Recent zbMATH articles in MSC 05Chttps://zbmath.org/atom/cc/05C2021-05-28T16:06:00+00:00WerkzeugOn the Hartsfield-Ringel hypothesis: connected unigraphs.https://zbmath.org/1459.051402021-05-28T16:06:00+00:00"Kalachev, V. N."https://zbmath.org/authors/?q=ai:kalachev.v-nSummary: The Hartsfield-Ringel hypothesis about the antimagicness of connected graphs is investigated in the class of connected unigraphs. It is proven that all connected unigraphs with no less than three vertices are antimagic.Stable polynomials and crystalline measures.https://zbmath.org/1459.051772021-05-28T16:06:00+00:00"Kurasov, P."https://zbmath.org/authors/?q=ai:kurasov.p-a|kurasov.pavel-b"Sarnak, P."https://zbmath.org/authors/?q=ai:sarnak.peter|sarnak.peter-cSummary: Explicit examples of positive crystalline measures and Fourier quasicrystals are constructed using pairs of stable polynomials, answering several open questions in the area.On the number of cliques in graphs with a forbidden subdivision or immersion.https://zbmath.org/1459.051322021-05-28T16:06:00+00:00"Fox, Jacob"https://zbmath.org/authors/?q=ai:fox.jacob"Wei, Fan"https://zbmath.org/authors/?q=ai:wei.fanCharacterization of hereditary unigraphs based on canonical decomposition.https://zbmath.org/1459.052462021-05-28T16:06:00+00:00"Petrovich, R. A."https://zbmath.org/authors/?q=ai:petrovich.r-aTranslation of the Russian summary: The class of hereditary unigraphs is considered. A graph defined up to isomorphism by a list of degrees of its vertices is called a unigraph. A unigraph is called a hereditary unigraph if each of its vertex-generated subgraphs is also a unigraph. It is known that not all unigraphs are hereditary. In this paper, we characterize hereditary unigraphs on the basis of the theory of canonical decomposition and propose a scheme of the ``being hereditary unigraph'' property that is linear with respect to the number of vertices of the recognition algorithm.A proof for a conjecture on the regularity of binomial edge ideals.https://zbmath.org/1459.130192021-05-28T16:06:00+00:00"Malayeri, Mohammad Rouzbahani"https://zbmath.org/authors/?q=ai:malayeri.mohammad-rouzbahani"Madani, Sara Saeedi"https://zbmath.org/authors/?q=ai:madani.sara-saeedi"Kiani, Dariush"https://zbmath.org/authors/?q=ai:kiani.dariushLet \(\mathbb K\) be a field and let \(G\) be a finite simple graph on the vertex set \(\{1,\dots,n\}\) and the edge set \(E(G)\). The associated binomial edge ideal is the ideal \(J_G=(x_i y_j-x_j y_i \mid \{i,j\} \in E(G))\) of the polynomial ring \(R=\mathbb K [x_1, \dots, x_n, y_1, \dots, y_n]\).
In [\textit{S. Saeedi Madani} and \textit{D. Kiani}, Electron. J. Comb. 20, No. 1, Research Paper P48, 13 p. (2013; Zbl 1278.13007); J. Comb. Theory, Ser. A 139, 80--86 (2016; Zbl 1328.05087)] the second and the third author of the present paper conjectured that the Castelnuovo-Mumford regularity of \(R/J_G\) is bounded above by the number of maximal cliques, \(c(G)\), of \(G\). In this paper this conjecture is settled in full generality; in fact, a better bound is proved.
A set \(\mathcal{H} \subseteq E(G)\) is said to be a clique disjoint edge set in \(G\) if there are no two elements of \(\mathcal{H}\) belonging to the same clique of \(G\). The maximum cardinality of a clique disjoint edge set in \(G\) is denoted by \(\eta(G)\). In this paper it is proved that the Castelnuovo-Mumford regularity of \(R/J_G\) is bounded above by \(\eta(G)\) and this settles the conjecture above because \(\eta(G) \leq c(G)\). Moreover, the authors show a family of graphs \(G_n\) for which \(\lim_{n \rightarrow \infty} (c(G_n)-\eta(G_n)) = \infty\).
Reviewer: Francesco Strazzanti (Torino)Descents on quasi-Stirling permutations.https://zbmath.org/1459.050012021-05-28T16:06:00+00:00"Elizalde, Sergi"https://zbmath.org/authors/?q=ai:elizalde.sergiSummary: Stirling permutations were introduced by \textit{I. Gessel} and \textit{R. P. Stanley} [ibid. 24, 24--33 (1978; Zbl 0378.05006)], who enumerated them by the number of descents to give a combinatorial interpretation of certain polynomials related to Stirling numbers. A natural extension of these permutations are quasi-Stirling permutations, which can be viewed as labeled noncrossing matchings. They were recently introduced by \textit{K. Archer} et al. [Australas. J. Comb. 74, Part 3, 389--407 (2019; Zbl 1419.05004)], motivated by the fact that the Koganov-Janson correspondence between Stirling permutations and labeled increasing plane trees extends to a bijection between quasi-Stirling permutations and the same set of trees without the increasing restriction. In this paper we prove a conjecture of Archer et al. stating that there are \(( n + 1 )^{n - 1}\) quasi-Stirling permutations of size \(n\) having \(n\) descents. More generally, we give the generating function for quasi-Stirling permutations by the number of descents, expressed as a compositional inverse of the generating function of Eulerian polynomials. We also find the analogue for quasi-Stirling permutations of the main result from Gessel and Stanley's paper [loc. cit.]. We prove that the distribution of descents on these permutations is asymptotically normal, and that the roots of the corresponding quasi-Stirling polynomials are all real, in analogy to Bóna's results for Stirling permutations. Finally, we generalize our results to a one-parameter family of permutations that extends \(k\)-Stirling permutations, and we refine them by also keeping track of the number of ascents and the number of plateaus.Trees of tangles in abstract separation systems.https://zbmath.org/1459.053142021-05-28T16:06:00+00:00"Elbracht, Christian"https://zbmath.org/authors/?q=ai:elbracht.christian"Kneip, Jakob"https://zbmath.org/authors/?q=ai:kneip.jakob"Teegen, Maximilian"https://zbmath.org/authors/?q=ai:teegen.maximilianSummary: We prove canonical and non-canonical tree-of-tangles theorems for abstract separation systems that are merely structurally submodular. Our results imply all known tree-of-tangles theorems for graphs, matroids and abstract separation systems with submodular order functions, with greatly simplified and shortened proofs.Critical group structure from the parameters of a strongly regular graph.https://zbmath.org/1459.053512021-05-28T16:06:00+00:00"Ducey, Joshua E."https://zbmath.org/authors/?q=ai:ducey.joshua-e"Duncan, David L."https://zbmath.org/authors/?q=ai:duncan.david-l"Engelbrecht, Wesley J."https://zbmath.org/authors/?q=ai:engelbrecht.wesley-j"Madan, Jawahar V."https://zbmath.org/authors/?q=ai:madan.jawahar-v"Piato, Eric"https://zbmath.org/authors/?q=ai:piato.eric"Shatford, Christina S."https://zbmath.org/authors/?q=ai:shatford.christina-s"Vichitbandha, Angela"https://zbmath.org/authors/?q=ai:vichitbandha.angelaSummary: We give simple arithmetic conditions that force the Sylow \(p\)-subgroup of the critical group of a strongly regular graph to take a specific form. These conditions depend only on the parameters \((v, k, \lambda, \mu)\) of the strongly regular graph under consideration. We give many examples, including how the theory can be used to compute the critical group of Conway's 99-graph and to give an elementary argument that no \(\mathrm{srg}(28, 9, 0, 4)\) exists.Relaxation of the famous NP-complete polar graphs recognition problem leading to the fast polynomial-time algorithm.https://zbmath.org/1459.052722021-05-28T16:06:00+00:00"Petrovich, R. A."https://zbmath.org/authors/?q=ai:petrovich.r-aSummary: Considered is the class of polar graphs and some of its subclasses. Graph \(G=(V,E)\) is called polar if there exist a partition \(VG=A\cup B\) of its vertex set such that all connected components of subgraphs \(G(B)\) and \(\overline{G(A)}\) are cliques.Computation of the biclique partition number for graphs with specific blocks.https://zbmath.org/1459.052672021-05-28T16:06:00+00:00"Lepin, V. V."https://zbmath.org/authors/?q=ai:lepin.v-v"Duginov, O. I."https://zbmath.org/authors/?q=ai:duginov.o-iSummary: The biclique partition number of an undirected graph \(G=(V,E)\) is the smallest number of bicliques (complete bipartite subgraphs) of the graph \(G\) needed to partition the edge set \(E\). We present an efficient algorithm for finding the biclique partition number of a connected graph whose blocks are either complete graphs or complete bipartite graphs or cycles.On optimization problems for graphs and security of digital communications.https://zbmath.org/1459.051352021-05-28T16:06:00+00:00"Ustimenko, V. A."https://zbmath.org/authors/?q=ai:ustimenko.vasiliy-a|ustimenko.vasyl-alexSummary: The most developed field of the classical Extremal Graph Theory studies the maximal size of simple graphs without certain cycles. We discuss recent results on the evaluation of the maximal size of digraphs without certain commutative diagrams that satisfy certain restrictions on the number of inputs and outputs (balanced digraphs or regular directed graphs). These studies are connected with problems of constructing LDPS codes in coding theory and graph based stream ciphers and graph based public