod{10}\) is between 0 and 4 or when it is 5, 6, or 7 and \(t\) is sufficiently large) or \(|T_t|=t+\left\lfloor\frac{t}{10}\right\rfloor+2\) (when \(t\bmod{10}\) is 8 or 9 or when it is 5, 6, or 7 and \(t\) is sufficiently small) and its ABC-index is \(\left(\sqrt{\frac{10}{11}}+ \frac{1}{10}\sqrt{\frac{1}{11}}\right)\cdot t+O|(1)\).On peripheral Wiener index: line graphs, Zagreb index, and cut methodhttps://zbmath.org/1472.923282021-11-25T18:46:10.358925Z"Narayankar, Kishori P."https://zbmath.org/authors/?q=ai:narayankar.kishori-p"Kahsay, Afework T."https://zbmath.org/authors/?q=ai:kahsay.afework-t"Klavžar, Sandi"https://zbmath.org/authors/?q=ai:klavzar.sandiSummary: The peripheral Wiener index, \(PW(G)\) is the sum of the distances of all pairs of vertices in the periphery of a graph \(G\). In this paper it is shown that, an arbitrary graph \(G\) is an induced subgraph of a graph \(H\) for which \(PW(H)=PW(L(H))\) holds, where \(L(K)\) is the line graph of \(K\). Using Pell-like equations, infinite families of graphs \(G\) are constructed for which \(PW(G)=PW(L(G))\) holds. A connection between the peripheral Wiener index and the Zagreb index for graphs of small diameter is presented. It is also demonstrated that the partition distance approach applicable to the peripheral Wiener index, making some earlier results as very special cases of this approach.Lower bounds for the Laplacian resolvent energy via majorizationhttps://zbmath.org/1472.923292021-11-25T18:46:10.358925Z"Palacios, José Luis"https://zbmath.org/authors/?q=ai:palacios.jose-luisSummary: Using majorization we find two general lower bounds for the Laplacian resolvent energy of a graph, one in terms of the degrees of the vertices, the other in terms of the number of edges, and some particular lower bounds for \(c\)-cyclic graphs, \(0\le c\le 6\).Energy, matching number and rank of graphshttps://zbmath.org/1472.923302021-11-25T18:46:10.358925Z"Pan, Yingui"https://zbmath.org/authors/?q=ai:pan.yingui"Chen, Jing"https://zbmath.org/authors/?q=ai:chen.jing.4|chen.jing.5|chen.jing.2|chen.jing.3|chen.jing.1"Li, Jianping"https://zbmath.org/authors/?q=ai:li.jianping.1Summary: The energy \(\mathcal{E}(G)\) of a graph \(G\) is the sum of the absolute values of all eigenvalues of the adjacency matrix \(A(G)\) of \(G\). For graphs with vertex-disjoint cycles, we prove that \(\mathcal{E}(G)\ge 2\mu(G)+\frac 35c_1(G)\), where \(\mu(G)\) is the matching number of \(G\) and \(c_1(G)\) denotes the number of odd cycles in \(G\). This result improves the bound \(2\mu(G)+\frac{\sqrt 5}{5}c_1(G)\) obtained by \textit{D. Wong} et al. [Linear Algebra Appl. 549, 276--286 (2018; Zbl 1390.05139)]. Moreover, we give a new lower bound \(\sqrt{41}\) for the energy of connected graphs, which improves the result obtained by Zhou et al.Upper bounds of graph energy in terms of matching numberhttps://zbmath.org/1472.923312021-11-25T18:46:10.358925Z"Pan, Yingui"https://zbmath.org/authors/?q=ai:pan.yingui"Chen, Jing"https://zbmath.org/authors/?q=ai:chen.jing.3|chen.jing.2|chen.jing.1|chen.jing.4|chen.jing.5"Li, Jianping"https://zbmath.org/authors/?q=ai:li.jianping.1Summary: The energy of a graph \(G\), denoted by \(\mathcal{E}(G)\), is the sum of the absolute values of all the eigenvalues of its adjacency matrix \(A(G)\). In this paper, we give the upper bounds of graph energy in terms of matching number and characterize all the extremal graphs achieving the upper bounds.Graphs that minimizing symmetric division deg indexhttps://zbmath.org/1472.923322021-11-25T18:46:10.358925Z"Pan, Yingui"https://zbmath.org/authors/?q=ai:pan.yingui"Li, Jianping"https://zbmath.org/authors/?q=ai:li.jianping.1Summary: Recently, [\textit{B. Furtula} et al., ``Comparative analysis of symmetric division deg index as potentially useful molecular descriptor'', Int. J. Quantum Chem. 118, \#e25659 (2018)] found that the symmetric division deg (SDD) index is a potentially useful molecular descriptor in structure-property and structure-activity relationships studies. In this paper, we determine the \(n\)-vertex trees with the second and the third for \(n\ge 7\), and the fourth for \(n\ge 11\) minimum SDD indices, unicyclic graphs with the first for \(n\ge 3\), the second and the third for \(n\ge 5\), and the fourth for \(n\ge 8\) minimum SDD indices, and bicyclic graphs with the first for \(n\ge 4\), the second for \(n\ge 6\), and the third for \(n\ge 7\) minimum SDD indices. In addition, we establish an upper bound for the \(n\)-vertex chemical trees (the trees with maximum degree no more than four).Exponential vertex-degree-based topological indices and discriminationhttps://zbmath.org/1472.923332021-11-25T18:46:10.358925Z"Rada, Juan"https://zbmath.org/authors/?q=ai:rada.juanSummary: We discuss the discrimination property of vertex-degree-based (VDB) topological indices over \(\mathcal{G}_n\), the set of graphs with \(n\) non-isolated vertices. Concretely, we consider a partition of \(\mathcal{G}_n\) into equivalence classes induced by an equivalence relation on \(\mathcal{G}_n\). Then we say that a VDB topological index has the discrimination property with respect to an equivalence relation on \(\mathcal{G}_n\) if it discriminates equivalence classes. This is a generalization of the usual concept of discrimination in topological indices. If \(\varphi\) is a VDB topological index, then we introduce the \(\widehat{m}_\varphi\)-structure relation on \(\mathcal{G}_n\) and show that the exponential of the best known VDB topological indices have the discrimination property. In view of the nice mathematical properties \(e^\varphi\) has, we study extremal values of Randić's exponential index over \(\mathcal{G}_n\).Zagreb energy and Zagreb Estrada index of graphshttps://zbmath.org/1472.923342021-11-25T18:46:10.358925Z"Rad, Nader Jafari"https://zbmath.org/authors/?q=ai:jafari-rad.nader"Jahanbani, Akbar"https://zbmath.org/authors/?q=ai:jahanbani.akbar"Gutman, Ivan"https://zbmath.org/authors/?q=ai:gutman.ivan-mSummary: Let \(G\) be a graph of order \(n\) with vertices labeled as \(v_1,v_2,\dots,v_n\). Let \(d_i\) be the degree of the vertex \(v_i\), for \(i=1,2,\dots,n\). The (first) Zagreb matrix of \(G\) is the square matrix of order \(n\) whose \((i,j)\)-entry is equal to \(d_i+d_j\) if \(v_i\) is adjacent to \(v_j\), and zero otherwise. We introduce and investigate the Zagreb energy and Zagreb Estrada index of a graph, both base on the eigenvalues of the Zagreb matrix. In addition, we establish upper and lower bounds for these new graph invariants, and relations between them.Graphs equienergetic with their complementshttps://zbmath.org/1472.923362021-11-25T18:46:10.358925Z"Ramane, Harishchandra S."https://zbmath.org/authors/?q=ai:ramane.harishchandra-s"Parvathalu, B."https://zbmath.org/authors/?q=ai:parvathalu.bolle"Patil, Daneshwari D."https://zbmath.org/authors/?q=ai:patil.daneshwari-d"Ashoka, K."https://zbmath.org/authors/?q=ai:ashoka.kSummary: The energy \(E(G)\) of a graph \(G\) is the sum of the absolute values of its eigenvalues. In this paper, we present several classes of non-self-complementary graphs, satisfying \(E(G)=E(\overline{G})\), where \(\overline{G}\) is the complement of \(G\).Graph irregularity indices used as molecular descriptors in QSPR studieshttps://zbmath.org/1472.923382021-11-25T18:46:10.358925Z"Réti, Támas"https://zbmath.org/authors/?q=ai:reti.tamas"Sharafdini, Reza"https://zbmath.org/authors/?q=ai:sharafdini.reza"Dregelyi-Kiss, Ágota"https://zbmath.org/authors/?q=ai:dregelyi-kiss.agota"Haghbin, Hossein"https://zbmath.org/authors/?q=ai:haghbin.hosseinSummary: A comparative study based on the structure-property regression analysis is performed in order to test and evaluate the application possibilities of various graph irregularity indices for the prediction of physico-chemical properties of octane isomers. By restricting attention to single-variable linear regressions, we investigate the stochastic relationships between 18 preselected irregularity indices and 5 physico-chemical properties of octane isomers. These are: boiling point (Bp), standard