keys in cryptography. Finally, we show that the combinatorial optimization problems above can be formulated in the language of integer linear programming.Algebraic graph decomposition theory.https://zbmath.org/1459.052152021-05-28T16:06:00+00:00"Tyshkevich, R. I."https://zbmath.org/authors/?q=ai:tyshkevich.regina-iosifauna"Skums, P. V."https://zbmath.org/authors/?q=ai:skums.p-v"Suzdal', S. V."https://zbmath.org/authors/?q=ai:suzdal.s-vSummary: The survey of the results of the new algebraic theory of graph decomposition (\((P,Q)\)-decomposition) is presented. The examples of the effective applications of this theory to the well-known hard graph-theoretical problems are given.Injective \(L(2,1)\)-coloring of split indecomposable unigraphs.https://zbmath.org/1459.050852021-05-28T16:06:00+00:00"Maksimovich, O. V."https://zbmath.org/authors/?q=ai:maksimovich.oleg-valerievich"Tyshkevich, R. I."https://zbmath.org/authors/?q=ai:tyshkevich.regina-iosifaunaSummary: There are calculated estimates of the parameter \(\lambda'\) for all split indecomposable unigraphs. The corresponding algorithms of their optimal injective \(L(2,1)\)-coloring are developed.Algorithms for solving problems on graphs of bounded pathwidth.https://zbmath.org/1459.053162021-05-28T16:06:00+00:00"Lepin, V. V."https://zbmath.org/authors/?q=ai:lepin.v-vSummary: In this paper we present a space-efficient algorithmic framework for solving construction variants of problems on graphs with bounded pathwidth. Algorithms for solving the \(\lambda\)-path cover problems and the problem of finding a minimum-weight Hamiltonian cycle on this type of graphs in \(O(n\log n)\) time with \(O(1)\) additional memory are given. An algorithm for solving 3-SAT with formulas having pathwidth-\(k\) interaction graphs in \(O(n\log n)\) time with \(O(1)\) additional memory is present.Approximation algorithms for approximating graphs with bounded number of connected components.https://zbmath.org/1459.053152021-05-28T16:06:00+00:00"Il'ev, V. P."https://zbmath.org/authors/?q=ai:ilev.victor-petrovich"Il'eva, S. D."https://zbmath.org/authors/?q=ai:ileva.s-dSummary: The following graph approximation problem is studied: given an undirected graph, one has to find the nearest graph on the same vertex set all connected components of which are the complete graphs. The distance between graphs is equal to the number of noncoinciding edges. The brief survey of the known results is given. The constant-factor approximation algorithms for two versions of the graph approximation problem are proposed.On graphs the neighbourhoods of whose vertices are pseudo-geometric graphs for \(\mathrm{GQ}(3,3)\).https://zbmath.org/1459.052762021-05-28T16:06:00+00:00"Gutnova, A. K."https://zbmath.org/authors/?q=ai:gutnova.alina-kazbekovna"Makhnev, A. A."https://zbmath.org/authors/?q=ai:makhnev.aleksandr-aSummary: Let \(\mathcal{F}\) be a class of graphs. We call a graph \(\Gamma\) a locally \(\mathcal{F}\)-graph if \([a]\in\mathcal{F}\) for every vertex \(a\) of \(\Gamma.\) Earlier for the class \(\mathcal{F}\) consisting of pseudogeometrical graphs for \(\mathrm{pG}_{s-2}(s,t)\) the study of locally \(\mathcal{F}\)-graphs was reduced to investigating locally pseudo \(\mathrm{GQ}(3,t)\)-graphs, \(t\in\{3,5\}\). A description of completely regular locally pseudo \(\mathrm{GQ}(3,3)\)-graphs is obtained in the paper.Domination numbers and noncover complexes of hypergraphs.https://zbmath.org/1459.052252021-05-28T16:06:00+00:00"Kim, Jinha"https://zbmath.org/authors/?q=ai:kim.jinha"Kim, Minki"https://zbmath.org/authors/?q=ai:kim.minkiSummary: Let \(\mathcal{H}\) be a hypergraph on a finite set \(V\). A cover of \(\mathcal{H}\) is a set of vertices that meets all edges of \(\mathcal{H} \). If \(W\) is not a cover of \(\mathcal{H} \), then \(W\) is said to be a noncover of \(\mathcal{H} \). The noncover complex of \(\mathcal{H}\) is the abstract simplicial complex whose faces are the noncovers of \(\mathcal{H} \). In this paper, we study homological properties of noncover complexes of hypergraphs. In particular, we obtain an upper bound on their Leray numbers. The bound is in terms of hypergraph domination numbers. Also, our proof idea is applied to compute the homotopy type of the noncover complexes of certain uniform hypergraphs, called tight paths and tight cycles. This extends to hypergraphs known results on graphs.Chromatic symmetric functions from the modular law.https://zbmath.org/1459.053342021-05-28T16:06:00+00:00"Abreu, Alex"https://zbmath.org/authors/?q=ai:abreu.alex-c"Nigro, Antonio"https://zbmath.org/authors/?q=ai:nigro.antonioSummary: In this article we show how to compute the chromatic quasisymmetric function of indifference graphs from the modular law introduced in [\textit{M. Guay-Paquet}, ``A modular relation for the chromatic symmetric functions of (3+1)-free posets'', Preprint, \url{arXiv:1306.2400}]. We provide an algorithm which works for any function that satisfies this law, such as unicellular LLT polynomials. When the indifference graph has bipartite complement it reduces to a planar network, in this case, we prove that the coefficients of the chromatic quasisymmetric function in the elementary basis are positive unimodal polynomials and characterize them as certain \(q\)-hit numbers (up to a factor). Finally, we discuss the logarithmic concavity of the coefficients of the chromatic quasisymmetric function.Counting independent sets in regular hypergraphs.https://zbmath.org/1459.052292021-05-28T16:06:00+00:00"Balogh, József"https://zbmath.org/authors/?q=ai:balogh.jozsef"Bollobás, Béla"https://zbmath.org/authors/?q=ai:bollobas.bela"Narayanan, Bhargav"https://zbmath.org/authors/?q=ai:narayanan.bhargav-pSummary: Amongst \(d\)-regular \(r\)-uniform hypergraphs on \(n\) vertices, which ones have the largest number of independent sets? While the analogous problem for graphs (originally raised by Granville) is now well-understood, it is not even clear what the correct general conjecture ought to be; our goal here is to propose such a generalisation. Lending credence to our conjecture, we verify it within the class of `quasi-bipartite' hypergraphs (a generalisation of bipartite graphs that seems natural in this context) by adopting the entropic approach of \textit{J. Kahn} [Comb. Probab. Comput. 10, No. 3, 219--237 (2001; Zbl 0985.60088)].Maximum bisections of graphs without short even cycles.https://zbmath.org/1459.052692021-05-28T16:06:00+00:00"Lin, Jing"https://zbmath.org/authors/?q=ai:lin.jing"Zeng, Qinghou"https://zbmath.org/authors/?q=ai:zeng.qinghouSummary: A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differ by at most 1, and its size is the number of edges which go across the two parts. In this paper, motivated by several known results of \textit{N. Alon} et al. [J. Comb. Theory, Ser. B 88, No. 2, 329--346 (2003; Zbl 1030.05060)] about Max-Cut, we study maximum bisections of graphs without short even cycles. Let \(G\) be a graph on \(m\) edges without cycles of length 4 and 6. We first extend a well-known result of \textit{J. B. Shearer} [Random Struct. Algorithms 3, No. 2, 223--226 (1992; Zbl 0765.05057)] on maximum cuts to bisections and show that if \(G\) has a perfect matching and degree sequence \(d_1, \ldots, d_n\), then \(G\) admits a bisection of size at least \(m / 2 + \Omega ( \sum_{i = 1}^n \sqrt{ d_i} )\). This is tight for certain polarity graphs. Together with a technique of \textit{V. Nikiforov} [Comb. Probab. Comput. 10, No. 6, 543--555 (2001; Zbl 0996.05077)], we prove that if \(G\) also contains no cycle of length \(2 k \geq 6\) then \(G\) either has a large bisection or is nearly bipartite. As a corollary, if \(G\) has a matching of size \(\Theta(n)\), then \(G\) admits a bisection of size at least \(m / 2 + \Omega ( m^{( 2 k + 1 ) / ( 2 k + 2 )} )\) and that this is tight for \(2 k \in \{6, 10 \} \); if \(G\) has a matching of size \(o(n)\), then the bound remains valid for \(G\) with minimum degree at least 2.The configuration model for partially directed graphs.https://zbmath.org/1459.820592021-05-28T16:06:00+00:00"Spricer, Kristoffer"https://zbmath.org/authors/?q=ai:spricer.kristoffer"Britton, Tom"https://zbmath.org/authors/?q=ai:britton.tomSummary: The configuration model was originally defined for undirected networks and has recently been extended to directed networks. Many empirical networks are however neither undirected nor completely directed, but instead usually partially directed meaning that certain edges are directed and others are undirected. In the paper we define a configuration model for such networks where vertices have in-, out-, and undirected degrees that may be dependent. We prove conditions under which the resulting degree distributions converge to the intended degree distributions. The new model is shown to better approximate several empirical networks compared to undirected and completely directed networks.On triangles in derangement graphs.https://zbmath.org/1459.051222021-05-28T16:06:00+00:00"Meagher, Karen"https://zbmath.org/authors/?q=ai:meagher.karen"Razafimahatratra, Andriaherimanana