enthalpy of vaporization (DHVAP), acentric factor (AcenFac), enthalpy of vaporization (HVAP) and entropy. The degree of the intercorrelation was evaluated by traditional correlation coefficients. In physico-chemical applications, it is a widely accepted but theoretically not verified belief is that the use of graph irregularity indices are not to be efficient in QSPR studies of molecular graphs. Our observations refute this preconception. Presenting demonstrative counter-examples it is shown that there exist several irregularity indices by which four octane isomer properties (DHVAP, entropy, AcenFac and HVAP) can be predicted with a good accuracy.On path eigenvalues and path energy of graphshttps://zbmath.org/1472.923402021-11-25T18:46:10.358925Z"Shikare, Maruti M."https://zbmath.org/authors/?q=ai:shikare.maruti-mukinda"Malavadkar, Prashant P."https://zbmath.org/authors/?q=ai:malavadkar.prashant-p"Patekar, Shridhar C."https://zbmath.org/authors/?q=ai:patekar.shridhar-c"Gutman, Ivan"https://zbmath.org/authors/?q=ai:gutman.ivan-mSummary: Given a graph \(G\) with vertex set \(V(G)=\{v_1,v_2,\dots,v_n\}\), we associate to \(G\) a path matrix \(\mathbf{P}\) whose \((i,j)\)-entry is the maximum number of vertex disjoint paths between the vertices \(v_i\) and \(v_j\) when \(i\ne j\) and is zero when \(i=j\). We explore some properties of the eigenvalues and energy of \(\mathbf{P}\).The saturation number of carbon nanocones and nanotubeshttps://zbmath.org/1472.923412021-11-25T18:46:10.358925Z"Short, Taylor"https://zbmath.org/authors/?q=ai:short.taylorSummary: The saturation number of a graph is the cardinality of a smallest maximal matching. This paper presents bounds for the saturation number of carbon nanocones which are asymptotically equal. The same techniques are applied for the saturation number of a certain family of carbon nanotubes, which improve previous results and in one case, yields the exact value.On distance-based topological indices used in architectural researchhttps://zbmath.org/1472.923422021-11-25T18:46:10.358925Z"Stevanović, Sanja"https://zbmath.org/authors/?q=ai:stevanovic.sanja"Stevanović, Dragan"https://zbmath.org/authors/?q=ai:stevanovic.draganSummary: Distance-based topological indices have been used in studies of molecular graphs ever since Harry Wiener introduced his now famous index back in the 1940s, with tens of such indices studied actively in current mathematical chemistry literature. Interestingly, two further distance-based invariants, the difference factor and the intelligibility, have been used in parallel in studies of graphs associated to building and urban plans since the 1970s as well. These invariants are defined in terms of integration values that represent normalized values of the sums of distances from a given vertex to all other vertices in a graph. The difference factor is defined as an entropic measure that quantifies the diversity of the sequence of integration values, while the intelligibility is defined as the Pearson correlation coefficient between sequences of vertex degrees and integration values thus quantifying the extent to which integration values, for which one has to know the structure of the whole graph, can be predicted from vertex degrees, for which one has to know only how many neighbors a vertex has. We perform here a number of computational studies of the difference factor and the intelligibility that reveal to what extent these invariants can be used as topological indices in mathematical chemistry as well.Comparison of resolvent energies of Laplacian matriceshttps://zbmath.org/1472.923432021-11-25T18:46:10.358925Z"Sun, Shaowei"https://zbmath.org/authors/?q=ai:sun.shaowei"Das, Kinkar Chandra"https://zbmath.org/authors/?q=ai:das.kinkar-chandraSummary: Let \(\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_n\) be the eigenvalues of the adjacency matrix of a simple graph \(G\) of order \(n\). A graph-spectrum-based invariant, resolvent energy, put forward by \textit{I. Gutman} et al. [MATCH Commun. Math. Comput. Chem. 