Sarobidy"https://zbmath.org/authors/?q=ai:razafimahatratra.andriaherimanana-sarobidy"Spiga, Pablo"https://zbmath.org/authors/?q=ai:spiga.pabloSummary: Given a permutation group \(G\), the derangement graph \(\Gamma_G\) of \(G\) is the Cayley graph with connection set the set of all derangements of \(G\). We prove that, when \(G\) is transitive of degree at least \(3\), \(\Gamma_G\) contains a triangle. The motivation for this work is the question of how large can be the ratio of the independence number of \(\Gamma_G\) to the size of the stabilizer of a point in \(G\). We give examples of transitive groups where this ratio is maximum.Minimal graphs with disjoint dominating and paired-dominating sets.https://zbmath.org/1459.052402021-05-28T16:06:00+00:00"Henning, Michael A."https://zbmath.org/authors/?q=ai:henning.michael-anthony"Topp, Jerzy"https://zbmath.org/authors/?q=ai:topp.jerzySummary: A subset \(D \subseteq V_G\) is a dominating set of \(G\) if every vertex in \(V_G - D\) has a neighbor in \(D\), while \(D\) is a paired-dominating set of \(G\) if \(D\) is a dominating set and the subgraph induced by \(D\) contains a perfect matching. A graph \(G\) is a \(DPDP\)-graph if it has a pair \((D, P)\) of disjoint sets of vertices of \(G\) such that \(D\) is a dominating set and \(P\) is a paired-dominating set of \(G\). The study of the \(DPDP\)-graphs was initiated by \textit{J. Southey} and \textit{M. A. Henning} [Cent. Eur. J. Math. 8, No. 3, 459--467 (2010; Zbl 1215.05126); J. Comb. Optim. 22, No. 2, 217--234 (2011; Zbl 1232.05172)]. In this paper, we provide conditions which ensure that a graph is a \(DPDP\)-graph. In particular, we characterize the minimal \(DPDP\)-graphs.Proper rainbow connection number of graphs.https://zbmath.org/1459.050772021-05-28T16:06:00+00:00"Doan, Trung Duy"https://zbmath.org/authors/?q=ai:doan.trung-duy"Schiermeyer, Ingo"https://zbmath.org/authors/?q=ai:schiermeyer.ingoSummary: A path in an edge-coloured graph is called a rainbow path if its edges receive pairwise distinct colours. An edge-coloured graph is said to be rainbow connected if any two distinct vertices of the graph are connected by a rainbow path. The minimum \(k\) for which there exists such an edge-colouring is the rainbow connection number \(\operatorname{rc}(G)\) of \(G\). Recently,
\textit{S. Bau} et al. [Australas. J. Comb. 71, Part 3, 381--393 (2018; Zbl 1404.05100)] introduced this concept with the additional requirement that the edge-colouring must be proper. The proper rainbow connection number of \(G\), denoted by \(prc \operatorname{prc}(G)\), is the minimum number of colours needed in order to make it properly rainbow connected. Obviously, \(\operatorname{prc}(G) \geq \max\{\operatorname{rc}(G), \chi^\prime(G)\}\). In this paper we first prove an improved upper bound \(\operatorname{prc}(G) \leq n\) for every connected graph \(G\) of order \(n \geq 3\). Next we show that the difference \(\operatorname{prc}(G) - \max\{\operatorname{rc}(G), \chi^\prime(G)\}\) can be arbitrarily large. Finally, we present several sufficient conditions for graph classes satisfying \(\operatorname{prc}(G) = \chi^\prime(G)\).Colorings of plane graphs without long monochromatic facial paths.https://zbmath.org/1459.050562021-05-28T16:06:00+00:00"Czap, Július"https://zbmath.org/authors/?q=ai:czap.julius"Fabrici, Igor"https://zbmath.org/authors/?q=ai:fabrici.igor"Jendrol', Stanislav"https://zbmath.org/authors/?q=ai:jendrol.stanislavSummary: Let \(G\) be a plane graph. A facial path of \(G\) is a subpath of the boundary walk of a face of \(G\). We prove that each plane graph admits a 3-coloring (a 2-coloring) such that every monochromatic facial path has at most 3 vertices (at most 4 vertices). These results are in a contrast with the results of \textit{G. Chartrand} et al. [J. Comb. Theory, Ser. B 10, 12--41 (1971; Zbl 0223.05101)] and \textit{M. Axenovich} et al. [J. Graph Theory 85, No. 3, 601--618 (2017; Zbl 1367.05044)] which state that for any positive integer \(t\) there exists a 4-colorable (a 3-colorable) plane graph \(G_t\) such that in any its 3-coloring (2-coloring) there is a monochromatic path of length at least \(t\). We also prove that every plane graph is 2-list-colorable in such a way that every monochromatic facial path has at most 4 vertices.Domination number of graphs with minimum degree five.https://zbmath.org/1459.052322021-05-28T16:06:00+00:00"Bujtás, Csilla"https://zbmath.org/authors/?q=ai:bujtas.csillaSummary: We prove that for every graph \(G\) on \(n\) vertices and with minimum degree five, the domination number \(\gamma (G)\) cannot exceed \(n/3\). The proof combines an algorithmic approach and the discharging method. Using the same technique, we provide a shorter proof for the known upper bound \(4n/11\) on the domination number of graphs of minimum degree four.A new framework to approach Vizing's conjecture.https://zbmath.org/1459.052312021-05-28T16:06:00+00:00"Brešar, Boštjan"https://zbmath.org/authors/?q=ai:bresar.bostjan"Hartnell, Bert L."https://zbmath.org/authors/?q=ai:hartnell.bert-l"Henning, Michael A."https://zbmath.org/authors/?q=ai:henning.michael-anthony"Kuenzel, Kirsti"https://zbmath.org/authors/?q=ai:kuenzel.kirsti"Rall, Douglas F."https://zbmath.org/authors/?q=ai:rall.douglas-fSummary: We introduce a new setting for dealing with the problem of the domination number of the Cartesian product of graphs related to \textit{V. G. Vizing}'s conjecture [Russ. Math. Surv. 23, No. 6, 125--141 (1968; Zbl 0192.60502); translation from Usp. Mat. Nauk 23, No. 6(144), 117--134 (1968)]. The new framework unifies two different approaches to the conjecture. The most common approach restricts one of the factors of the product to some class of graphs and proves the inequality of the conjecture then holds when the other factor is any graph. The other approach utilizes the so-called Clark-Suen partition for proving a weaker inequality that holds for all pairs of graphs. We demonstrate the strength of our framework by improving the bound of \textit{S. Suen} and \textit{J. Tarr} [Electron. J. Comb. 19, No. 1, Research Paper P8, 4 p. (2012; Zbl 1243.05190)] as follows: \( \gamma (X \square Y) \geq \max \{\frac12\gamma(X)\gamma_t(Y),\frac12\gamma_t(X)\gamma(Y)\}\), where \(\gamma\) stands for the domination number, \( \gamma_t\) is the total domination number, and \(X \square Y\) is the Cartesian product of graphs \(X\) and \(Y\).Independence number and packing coloring of generalized Mycielski graphs.https://zbmath.org/1459.052302021-05-28T16:06:00+00:00"Bidine, Ez Zobair"https://zbmath.org/authors/?q=ai:bidine.ez-zobair"Gadi, Taoufiq"https://zbmath.org/authors/?q=ai:gadi.taoufiq"Kchikech, Mustapha"https://zbmath.org/authors/?q=ai:kchikech.mustaphaSummary: For a positive integer \(k \geq 1\), a graph \(G\) with vertex set \(V\) is said to be \(k\)-packing colorable if there exists a mapping \(f : V \mapsto \{1, 2, \dots, k\}\) such that any two distinct vertices \(x\) and \(y\) with the same color \(f(x) = f(y)\) are at distance at least \(f(x) + 1\). The packing chromatic number of a graph \(G\), denoted by \(\chi_\rho (G)\), is the smallest integer \(k\) such that \(G\) is \(k\)-packing colorable.
In this work, we study both independence and packing colorings in the \(m\)-generalized Mycielskian of a graph \(G\), denoted \(\mu_m (G)\). We first give an explicit formula for \(\alpha (\mu_m (G))\) when \(m\) is odd and bounds when \(m\) is even. We then use these results to give exact values of \(\alpha(\mu_m (K_n))\) for any \(m\) and \(n\). Next, we give bounds on the packing chromatic number, \( \chi_\rho \), of \(\mu_m (G)\). We also prove the existence of large planar graphs whose packing chromatic number is 4. The rest of the paper is focused on packing chromatic numbers of the Mycielskian of paths and cycles.Achromatic numbers for circulant graphs and digraphs.https://zbmath.org/1459.050972021-05-28T16:06:00+00:00"Araujo-Pardo, Gabriela"https://zbmath.org/authors/?q=ai:araujo-pardo.gabriela"Montellano-Ballesteros, Juan José"https://zbmath.org/authors/?q=ai:montellano-ballesteros.juan-jose"Olsen, Mika"https://zbmath.org/authors/?q=ai:olsen.mika"Rubio-Montiel, Christian"https://zbmath.org/authors/?q=ai:rubio-montiel.christianSummary: In this paper, we determine the achromatic and diachromatic numbers of some circulant graphs and digraphs each one with two lengths and give bounds for other circulant graphs and digraphs with two lengths. In particular, for the achromatic number we state that \(\alpha (C_{16q^2+20q+7} (1, 2)) = 8q + 5\), and for the diachromatic number we state that \(\operatorname{dac}(\vec{C}_{32q^2+24q+5} (1, 2)) = 8 q + 3\). In general, we give the lower bounds \(\alpha (C_{4q^2+aq+1}(1, a)) \geq 4q + 1\) and \(\operatorname{dac}(\vec{C}_{8q^2+2(a+4)q+a+3}(1, a)) \geq 4q + 3\) when \(a\) is a non quadratic residue of \(\mathbb{Z}_{4q+1}\) for graphs and \(\mathbb{Z}_{4q+3}\) for digraphs, and the equality is attained, in both cases, for \(a = 3\).
Finally, we determine the achromatic index for circulant graphs of \(q^2 +q + 1\) vertices when the projective cyclic plane of odd order \(q\) exists.Graph operations and neighborhood polynomials.https://zbmath.org/1459.052782021-05-28T16:06:00+00:00"Alipour, Maryam"https://zbmath.org/authors/?q=ai:alipour.maryam"Tittmann, Peter"https://zbmath.org/authors/?q=ai:tittmann.peterSummary: The neighborhood polynomial of graph \(G\) is the generating function for the number of vertex subsets of \(G\) of which the vertices have a common neighbor in \(G\). In this paper, we investigate the behavior of this polynomial under several graph operations. Specifically, we provide an explicit formula for the neighborhood polynomial of the graph obtained from a given graph \(G\) by vertex attachment. We use this result to propose a recursive algorithm for the calculation of the neighborhood polynomial. Finally, we prove that the neighborhood polynomial can be found in polynomial-time in the class of \(k\)-degenerate graphs.On Hb-graphs and their application to general hypergraph e-adjacency tensor.https://zbmath.org/1459.052272021-05-28T16:06:00+00:00"Ouvrard, Xavier"https://zbmath.org/authors/?q=ai:ouvrard.xavier"Le Goff, Jean-Marie"https://zbmath.org/authors/?q=ai:le-goff.jean-marie"Marchand-Maillet, Stéphane"https://zbmath.org/authors/?q=ai:marchand-maillet.stephaneSummary: Working on general hypergraphs requires to properly redefine the concept of adjacency in a way that it captures the information of the hyperedges independently of their size. Coming to represent this information in a tensor imposes to go through a uniformisation process of the hypergraph. Hypergraphs limit the way of achieving it as redundancy is not permitted. Hence, our introduction of Hb-graphs, families of multisets on a common universe corresponding to the vertex set, that we present in details in this article, allowing us to have a construction of adequate adjacency tensor that is interpretable in term of \(m\)-uniformisation of a general Hb-graph. As hypergraphs appear as particular Hb-graphs, we deduce two new (e-)adjacency tensors for general hypergraphs. We conclude this article by giving some first results on hypergraph spectral analysis of these tensors and a comparison with the existing tensors for general hypergraphs, before making a final choice.On the maximal-adjacency-spectrum unicyclic graphs with given maximum degree.https://zbmath.org/1459.051872021-05-28T16:06:00+00:00"Song, Haizhou"https://zbmath.org/authors/?q=ai:song.haizhou"Tian, Lulu"https://zbmath.org/authors/?q=ai:tian.luluSummary: In this paper, we study the properties and structure of the maximal-adjacency-spectrum unicyclic graphs with given maximum degree. We obtain some necessary conditions on the maximal-adjacency-spectrum unicyclic graphs in the set of unicyclic graphs with \(n\) vertices and maximum degree \(\Delta\) and describe the structure of the maximal-adjacency-spectrum unicyclic graphs in the set. Besides, we also give a new upper bound on the adjacency spectral radius of unicyclic graphs, and this new upper bound is the best upper bound expressed by vertices \(n\) and maximum degree \(\Delta\) from now on.\(L(h,k)\) labelings of \(K_n-M\) and \(K_n-P_m\) for all values of \(h\) and \(k^\ast\).https://zbmath.org/1459.052872021-05-28T16:06:00+00:00"Jacob, Jobby"https://zbmath.org/authors/?q=ai:jacob.jobby"Mattes, Connor"https://zbmath.org/authors/?q=ai:mattes.connor"Witt, Marika"https://zbmath.org/authors/?q=ai:witt.marikaSummary: An \(L(h,k)\) labeling of a graph \(G\) is an integer labeling of the vertices where the labels of adjacent vertices differ by at least \(h\), and the labels of vertices that are at distance two from each other differ by at least \(k\). The span of an \(L(h,k)\) labeling \(f\) on a graph \(G\) is the largest label minus the smallest label under \(f\). The \(L(h,k)\) span of a graph \(G\), denoted \(\lambda_{h,k}(G)\), is the minimum span of all \(L(h,k)\) labelings of \(G\).
We study \(L(h,k)\) labelings of some dense graphs obtained by deleting either a maximum matching or the edges of an arbitrary path from a complete graph. For all non-negative integer values of \(h\) and \(k\), we establish the \(L(h,k)\) spans of these graphs.Decomposition of complete graphs into tri-cyclic graphs with eight edges.https://zbmath.org/1459.052592021-05-28T16:06:00+00:00"Froncek, Dalibor"https://zbmath.org/authors/?q=ai:froncek.dalibor"Kubik, Bethany"https://zbmath.org/authors/?q=ai:kubik.bethanySummary: In this paper, we use standard graph labeling techniques to prove that each tri-cyclic graph with eight edges decomposes the complete graph \(K_n\) if and only if \(n\equiv 0,1\pmod{16}\). We apply \(\rho\)-tripartite labelings and 1-rotational \(\rho\)-tripartite labelings.Sequential metric dimension.https://zbmath.org/1459.052082021-05-28T16:06:00+00:00"Bensmail, Julien"https://zbmath.org/authors/?q=ai:bensmail.julien"Mazauric, Dorian"https://zbmath.org/authors/?q=ai:mazauric.dorian"Mc Inerney, Fionn"https://zbmath.org/authors/?q=ai:mc-inerney.fionn"Nisse, Nicolas"https://zbmath.org/authors/?q=ai:nisse.nicolas"Pérennes, Stéphane"https://zbmath.org/authors/?q=ai:perennes.stephaneSummary: \textit{S. M. Seager} [Ars Comb. 110, 45--54 (2013; Zbl 1313.05099)] introduced the following game. An invisible and immobile target is hidden at some vertex of a graph \(G\). Every step, one vertex \(v\) of \(G\) can be probed which results in the knowledge of the distance between \(v\) and the target. The objective of the game is to minimize the number of steps needed to locate the target, wherever it is.{
}We address the generalization of this game where \(k\geq 1\) vertices can be probed at every step. Our game also generalizes the notion of the metric dimension of a graph. Precisely, given a graph \(G\) and two integers \(k,\ell\geq 1\), the Localization Problem asks whether there exists a strategy to locate a target hidden in \(G\) in at most \(\ell\) steps by probing at most \(k\) vertices per step. We show this problem is NP-complete when \(k\) (resp., \(\ell\)) is a fixed parameter.{
}Our main results are for the class of trees where we prove this problem is NP-complete when \(k\) and \(\ell\) are part of the input but, despite this, we design a polynomial-time \((+1)\)-approximation algorithm in trees which gives a solution using at most one more step than the optimal one. It follows that the Localization problem is polynomial-time solvable in trees if \(k\) is fixed.
For the entire collection see [Zbl 1400.68022].Decomposition of complete graphs into unicyclic graphs with eight edges.https://zbmath.org/1459.052572021-05-28T16:06:00+00:00"Freyberg, Bryan"https://zbmath.org/authors/?q=ai:freyberg.bryan"Froncek, Dalibor"https://zbmath.org/authors/?q=ai:froncek.daliborSummary: Let \(G\) be a tripartite unicyclic graph with eight edges that either (i) contains a triangle or heptagon, or (ii) contains a pentagon and is disconnected. We prove that \(G\) decomposes the complete graph \(K_n\) whenever the necessary conditions are satisfied. We combine this result with other known results to prove that every unicyclic graph with eight edges other than \(C_s\) decomposes \(K_n\) if and only if \(n\equiv 0,1\pmod{16}\).Color-induced graph colorings.https://zbmath.org/1459.050742021-05-28T16:06:00+00:00"Chartrand, Gary"https://zbmath.org/authors/?q=ai:chartrand.gary"Hallas, James"https://zbmath.org/authors/?q=ai:hallas.james"Zhang, Ping"https://zbmath.org/authors/?q=ai:zhang.ping.1|zhang.ping.6Summary: For a positive integer \(k\), let \(\mathcal{P}^\ast([k])\) denote the set of nonempty subsets of \([k]=\{1,2,\dots,k\}\). For a graph \(G\) without isolated vertices, let \(c:E(G)\to\mathcal{P}^\ast([k])\) be an edge coloring of \(G\) where adjacent edges may be colored the same. The induced vertex coloring \(c\prime:V(G)\to\mathcal{P}^\ast([k])\) is defined by \(c'(v)= \cap_{e\in E_v}c(e)\), where \(E_v\) is the set of edges incident with \(v\). If \(c^\prime\) is a proper vertex coloring of \(G\), then \(c\) is called a regal \(k\)-edge coloring of \(G\). The minimum positive integer \(k\) for which a graph \(G\) has a regal \(k\)-edge coloring is the regal index of \(G\). If \(c^\prime\) is vertex-distinguishing, then \(c\) is a strong regal \(k\)-edge coloring of \(G\). The minimum positive integer \(k\) for which a graph \(G\) has a strong regal \(k\)-edge coloring is the strong regal index of \(G\). The regal index (and, consequently, the strong regal index) is determined for each complete graph and for each complete multipartite graph. Sharp bounds for regal indexes and strong regal indexes of connected graphs are established. Strong regal indexes are also determined for several classes of trees. Other results and problems are also presented.On 2-fold graceful labelings.https://zbmath.org/1459.052842021-05-28T16:06:00+00:00"Bunge, Ryan C."https://zbmath.org/authors/?q=ai:bunge.ryan-c"Cornett, Megan"https://zbmath.org/authors/?q=ai:cornett.megan"El-Zanati, Saad I."https://zbmath.org/authors/?q=ai:el-zanati.saad-i-el-zanati"Jeffries, Joel"https://zbmath.org/authors/?q=ai:jeffries.joel"Rattin, Ellen"https://zbmath.org/authors/?q=ai:rattin.ellenSummary: A long-standing conjecture by Kotzig, Ringel, and Rosa states that every tree admits a graceful labeling. That is, for any tree \(T\) with \(n\) edges, it is conjectured that there exists a labeling \(f:V(T)\to \{0,1,\dots,n\}\) such that the set of induced edge labels \(\{|f(u)-f(v)|: \{u,v\}\in E(T)\}\) is exactly \(\{1,2,\dots,n\}\). We extend this concept to allow for multigraphs with edge multiplicity at most 2. A 2-fold graceful labeling of a graph (or multigraph) \(G\) with \(n\) edges is a one-to-one function \(f:V(G)\to\{0,1,\dots,n\}\) such