75, No. 2, 279--290 (2016; Zbl 1461.05126)], is defined as \(ER(G)= \sum^n_{i=1}(n-\lambda_i)^{-1}\). After that two more resolvent energies defined in the literature, first one is Laplacian resolvent energy \((RL)\) and the second one is signless Laplacian resolvent energy \((RQ)\). In this paper we define normalized Laplacian resolvent energy \((ERN)\), and give some lower and upper bounds on \(ER, RL\) and \(ERN\) of graphs, and characterize the extremal graphs. In particular, we obtain some relations between Laplacian resolvent energy \((RL)\) with popular graph invariants, like Kirchhoff index and the number of spanning trees of graphs. Moreover we compare between resolvent energies of different graph matrices.Ordering unbranched catacondensed benzenoid hydrocarbons by the number of Kekule structureshttps://zbmath.org/1472.923442021-11-25T18:46:10.358925Z"Tang, Yaqian"https://zbmath.org/authors/?q=ai:tang.yaqian"Zuo, Yang"https://zbmath.org/authors/?q=ai:zuo.yang"Tang, Zikai"https://zbmath.org/authors/?q=ai:tang.zikai"Deng, Hanyuan"https://zbmath.org/authors/?q=ai:deng.hanyuanSummary: In this paper, we first determine the order of caterpillar trees (Gutman trees) in terms of the Hosoya index, and then by using a connection between the Hosoya index of caterpillar trees and the number of Kekulé structures of hexagonal chains, polyomino chains, square-hexagonal chains and pentagonal chains, we present the first ten hexagonal chains, polyomino chains, square-hexagonal chains and pentagonal chains with the minimal numbers of Kekulé structures, the first five hexagonal chains, polyomino chains, square-hexagonal chains with the maximal numbers of Kekulé structures among all of these polycyclic molecules with given number of polygons, respectively.Unified extremal results for vertex-degree-based graph invariants with given diameterhttps://zbmath.org/1472.923472021-11-25T18:46:10.358925Z"Yao, Yuedan"https://zbmath.org/authors/?q=ai:yao.yuedan"Liu, Muhuo"https://zbmath.org/authors/?q=ai:liu.muhuo"Gu, Xiaofeng"https://zbmath.org/authors/?q=ai:gu.xiaofengSummary: In this paper, we determine all unified extremal graphs for maximum (resp., minimum) first general Zagreb index \(R^0_\alpha(G)\) for \(\alpha<0\) or \(\alpha>1\) (resp., \(0<\alpha<1)\), maximum (resp., minimum) general first multiplicative Zagreb index \(\prod^\alpha(G)\) for \(\alpha<0\) (resp., \(\alpha>0)\), maximum second multiplicative Zagreb index \(\prod_2(G)\), and minimum first Zagreb coindex \(\overline{M_1 (G)}\) among the class of trees, unicyclic graphs and bicyclic graphs with given diameter, respectively.On the inverse Steiner Wiener problemhttps://zbmath.org/1472.923482021-11-25T18:46:10.358925Z"Zhang, Jie"https://zbmath.org/authors/?q=ai:zhang.jie.5|zhang.jie.3|zhang.jie.1|zhang.jie.4|zhang.jie|zhang.jie.2"Gentry, Matthew"https://zbmath.org/authors/?q=ai:gentry.matthew-l"Wang, Hua"https://zbmath.org/authors/?q=ai:wang.hua.2|wang.hua|wang.hua.1"Jin, Ya-Lei"https://zbmath.org/authors/?q=ai:jin.yalei"Zhang, Xiao-Dong"https://zbmath.org/authors/?q=ai:zhang.xiaodongSummary: The well-known Wiener index is defined as the sum of all distances between pairs of vertices. Motivated from applications in biochemistry the inverse Wiener problem asks, for any given positive integer, the structure that has its Wiener index of this value. This problem was completely solved through a series of studies. When distances are replaced with the Steiner distances, the \(k\)-Steiner Wiener index was introduced recently. Naturally, the inverse Steiner Wiener problem was alsobrought forward. In this paper we show that all but finitely many positive integers are Steiner 3-Wiener indices of some graphs, consequently solving the inverse Steiner Wiener problem for \(k=3\). We then generalize our approach to show, that pending some initial condition, it is likely that all but finitely many positive integers are Steiner \(k\)-Wiener indices of graphs. This is confirmed for small values of \(k\) with the help of computer. We also comment on