that the multiset of induced edge labels is comprised of two copies of each element in \(\{1,2,\dots, \lfloor n/2\rfloor\}\) and, if \(n\) is odd, one copy of \(\lceil n/2\rceil\}\). When \(n\) is even, this concept is similar to that of 2-equitable labelings which were introduced by Bloom and have been studied for several classes of graphs. We show that caterpillars, cycles of length \(n\not\equiv 1\pmod 4\), and complete bipartite graphs admit 2-fold graceful labelings. We also show that under certain conditions, the join of a tree and an empty graph (i.e., a graph with vertices but no edges) is 2-fold graceful.Rényi ordering of tournaments.https://zbmath.org/1459.051002021-05-28T16:06:00+00:00"Brown, David E."https://zbmath.org/authors/?q=ai:brown.david-e"Frederickson, Bryce"https://zbmath.org/authors/?q=ai:frederickson.bryceSummary: We develop an ordering function on the class of tournament digraphs (complete antisymmetric digraphs) that is based on the Rényi \(\alpha\)-entropy. This ordering function partitions tournaments on \(n\) vertices into equivalence classes that are approximately sorted from transitive (the arc relation is transitive -- the score sequence is \((0,1,2,\dots,n-1))\) to regular (score sequence \((\frac{n-1}{2}, \dots,\frac{n-1}{2}))\). But the diversity among regular tournaments (there are for example 1123 regular tournaments on 11 vertices, and 1,495,297 regular tournaments on 13 vertices up to isomorphism) is captured to an extent.The spectrum problem for two multigraphs with four vertices and seven edges.https://zbmath.org/1459.051602021-05-28T16:06:00+00:00"Bermudez, H."https://zbmath.org/authors/?q=ai:bermudez.hernando"Bunge, R. C."https://zbmath.org/authors/?q=ai:bunge.ryan-c"Cornelius, E. D."https://zbmath.org/authors/?q=ai:cornelius.e-d"El-Zanati, S. I."https://zbmath.org/authors/?q=ai:el-zanati.saad-i-el-zanati"Mamboleo, W. H."https://zbmath.org/authors/?q=ai:mamboleo.w-h"Nguyen, N. T."https://zbmath.org/authors/?q=ai:nguyen.nam-tuan|nguyen.nhu-thang|nguyen.nhan-tam|nguyen.nhu-tuan|nguyen.ngoc-thinh|nguyen.ngoc-thach|nguyen.nha-thanh|nguyen.nhan-t|nguyen.ngoc-tuan|nguyen.nam-trung|nguyen.ngoc-tuy|nguyen.ngoc-trung|nguyen.nguyen-t|nguyen.nga-t-t|nguyen.nang-tam|nguyen.nguyet-thanh|nguyen.nhan-thanh|nguyen-thanh-ngoc."Roberts, D. P."https://zbmath.org/authors/?q=ai:roberts.dan-pSummary: Let \(G\) be one of the two multigraphs obtained from \(K_4-e\) by replacing two edges with a double-edge while maintaining a minimum degree of 2. We find necessary and sufficient conditions on \(n\) and \(\lambda\) for the existence of a \(G\)-decomposition of \(^\lambda K_n\).On tree-connection of generalized line graphs.https://zbmath.org/1459.052802021-05-28T16:06:00+00:00"Hallas, Jamie"https://zbmath.org/authors/?q=ai:hallas.jamie"Zayed, Mohra"https://zbmath.org/authors/?q=ai:zayed.mohra"Zhang, Ping"https://zbmath.org/authors/?q=ai:zhang.ping.1Summary: The line graph \(L(G)\) of a nonempty graph \(G\) has the set of edges in \(G\) as its vertex set where two vertices of \(L(G)\) are adjacent if the corresponding edges of \(G\) are adjacent. Let \(k\ge 2\) be an integer and let \(G\) be a graph containing \(k\)-paths (paths of order \(k)\). The \(k\)-path graph \(\mathcal{P}_k(G)\) of \(G\) has the set of \(k\)-paths of \(G\) as its vertex set where two distinct vertices of \(\mathcal{P}_k(G)\) are adjacent if the corresponding \(k\)-paths of \(G\) have a \((k-1)\)-path in common. Thus, \(\mathcal{P}_2(G)=L(G)\) and \(\mathcal{P}_3(G)= L(L(G))\). Hence, the \(k\)-path graph \(\mathcal{P}_k(G)\) of a graph \(G\) is a generalization of the line graph \(L(G)\). Let \(G\) be a connected graph of order \(n\ge 3\) and let \(k\) be an integer with \(2\le k\le n-1\). The graph \(G\) is \(k\)-tree-connected if for every set \(S\) of \(k\) distinct vertices of \(G\), there exists a spanning tree \(T\) of \(G\) whose set of end-vertices is \(S\). Thus, \(G\) is 2-tree-connected if and only if \(G\) is Hamiltonian-connected. It was conjectured that if \(T\) is a tree of sufficiently large order containing no vertices of any of the degrees \(2,3,\dots,k+1\) for an integer \(k\ge 2\), then \(\mathcal{P}_3(T)\) is \(k\)-tree-connected. This conjecture was verified for \(k=2,3\). In this work, we show that if \(T\) is a tree of order at least 6 containing no vertices of degree 2, 3, 4, or 5, then \(\mathcal{P}_3(T)\) is 4-tree-connected and so verify the conjecture for the case when \(k=4\).Perpendicular graph of modules.https://zbmath.org/1459.051252021-05-28T16:06:00+00:00"Shirali, Maryam"https://zbmath.org/authors/?q=ai:shirali.maryam"Momtahan, Ehsan"https://zbmath.org/authors/?q=ai:momtahan.ehsan"Safaeeyan, Saeed"https://zbmath.org/authors/?q=ai:safaeeyan.saeedSummary: Let \(R\) be a ring and \(M\) be an \(R\)-module. Two modules \(A\) and \(B\) are called orthogonal, written \(A\perp B\), if they do not have non-zero isomorphic submodules. We associate a graph \(\Gamma_{\bot}(M)\) to \(M\) with vertices \(\mathcal{M}_{\perp}=\{(0)\neq A\leq M\mid\exists B\neq (0)\) such that \(A\perp B\}\), and for distinct \(A,B\in\mathcal{M}_{\perp}\), the vertices \(A\) and \(B\) are adjacent if and only if \(A\perp B\). The main object of this article is to study the interplay of module-theoretic properties of \(M\) with graph-theoretic properties of \(\Gamma_{\bot}(M)\). An algorithm is given to generate perpendicular graphs of \(\mathbb{Z}_n\).Optimal Cheeger cuts and bisections of random geometric graphs.https://zbmath.org/1459.053062021-05-28T16:06:00+00:00"Müller, Tobias"https://zbmath.org/authors/?q=ai:muller.tobias-m|muller.tobias"Penrose, Mathew D."https://zbmath.org/authors/?q=ai:penrose.mathew-dSummary: Let \(d \geq 2\). The Cheeger constant of a graph is the minimum surface-to-volume ratio of all subsets of the vertex set with relative volume at most 1/2. There are several ways to define surface and volume here: the simplest method is to count boundary edges (for the surface) and vertices (for the volume). We show that for a geometric (possibly weighted) graph on \(n\) random points in a \(d\)-dimensional domain with Lipschitz boundary and with distance parameter decaying more slowly (as a function of \(n)\) than the connectivity threshold, the Cheeger constant (under several possible definitions of surface and volume), also known as conductance, suitably rescaled, converges for large \(n\) to an analogous Cheeger-type constant of the domain. Previously, \textit{N. García Trillos} et al. [J. Mach. Learn. Res. 17, Paper No. 181, 46 p. (2016; Zbl 1392.62180)] had shown this for \(d \geq 3\) but had required an extra condition on the distance parameter when \(d = 2\).Ramanujan graphs for post-quantum cryptography.https://zbmath.org/1459.941182021-05-28T16:06:00+00:00"Jo, Hyungrok"https://zbmath.org/authors/?q=ai:jo.hyungrok"Sugiyama, Shingo"https://zbmath.org/authors/?q=ai:sugiyama.shingo"Yamasaki, Yoshinori"https://zbmath.org/authors/?q=ai:yamasaki.yoshinoriSummary: We introduce a cryptographic hash function based on expander graphs, suggested by \textit{D. X. Charles} et al. [J. Cryptology 22, No. 1, 93--113 (2009; Zbl 1166.94006); in: Groups and symmetries. From Neolithic Scots to John McKay. Selected papers of the conference, Montreal, Canada, April 27--29, 2007. Providence, RI: American Mathematical Society (AMS). 53--80 (2009; Zbl 1250.05057)], as one prominent candidate in post-quantum cryptography. We propose a generalized version of explicit constructions of Ramanujan graphs, which are seen as an optimal structure of expander graphs in a spectral sense, from the previous works of \textit{A. Lubotzky} et al. [Combinatorica 8, No. 3, 261--277 (1988; Zbl 0661.05035)] and \textit{P. Chiu} [ibid. 12, No. 3, 275--285 (1992; Zbl 0770.05062)]. We also describe the relationship between the security of Cayley hash functions and word problems for group theory. We also give a brief comparison of LPS-type graphs and Pizer's graphs to draw attention to the underlying hard problems in cryptography.
For the entire collection see [Zbl 1457.94002].Divisible design digraphs and association schemes.https://zbmath.org/1459.051052021-05-28T16:06:00+00:00"Kharaghani, Hadi"https://zbmath.org/authors/?q=ai:kharaghani.hadi"Suda, Sho"https://zbmath.org/authors/?q=ai:suda.shoSummary: Divisible design digraphs are constructed from skew balanced generalized weighing matrices and generalized Hadamard matrices. Commutative and non-commutative association schemes are shown to be attached to the constructed divisible design digraphs.Characterizing star-PCGs.https://zbmath.org/1459.053172021-05-28T16:06:00+00:00"Xiao, Mingyu"https://zbmath.org/authors/?q=ai:xiao.mingyu"Nagamochi, Hiroshi"https://zbmath.org/authors/?q=ai:nagamochi.hiroshiSummary: A graph \(G\) is called a pairwise compatibility graph (PCG, for short) if it admits a tuple \((T,w,d_{\min},d_{\max})\) of a tree \(T\) whose leaf set is equal to the vertex set of \(G\), a non-negative edge weight \(w\), and two non-negative reals \(d_{\min}\leq d_{\max}\) such that \(G\) has an edge between two vertices \(u,v\in V\) if and only if the distance between the two leaves \(u\) and \(v\) in the weighted tree \((T,w)\) is in the interval \([d_{\min},d_{\max}]\). The tree \(T\) is also called a witness tree of the PCG \(G\). The problem of testing if a given graph is a PCG is not known to be NP-hard yet. To obtain a complete characterization of PCGs is a wide open problem in computational biology and graph theory. In the literature, most witness trees admitted by known PCGs are stars and caterpillars. In this paper, we give a complete characterization for a graph to be a star-PCG (a PCG that admits a star as its witness tree), which provides us the first polynomial-time algorithm for recognizing star-PCGs.
For the entire collection see [Zbl 1390.68029].Relation between the Hermitian energy of a mixed graph and the matching number of its underlying graph.https://zbmath.org/1459.051942021-05-28T16:06:00+00:00"Wei, Wei"https://zbmath.org/authors/?q=ai:wei.wei.6|wei.wei.7|wei.wei.3|wei.wei.5|wei.wei.4|wei.wei.2"Li, Shuchao"https://zbmath.org/authors/?q=ai:li.shuchaoSummary: Given a graph \(G\), the mixed graph \(D_G\) is obtained from \(G\) by orienting some of its edges (\(G\) is also called the underlying graph of \(D_G\)). Let \(\mathcal{E}_H(D_G)\) be the Hermitian energy of \(D_G\) and let \(\alpha^\prime (G)\) be the matching number of the underlying graph \(G\). In this paper, we first establish the inequality \(\mathcal{E}_H(D_G) \geqslant 2 \alpha^\prime (G)\) and characterize all the mixed graphs \(D_G\) which make \(\mathcal{E}_H (D_G) = 2 \alpha^\prime (G)\) hold. Furthermore, we obtain \(2 \beta(G) - 2c(G) \leqslant \mathcal{E}_H (D_G) \leqslant 2 \beta\), where \(\beta(G), c(G)\) and \(\Delta(G)\) are, respectively, the vertex cover number, the number of odd cycles and the maximum degree of graph \(G\). The lower bound is attained if and only if \(D_G\) is switching equivalent to its underlying graph \(G\), where \(G\) is the disjoint union of some complete bipartite graphs each of which contains a perfect matching, together with some isolated vertices. The upper bound is best possible. By our results in this paper, some main results in [\textit{F. Tian} and \textit{D. Wong}, Discrete Appl. Math. 222, 179--184 (2017; Zbl 1396.05078); \textit{L. Wang} and \textit{X. Ma}, Linear Algebra Appl. 517, 207--216 (2017; Zbl 1353.05082); \textit{D. Wong} et al., Linear Algebra Appl. 549, 276--286 (2018; Zbl 1390.05139)] can be deduced in a unified approach.Amalgamations and equitable block-colorings.https://zbmath.org/1459.052712021-05-28T16:06:00+00:00"Matson, E. B."https://zbmath.org/authors/?q=ai:matson.e-b"Rodger, C. A."https://zbmath.org/authors/?q=ai:rodger.christopher-a|rodger.chris-aSummary: An \(H\)-decomposition of \(G\) is a partition \(P\) of \(E(G)\) into blocks, each element of which induces a copy of \(H\). Amalgamations of graphs have proved to be a valuable tool in the construction of \(H\)-decompositions. The method can force decompositions to satisfy fairness notions. Here, the use of the method is further applied to \((s, p)\)-equitable block-colorings of \(H\)-decompositions: a coloring of the blocks using exactly \(s\) colors such that each vertex \(v\) is incident with blocks colored with exactly \(p\) colors, the blocks containing \(v\) being shared out as evenly as possible among the \(p\) color classes. Recently interest has turned to the color vector \(V(E)=(c_1(E), c_2(E),\dots , c_s(E))\) of such colorings. Amalgamations are used to construct \((s, p)\)-equitable block-colorings of \(C_4\)-decompositions of \(K_n - F\) and \(K_2\)-decompositions of \(K_n\), focusing on one unsolved case with each where \(c_1\) is as small as possible and \(c_2\) is as large as possible.
For the entire collection see [Zbl 1411.65006].Adaptive directional Haar tight framelets on bounded domains for digraph signal representations.https://zbmath.org/1459.420492021-05-28T16:06:00+00:00"Xiao, Yuchen"https://zbmath.org/authors/?q=ai:xiao.yuchen"Zhuang, Xiaosheng"https://zbmath.org/authors/?q=ai:zhuang.xiaoshengSummary: Based on hierarchical partitions, we provide the construction of Haar-type tight framelets on any compact set \(K\subseteq \mathbb{R}^d\). In particular, on the unit block \([0,1]^d\), such tight framelets can be built to be with adaptivity and directionality. We show that the adaptive directional Haar tight framelet systems can be used for digraph signal representations. Some examples are provided to illustrate results in this paper.Spectral radii of sparse random matrices.https://zbmath.org/1459.150362021-05-28T16:06:00+00:00"Benaych-Georges, Florent"https://zbmath.org/authors/?q=ai:benaych-georges.florent"Bordenave, Charles"https://zbmath.org/authors/?q=ai:bordenave.charles"Knowles, Antti"https://zbmath.org/authors/?q=ai:knowles.anttiThe paper studies spectral radii of classes of random matrices of combinatorial interest, including the adjacency matrices of (inhomogeneous) Erdős-Rényi random graphs. It was already known from [\textit{Z. Füredi} and \textit{J. Komloś}, Combinatorica 1, 233--241 (1981; Zbl 0494.15010); \textit{V. H. Vu}, Combinatorica 27, No. 6, 721--736 (2007; Zbl 1164.05066)] that, for example, in the sparse Erdős-Rényi random graph \(G(n, d/n)\), the second and smallest adjacency eigenvalues converge to the edges of the support of the asymptotic eigenvalue distribution provided \(d/\log(n)^{4}\rightarrow \infty\).
In this paper, these results are extended to a proof that the same statement holds under the weaker assumption that \(d/\log(n)\rightarrow\infty\). A companion paper of the authors shows that in the other regime \(d/\log(n)\rightarrow 0\) the behavior is different [Ann. Probab. 47, No. 3, 1653--1676 (2019; Zbl 1447.60017)].
The main new tool is a refined use of the non-backtracking matrix. It is important to emphasize that the results apply to a much more general class of random graphs, including block stochastic models and inhomogeneous Erdős-Rényi graphs.
Reviewer: David B. Penman (Colchester)Exceptional graphs for the random walk.https://zbmath.org/1459.053092021-05-28T16:06:00+00:00"Aru, Juhan"https://zbmath.org/authors/?q=ai:aru.juhan"Groenland, Carla"https://zbmath.org/authors/?q=ai:groenland.carla"Johnston, Tom"https://zbmath.org/authors/?q=ai:johnston.tom"Narayanan, Bhargav"https://zbmath.org/authors/?q=ai:narayanan.bhargav-p"Roberts, Alex"https://zbmath.org/authors/?q=ai:roberts.alex"Scott, Alex"https://zbmath.org/authors/?q=ai:scott.alexander-dThis paper studies exceptional graphs for random walks on \(\mathbb{Z}^2\). If \(W\) is a simple random walk on \(\mathbb{Z}^2\), then there almost surely exists a random exceptional subgraph \(H \subseteq\mathbb{Z}^2\) for which the induced walk \(W_H\) visits each node reachable from the origin in \(H\) infinitely many times and fails to visit infinitely many nodes reachable from the origin in \(H\). The question of whether a countably infinite independent set of simple random walks almost surely has the property that at least one of these walks induces a recurrent walk over a given \(G\subseteq\mathbb{Z}^2\) is negatively answered in this paper either. Moreover, for \(d\in\mathbb{N}\) and \(W_i\) \((i\in S)\) being a set of random walks created by a branching random walk over \(\mathbb{Z}^d\) with a nontrivial offspring distribution, it is shown that almost surely for every spanning subgraph \(G\subseteq\mathbb{Z}^d\), there is a \(j\in S\) for which the induced walk \(W_j\) is recurrent.