potential future work.On spectral radius and energy of arithmetic-geometric matrix of graphshttps://zbmath.org/1472.923492021-11-25T18:46:10.358925Z"Zheng, Lu"https://zbmath.org/authors/?q=ai:zheng.lu"Tian, Gui-Xian"https://zbmath.org/authors/?q=ai:tian.guixian"Cui, Shu-Yu"https://zbmath.org/authors/?q=ai:cui.shuyuSummary: Let \(G=(V,E)\) be a simple graph of order \(n\) with vertex set \(V=\{v_1,v_2,\dots,v_n\}\) and edge set \(E\). Denote the sequence of its vertex degrees by \(d_1,d_2,\dots,d_n\). The arithmetic-geometric matrix \(A_{AG}(G)=(a_{i,j}\) of \(G\) is the square matrix of order \(n\), where \(a_{i,j}=\frac 12\left(\sqrt{\frac{d_i}{d_j}}+\sqrt{\frac{d_i} {d_i}}\right)\) if \(v_iv_j\in E(G)\) and 0 otherwise. We give some bounds for the arithmetic-geometric spectral radius in terms of the maximum degree and minimum degree of \(G\), the Randić index \(R_{-1}\), and the first Zagreb index \(M_1\). We also obtain some bounds for the arithmetic-geometric energy in terms of ordinary energy, the sum of 2-degrees of \(G\), symmetric division deg index, the forgotten index, the second Zagreb index, and so on. Finally, some families of arithmetic-geometric equienergetic graphs are constructed by graph operations.Bounds on the general atom-bond connectivity indiceshttps://zbmath.org/1472.923502021-11-25T18:46:10.358925Z"Zheng, Ruiling"https://zbmath.org/authors/?q=ai:zheng.ruiling"Liu, Jianping"https://zbmath.org/authors/?q=ai:liu.jianping"Chen, Jinsong"https://zbmath.org/authors/?q=ai:chen.jinsong"Liu, Bolian"https://zbmath.org/authors/?q=ai:liu.bolianSummary: The general atom-bond connectivity index \((ABC_\alpha)\) of a graph \(G=(V,E)\) is defined as \(ABC_\alpha(G)=\sum_{uv\in E}\left( \frac{d_u+d_v-2}{d_ud_v}\right)^n\), where \(uv\) is an edge of \(G\), \(d_u\) is the degree of the vertex \(u,\alpha\) is an arbitrary nonzero real number, and \(G\) has no isolated \(K_2\) if \(\alpha<0\). In this paper, we will determine the upper bound (resp. the lower bound) of \(ABC_\alpha\) index for \(\alpha\in(0,1]\) (resp. for \(\alpha\in(-\infty,0))\) among all connected graphs with fixed maximum degree, and characterize the corresponding extremal graphs.Convergence of distributed gradient-tracking-based optimization algorithms with random graphshttps://zbmath.org/1472.930062021-11-25T18:46:10.358925Z"Wang, Jiexiang"https://zbmath.org/authors/?q=ai:wang.jiexiang"Fu, Keli"https://zbmath.org/authors/?q=ai:fu.keli"Gu, Yu"https://zbmath.org/authors/?q=ai:gu.yu.1"Li, Tao"https://zbmath.org/authors/?q=ai:li.tao.3Summary: This paper studies distributed convex optimization over a multi-agent system, where each agent owns only a local cost function with convexity and Lipschitz continuous gradients. The goal of the agents is to cooperatively minimize a sum of the local cost functions. The underlying communication networks are modelled by a sequence of random and balanced digraphs, which are not required to be spatially or temporally independent and have any special distributions. The authors use a distributed gradient-tracking-based optimization algorithm to solve the optimization problem. In the algorithm, each agent makes an estimate of the optimal solution and an estimate of the average of all the local gradients. The values of the estimates are updated based on a combination of a consensus method and a gradient tracking method. The authors prove that the algorithm can achieve convergence to the optimal solution at a geometric rate if the conditional graphs are uniformly strongly connected, the global cost function is strongly convex and the step-sizes don't exceed some upper bounds.On the index of convergence of a class of Boolean matrices with structural propertieshttps://zbmath.org/1472.930822021-11-25T18:46:10.358925Z"Ramos, Guilherme"https://zbmath.org/authors/?q=ai:ramos.guilherme"Pequito, Sérgio"https://zbmath.org/authors/?q=ai:pequito.sergio"Caleiro, Carlos"https://zbmath.org/authors/?q=ai:caleiro.carlosSummary: Boolean matrices are of prime importance in the study of discrete event