Reviewer: Yilun Shang (Newcastle)Algebraic and logical descriptions of generalized trees.https://zbmath.org/1459.030072021-05-28T16:06:00+00:00"Courcelle, Bruno"https://zbmath.org/authors/?q=ai:courcelle.brunoSummary: \textit{Quasi-trees} generalize trees in that the unique ``path'' between two nodes may be infinite and have any countable order type. They are used to define the rank-width of a countable graph in such a way that it is equal to the least upper-bound of the rank-widths of its finite induced subgraphs. \textit{Join-trees} are the corresponding directed trees. They are useful to define the modular decomposition of a countable graph. We also consider \textit{ordered join-trees}, that generalize rooted trees equipped with a linear order on the set of sons of each node. We define algebras with finitely many operations that generate (via infinite terms) these generalized trees. We prove that the associated regular objects (those defined by regular terms) are exactly the ones that are the unique models of monadic second-order sentences. These results use and generalize a similar result by \textit{W. Thomas} [RAIRO, Inform. Théor. Appl. 20, 371--381 (1986; Zbl 0639.68071)] for countable linear orders.How to contract a vertex transitive 5-connected graph.https://zbmath.org/1459.051412021-05-28T16:06:00+00:00"Qin, Chengfu"https://zbmath.org/authors/?q=ai:qin.chengfu"Yang, Weihua"https://zbmath.org/authors/?q=ai:yang.weihua"Guo, Xiaofeng"https://zbmath.org/authors/?q=ai:guo.xiaofengSummary: M. Kriesell conjectured that there existed \(b, h\) such that every 5-connected graph \(G\) with at least \(b\) vertices can be contracted to a 5-connected graph \(G_0\) such that \(0<\left| V \left( G\right)\right|-\left( V \left( G_0\right)\right)<h\). We show that this conjecture holds for vertex transitive 5-connected graphs.On the length spectra of simple regular periodic graphs.https://zbmath.org/1459.051612021-05-28T16:06:00+00:00"Bhagwat, Chandrasheel"https://zbmath.org/authors/?q=ai:bhagwat.chandrasheel"Fatima, Ayesha"https://zbmath.org/authors/?q=ai:fatima.ayeshaSummary: One can define the notion of primitive length spectrum for a simple regular periodic graph via counting the orbits of closed reduced primitive cycles under an action of a discrete group of automorphisms [\textit{D. Guido} et al., J. Funct. Anal. 255, No. 6, 1339--1361 (2008; Zbl 1233.11095)]. We prove that this primitive length spectrum satisfies an analogue of the `Multiplicity one' property. We show that if all but finitely many primitive cycles in two simple regular periodic graphs have equal lengths, then all the primitive cycles have equal lengths. This is a graph-theoretic analogue of a similar theorem in the context of geodesics on hyperbolic spaces [\textit{C. Bhagwat} and \textit{C. S. Rajan}, J. Number Theory 131, No. 11, 2239--2244 (2011; Zbl 1261.11042)]. We also prove, in the context of actions of finitely generated abelian groups on a graph, that if the adjacency operators [\textit{B. Clair}, J. Comb. Theory, Ser. B 99, No. 1, 48--61 (2009; Zbl 1220.05056)] for two actions of such a group on a graph are similar, then corresponding periodic graphs are length isospectral.A new method to find the Wiener index of hypergraphs.https://zbmath.org/1459.051342021-05-28T16:06:00+00:00"Li, Yalan"https://zbmath.org/authors/?q=ai:li.yalan"Deng, Bo"https://zbmath.org/authors/?q=ai:deng.boSummary: The Wiener index is defined as the summation of distances between all pairs of vertices in a graph or in a hypergraph. Both models -- graph-theoretical and hypergraph-theoretical -- are used in mathematical chemistry for quantitatively studying physical and chemical properties of classical and nonclassical organic compounds. In this paper, we consider relationships between hypertrees and trees and hypercycles and cycles with respect to their Wiener indices.Maximum reciprocal degree resistance distance index of unicyclic graphs.https://zbmath.org/1459.050602021-05-28T16:06:00+00:00"Cai, Gai-Xiang"https://zbmath.org/authors/?q=ai:cai.gaixiang"Li, Xing-Xing"https://zbmath.org/authors/?q=ai:li.xingxing"Yu, Gui-Dong"https://zbmath.org/authors/?q=ai:yu.guidongSummary: The reciprocal degree resistance distance index of a connected graph \(G\) is defined as \(\mathrm{RDR}\left( G\right)={\sum_{\left\{ u, v\right\} \subseteq V \left( G\right)} \left( \left( d_G \left( u\right) + d_G \left( v\right)\right) / \left( r_G \left( u, v\right)\right)\right)} \), where \(r_G\left( u, v\right)\) is the resistance distance between vertices \(u\) and \(v\) in \(G\). Let \(\mathcal{U}_n\) denote the set of unicyclic graphs with \(n\) vertices. We study the graph with maximum reciprocal degree resistance distance index among all graphs in \(\mathcal{U}_n\) and characterize the corresponding extremal graph.Interacting diffusions on sparse graphs: hydrodynamics from local weak limits.https://zbmath.org/1459.602072021-05-28T16:06:00+00:00"Oliveira, Roberto I."https://zbmath.org/authors/?q=ai:oliveira.roberto-imbuzeiro"Reis, Guilherme H."https://zbmath.org/authors/?q=ai:reis.guilherme-h"Stolerman, Lucas M."https://zbmath.org/authors/?q=ai:stolerman.lucas-mSummary: We prove limit theorems for systems of interacting diffusions on sparse graphs. For example, we deduce a hydrodynamic limit and the propagation of chaos property for the stochastic Kuramoto model with interactions determined by Erdős-Rényi graphs with constant mean degree. The limiting object is related to a potentially infinite system of SDEs defined over a Galton-Watson tree. Our theorems apply more generally, when the sequence of graphs (``decorated'' with edge and vertex parameters) converges in the local weak sense. Our main technical result is a locality estimate bounding the influence of far-away diffusions on one another. We also numerically explore the emergence of synchronization phenomena on Galton-Watson random trees, observing rich phase transitions from synchronized to desynchronized activity among nodes at different distances from the root.Some properties of double Roman domination.https://zbmath.org/1459.052502021-05-28T16:06:00+00:00"Yang, Hong"https://zbmath.org/authors/?q=ai:yang.hong"Zhou, Xiaoqing"https://zbmath.org/authors/?q=ai:zhou.xiaoqingSummary: A \textit{double Roman dominating function} on a graph \(G\) is a function \(f:V\left( G\right)\longrightarrow\left\{ 0,1,2,3\right\}\) satisfying the conditions that every vertex \(u\) for which \(f\left( u\right)=0\) is adjacent to at least one vertex \(v\) for which \(f\left( v\right)=3\) or two vertices \(v_1\) and \(v_2\) for which \(f\left( v_1\right)=f\left( v_2\right)=2\) and every vertex \(u\) for which \(f\left( u\right)=1\) is adjacent to at least one vertex \(v\) for which \(f\left( v\right)\geq2\). The weight of a double Roman dominating function \(f\) is the value \(f\left( V\right)={\sum\nolimits_{u \in V} f \left( u\right)}\). The minimum weight of a double Roman dominating function on a graph \(G\) is called the \textit{double Roman domination number} \( \gamma_{d R}\left( G\right)\) of \(G\). A graph with \(\gamma_{d R}\left( G\right)=3\gamma\left( G\right)\) is called a \textit{double Roman} graph. In this paper, we study properties of double Roman domination in graphs. Moreover, we find a class of double Roman graphs and give characterizations of trees with \(\gamma_{d R}\left( T\right)= \gamma_R\left( T\right)+k\) for \(k=1,2\).On face index of silicon carbides.https://zbmath.org/1459.921702021-05-28T16:06:00+00:00"Zhang, Xiujun"https://zbmath.org/authors/?q=ai:zhang.xiujun"Raza, Ali"https://zbmath.org/authors/?q=ai:raza.ali"Fahad, Asfand"https://zbmath.org/authors/?q=ai:fahad.asfand"Jamil, Muhammad Kamran"https://zbmath.org/authors/?q=ai:jamil.muhammad-kamran"Chaudhry, Muhammad Anwar"https://zbmath.org/authors/?q=ai:chaudhry.muhammad-anwar"Iqbal, Zahid"https://zbmath.org/authors/?q=ai:iqbal.zahidSummary: Several graph invariants have been defined and studied, which present applications in nanochemistry, computer networks, and other areas of science. One vastly studied class of the graph invariants is the class of the topological indices, which helps in the studies of chemical, biological, and physical properties of a chemical structure. One recently introduced graph invariant is the face index, which can assist in predicting the energy and the boiling points of the certain chemical structures. In this paper, we drive the analytical closed formulas of face index of silicon carbides \(\mathrm{S i_2 C_3-I\left[ a, b\right]}\), \(\mathrm{S i_2 C_3-II\left[ a, b\right]}\), \(\mathrm{S i_2 C_3-III\left[ a, b\right]}\), and \(\mathrm{Si C_3-III\left[ a, b\right]}\).Minimum variable connectivity index of trees of a fixed order.https://zbmath.org/1459.050532021-05-28T16:06:00+00:00"Yousaf, Shamaila"https://zbmath.org/authors/?q=ai:yousaf.shamaila"Bhatti, Akhlaq Ahmad"https://zbmath.org/authors/?q=ai:bhatti.akhlaq-ahmad"Ali, Akbar"https://zbmath.org/authors/?q=ai:ali.akbarSummary: The connectivity index, introduced by the chemist Milan Randić in 1975, is one of the topological indices with many applications. In the first quarter of 1990s, Randić proposed the variable connectivity index by extending the definition of the connectivity index. The variable connectivity index for graph \(G\) is defined as \({\sum\nolimits_{v w \in E \left( G\right)} \left( \left( d \left( v\right) + \gamma\right) \left( d \left( w\right) + \gamma\right)\right)^{- 1 / 2}}\), where \(\gamma\) is a nonnegative real number, \(E\left( G\right)\) is the edge set of \(G\), and \(d\left( t\right)\) denotes the degree of an arbitrary vertex \(t\) in \(G\). Soon after the innovation of the variable connectivity index, its various chemical applications have been reported in different papers. However, to the best of the authors' knowledge, mathematical properties of the variable connectivity index, for \(\gamma>0\), have not yet been discussed explicitly in any paper. The main purpose of the present paper is to fill this gap by studying this topological index in a mathematical point of view. More precisely, in this paper, we prove that the star graph has the minimum variable connectivity index among all trees of a fixed order \(n\), where \(n\geq4\).Unimodality of independence polynomials of rooted products of