systems (DES), which allow us to model systems across a variety of applications. The index of convergence (i.e. the number of distinct powers of a Boolean matrix) is a crucial characteristic in that it assesses the transient behaviour of the system until reaching a periodic course. In this paper, adopting a graph-theoretic approach, we present bounds for the index of convergence of Boolean matrices for a diverse class of systems, with a certain decomposition. The presented bounds are an extension of the bound on irreducible Boolean matrices, and we provide non-trivial bounds that were unknown for classes of systems. Furthermore, the proposed method is able to determine the bounds in polynomial time. Lastly, we illustrate how the new bounds compare with the previously known bounds and we show their effectiveness in cases such as the benchmark IEEE 5-bus power system.Graph-theoretic stability conditions for Metzler matrices and monotone systemshttps://zbmath.org/1472.931432021-11-25T18:46:10.358925Z"Duan, Xiaoming"https://zbmath.org/authors/?q=ai:duan.xiaoming"Jafarpour, Saber"https://zbmath.org/authors/?q=ai:jafarpour.saber"Bullo, Francesco"https://zbmath.org/authors/?q=ai:bullo.francescoOn ideal homomorphic secret sharing schemes and their decompositionhttps://zbmath.org/1472.940682021-11-25T18:46:10.358925Z"Ghasemi, Fatemeh"https://zbmath.org/authors/?q=ai:ghasemi.fatemeh"Kaboli, Reza"https://zbmath.org/authors/?q=ai:kaboli.reza"Khazaei, Shahram"https://zbmath.org/authors/?q=ai:khazaei.shahram"Parviz, Maghsoud"https://zbmath.org/authors/?q=ai:parviz.maghsoud"Rafiei, Mohammad-Mahdi"https://zbmath.org/authors/?q=ai:rafiei.mohammad-mahdiSummary: \textit{Y. Frankel} and \textit{Y. Desmedt} [Lect. Notes Comput. Sci. 658, 25--34 (1993; Zbl 0787.94016)] introduced a technique that enables one to reduce the secret space of an ideal homomorphic secret sharing scheme (IHSSS) into any of its characteristic subgroups. In this paper, we propose a similar technique to reduce the secret space of IHSSSs called the quotient technique. By using the quotient technique, we show that it is possible to yield an ideal linear scheme from an IHSSS for the same access structure, providing an alternative proof of a recent result by \textit{A. Jafari} and \textit{S. Khazaei} [``Partial secret sharing schemes'', Preprint, \url{https://eprint.iacr.org/2020/448.pdf}]. Moreover, we introduce the concept of decomposition of secret sharing schemes. We give a decomposition for IHSSSs, and as an application, we present a necessary and sufficient condition for an IHSSS to be mixed-linear. Continuing this line of research, we explore the decomposability of some other scheme classes.Generalized threshold secret sharing and finite geometryhttps://zbmath.org/1472.940702021-11-25T18:46:10.358925Z"Ligeti, Peter"https://zbmath.org/authors/?q=ai:ligeti.peter"Sziklai, Peter"https://zbmath.org/authors/?q=ai:sziklai.peter"Takáts, Marcella"https://zbmath.org/authors/?q=ai:takats.marcellaSummary: In the history of secret sharing schemes many constructions are based on geometric objects. In this paper we investigate generalizations of threshold schemes and related finite geometric structures. In particular, we analyse compartmented and hierarchical schemes, and deduce some more general results, especially bounds for special arcs and novel constructions for conjunctive 2-level and 3-level hierarchical schemes.New extremal Type II \(\mathbb Z_4\)-codes of length 32 obtained from Hadamard matriceshttps://zbmath.org/1472.940712021-11-25T18:46:10.358925Z"Ban, Sara"https://zbmath.org/authors/?q=ai:ban.sara"Crnković, Dean"https://zbmath.org/authors/?q=ai:crnkovic.dean"Mravić, Matteo"https://zbmath.org/authors/?q=ai:mravic.matteo"Rukavina, Sanja"https://zbmath.org/authors/?q=ai:rukavina.sanjaProperties and parameters of codes from line graphs of circulant graphshttps://zbmath.org/1472.940852021-11-25T18:46:10.358925Z"Seneviratne, Pani"https://zbmath.org/authors/?q=ai:seneviratne.pani"Melendez, Jennifer D."https://zbmath.org/authors/?q=ai:melendez.jennifer-d"Westbrooks, Alexander N."https://zbmath.org/authors/?q=ai:westbrooks.alexander-nThe rows of an adjacency matrix of a graph may by viewed as codewords of a binary code, and the linear code generated by these may be studied. This is here done for line graphs of circulant graphs. Some good and even optimal (known) linear codes can be obtained in this way. The paper contains some old results on properties of codes of this type as well as a couple of proofs of new results.Asymptotically good homological error correcting codeshttps://zbmath.org/1472.940992021-11-25T18:46:10.358925Z"McCullough, Jason"https://zbmath.org/authors/?q=ai:mccullough.jason"Newman, Heather"https://zbmath.org/authors/?q=ai:newman.heather-aSummary: Let \(\Delta\) be an abstract simplicial complex. We study classical homological error correcting codes associated to \(\Delta \), which generalize the cycle codes of simple graphs. It is well-known that cycle codes of graphs do not yield asymptotically good families of codes. We show that asymptotically good families of codes do exist for homological codes associated to simplicial complexes of dimension at least \(2\). We also prove general bounds and formulas for (co-)cycle and (co-)boundary codes for arbitrary simplicial complexes over arbitrary fields.Analysis of self-equilibrated networks through cellular modellinghttps://zbmath.org/1472.941022021-11-25T18:46:10.358925Z"Aloui, O."https://zbmath.org/authors/?q=ai:aloui.omar"Orden, D."https://zbmath.org/authors/?q=ai:orden.david"Ali, N. Bel Hadj"https://zbmath.org/authors/?q=ai:ali.nizar-bel-hadj"Rhode-Barbarigos, L."https://zbmath.org/authors/?q=ai:rhode-barbarigos.landolfSummary: Network equilibrium models represent a versatile tool for the analysis of interconnected objects and their relationships. They have been widely employed in both science and engineering to study the behaviour of complex systems under various conditions, including external perturbations and damage. In this paper, network equilibrium models are revisited through graph-theory laws and attributes with special focus on systems that can sustain equilibrium in the absence of external perturbations (self-equilibrium). A new approach for the analysis of self-equilibrated networks is proposed; they are modelled as a collection of cells, predefined elementary network units that have been mathematically shown to compose any self-equilibrated network. Consequently, the equilibrium state of complex self-equilibrated systems can be obtained through the study of individual cell equilibria and their interactions. A series of examples that highlight the flexibility of network equilibrium models are included in the paper. The examples attest how the proposed approach, which combines topological as well as geometrical considerations, can be used to decipher the state of complex systems.Minimal descriptions of cyclic memorieshttps://zbmath.org/1472.941042021-11-25T18:46:10.358925Z"Paulsen, Joseph D."https://zbmath.org/authors/?q=ai:paulsen.joseph-d"Keim, Nathan C."https://zbmath.org/authors/?q=ai:keim.nathan-cSummary: Many materials that are out of equilibrium can `learn' one or more inputs that are repeatedly applied. Yet, a common framework for understanding such memories is lacking. Here, we construct minimal representations of cyclic memory behaviours as directed graphs, and we construct simple physically motivated models that produce the same graph structures. We show how a model of worn grass between park benches can produce multiple transient memories -- a behaviour previously observed in dilute suspensions of particles and charge-density-wave conductors -- and the Mullins effect. Isolating these behaviours in our simple model allows us to assess the necessary ingredients for these kinds of memory, and to quantify memory capacity. We contrast these behaviours with a simple Preisach model that produces return-point memory. Our analysis provides a unified method for comparing and diagnosing cyclic memory behaviours across different materials.