graphs.https://zbmath.org/1459.052512021-05-28T16:06:00+00:00"Zhu, Bao-Xuan"https://zbmath.org/authors/?q=ai:zhu.baoxuan"Wang, Qingxiu"https://zbmath.org/authors/?q=ai:wang.qingxiuSummary: \textit{Y. Alavi} et al. [Congr. Numerantium 58, 15--23 (1987; Zbl 0679.05061)] conjectured that the independence polynomial of any tree is unimodal. Although it attracts many researchers' attention, it is still open. Motivated by this conjecture, in this paper, we prove that rooted products of some graphs preserve real rootedness of independence polynomials. As application, we not only give a unified proof for some known results, but also we can apply them to generate infinite kinds of trees whose independence polynomials have only real zeros. Thus their independence polynomials are unimodal.On computation of face index of certain nanotubes.https://zbmath.org/1459.053212021-05-28T16:06:00+00:00"Ye, Ansheng"https://zbmath.org/authors/?q=ai:ye.ansheng"Javed, Aisha"https://zbmath.org/authors/?q=ai:javed.aisha"Jamil, Muhammad Kamran"https://zbmath.org/authors/?q=ai:jamil.muhammad-kamran"Abdul Sattar, Kanza"https://zbmath.org/authors/?q=ai:abdul-sattar.kanza"Aslam, Adnan"https://zbmath.org/authors/?q=ai:aslam.adnan"Iqbal, Zahid"https://zbmath.org/authors/?q=ai:iqbal.zahid"Fahad, Asfand"https://zbmath.org/authors/?q=ai:fahad.asfandSummary: Topological index is a number that can be used to characterize the graph of a molecule. Topological indices describe the physical, chemical, and biological properties of a chemical structure. In this paper, we derive the analytical closed formulas of face index of some planar molecular structures such as \(\mathrm{TUC_4}\), \(\mathrm{TUC_4 C_8}\left( S\right)\), \(\mathrm{TUH C_6}\), \(\mathrm{TUC_4 C_8}\left( R\right)\), and armchair \(\mathrm{TUV C_6}\).Rank-constrained nonnegative matrix factorization for data representation.https://zbmath.org/1459.681872021-05-28T16:06:00+00:00"Shu, Zhenqiu"https://zbmath.org/authors/?q=ai:shu.zhenqiu"Wu, Xiao-Jun"https://zbmath.org/authors/?q=ai:wu.xiaojun"You, Congzhe"https://zbmath.org/authors/?q=ai:you.congzhe"Liu, Zhen"https://zbmath.org/authors/?q=ai:liu.zhen.1|liu.zhen"Li, Peng"https://zbmath.org/authors/?q=ai:li.peng.1|li.peng|li.peng.2|li.peng.4|li.peng.3"Fan, Honghui"https://zbmath.org/authors/?q=ai:fan.honghui"Ye, Feiyue"https://zbmath.org/authors/?q=ai:ye.feiyueSummary: Graph-based regularized nonnegative matrix factorization (NMF) methods performed well in many real-world applications. However, it is still an open problem to construct an optimal graph to effectively discover the intrinsic geometric structure of data. In this paper, we propose a new data representation framework, called rank-constrained nonnegative matrix factorization (RCNMF). We impose the rank constraint on the Laplacian matrix of the learned graph, so it can ensure that the number of connected components is consistent with the number of sample categories. Instead of a fixed graph-based regularization, the proposed framework can adaptively adjust the weight of the affinity matrix in each iteration. We develop two versions of RCNMF based on the \(l_1\) and \(l_2\) norms, and introduce their optimization schemes. In addition, their convergence and the complexity analyses are also provided. Experimental results on four benchmark datasets show that our methods outperform state-of-the-art methods in clustering.On the minimum variable connectivity index of unicyclic graphs with a given order.https://zbmath.org/1459.050522021-05-28T16:06:00+00:00"Yousaf, Shamaila"https://zbmath.org/authors/?q=ai:yousaf.shamaila"Bhatti, Akhlaq Ahmad"https://zbmath.org/authors/?q=ai:bhatti.akhlaq-ahmad"Ali, Akbar"https://zbmath.org/authors/?q=ai:ali.akbarSummary: The variable connectivity index, introduced by the chemist Milan Randić in the first quarter of 1990s, for a graph \(G\) is defined as \({\sum\nolimits_{v w \in E \left( G\right)} \left( \left( d_v + \gamma\right) \left( d_w + \gamma\right)\right)^{- 1 / 2}}\), where \(\gamma\) is a non-negative real number and \(d_w\) is the degree of a vertex \(w\) in \(G\). We call this index as the variable Randić index and denote it by \({\displaystyle{}^vR}_\gamma \). In this paper, we show that the graph created from the star graph of order \(n\) by adding an edge has the minimum \({\displaystyle{}^vR}_\gamma\) value among all unicyclic graphs of a fixed order \(n\), for every \(n\geq4\) and \(\gamma\geq0\).Random deposition with surface relaxation model in \(uv\) flower networks.https://zbmath.org/1459.822682021-05-28T16:06:00+00:00"Kim, Jin Min"https://zbmath.org/authors/?q=ai:kim.jinminDegree-ordered-percolation on uncorrelated networks.https://zbmath.org/1459.821152021-05-28T16:06:00+00:00"Caligiuri, Annalisa"https://zbmath.org/authors/?q=ai:caligiuri.annalisa"Castellano, Claudio"https://zbmath.org/authors/?q=ai:castellano.claudioBiased measures for random constraint satisfaction problems: larger interaction range and asymptotic expansion.https://zbmath.org/1459.052962021-05-28T16:06:00+00:00"Budzynski, Louise"https://zbmath.org/authors/?q=ai:budzynski.louise"Semerjian, Guilhem"https://zbmath.org/authors/?q=ai:semerjian.guilhemThe coloring of the cozero-divisor graph of a commutative ring.https://zbmath.org/1459.053542021-05-28T16:06:00+00:00"Akbari, S."https://zbmath.org/authors/?q=ai:akbari.samin|akbari.saeeid|akbari.shahabeddin|akbari.samira|akbari.saieed|akbari.soheil"Khojasteh, S."https://zbmath.org/authors/?q=ai:khojasteh.sohiela|khojasteh.soheilaCommunity enhancement network embedding based on edge reweighting preprocessing.https://zbmath.org/1459.681622021-05-28T16:06:00+00:00"Lv, Shaoqing"https://zbmath.org/authors/?q=ai:lv.shaoqing"Xiang, Ju"https://zbmath.org/authors/?q=ai:xiang.ju"Feng, Jingyu"https://zbmath.org/authors/?q=ai:feng.jingyu"Wang, Honggang"https://zbmath.org/authors/?q=ai:wang.honggang"Lu, Guangyue"https://zbmath.org/authors/?q=ai:lu.guangyue"Li, Min"https://zbmath.org/authors/?q=ai:li.min.9|li.min.2|li.min.10|li.min|li.min.3|li.min.6|li.min.7|li.min.5|li.min.4|li.min.8|li.min.1Godsil-McKay switching for mixed and gain graphs over the circle group.https://zbmath.org/1459.051102021-05-28T16:06:00+00:00"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: In this paper we describe two methods, both inspired from Godsil-McKay switching on simple graphs, to build cospectral gain graphs whose gain group consists of the complex numbers of modulus 1 (the circle group). The results obtained here can be also applied to the Hermitian matrix of mixed graphs.Full cycle extendability of locally connected \(K_{1,4}\)-restricted graphs.https://zbmath.org/1459.051392021-05-28T16:06:00+00:00"Irzhavskiĭ, P. A."https://zbmath.org/authors/?q=ai:irzhavskii.p-a"Orlovich, Yu. L."https://zbmath.org/authors/?q=ai:orlovich.yu-l|orlovich.yury-lSummary: In this paper we show that a connected locally connected \(K_{1,4}\)-restricted graph on at least three vertices is either fully cycle extendable or isomorphic to one of the five exceptional (non-Hamiltonian) graphs. This result generalizes several known results on the existence of Hamiltonian cycles in locally connected graphs. We also propose a polynomial time algorithm for finding a Hamiltonian cycle in graphs under consideration.Pseudorandom number generator by \(p\)-adic chaos and Ramanujan expander graphs.https://zbmath.org/1459.053072021-05-28T16:06:00+00:00"Naito, Koichiro"https://zbmath.org/authors/?q=ai:naito.koichiroSummary: In our previous paper [the author, ``Randomness of \(p\)-adic discrete dynamical systems and its applications to cryptosystems'', in: Proceedings of the 10th international conference on nonlinear analysis and convex analysis, NACA 2017, Chitose, Japan, 2017. Yokohama: Yokohama Publishers (to appear)], applying chaotic properties of the \(p\)-adic dynamical system given by the \(p\)-adic logistic map, we constructed a new pseudorandom number generator. In this paper, using the pseudorandom sequences given by this generator, we construct random adjacency matrices and their random graphs. Then we numerically show that the eigenvalue distributions of these random matrices have the characteristical properties of the adjacency matrices of Ramanujan graphs.Perfect matchings and \(K_{1,p}\)-restricted graphs.https://zbmath.org/1459.052612021-05-28T16:06:00+00:00"Irzhavski, Pavel A."https://zbmath.org/authors/?q=ai:irzhavski.pavel-a"Orlovich, Yury L."https://zbmath.org/authors/?q=ai:orlovich.yury-lSummary: A graph is called \(K_{1,p}\)-restricted \((p \geq 3)\) if for every vertex of the graph there are at least \(p - 2\) edges between any \(p\) of its neighbours. We establish sufficient conditions for the existence of a perfect matching in \(K_{1,p}\)-restricted graphs in terms of their connectivity and vertex degrees. These conditions imply, in particular, the classical Petersen's result: any 2-edge-connected 3-regular graph contains a perfect matching.On leaf node edge switchings in spanning trees of de Bruijn graphs.https://zbmath.org/1459.051522021-05-28T16:06:00+00:00"Roy, Suman"https://zbmath.org/authors/?q=ai:roy.suman"Krishnaswamy, Srinivasan"https://zbmath.org/authors/?q=ai:krishnaswamy.srinivasan"Kumar, P. Vinod"https://zbmath.org/authors/?q=ai:kumar.p-vinodSummary: An \(n\)-th order \(k\)-ary de Bruijn sequence is a cyclic sequence of length \(k^n\) which contains every possible \(k\)-ary subsequence of length \(n\) exactly once during each period. In this paper, we show that, if we fix the initial \(n\) bits, any \(n\)-th order de Bruijn sequence can be transformed to another using a sequence of transformations.
For the entire collection see [Zbl 1411.65006].Zero forcing number of a graph in terms of the number of pendant vertices.https://zbmath.org/1459.052482021-05-28T16:06:00+00:00"Wang, Xinlei"https://zbmath.org/authors/?q=ai:wang.xinlei"Wong, Dein"https://zbmath.org/authors/?q=ai:wong.dein"Zhang, Yuanshuai"https://zbmath.org/authors/?q=ai:zhang.yuanshuaiSummary: The zero forcing number of a graph has recently become an interesting graph parameter studied in its own right since its introduction by the `AIM Minimum Rank -- Special Graphs Work Group'. In this article, we are interested in bounding the zero forcing number of a graph by the number of pendant vertices. Let \(p(G)\) and \(\varphi(G)\) be the number of pendant vertices and the cyclomatic number of \(G\). If \(G\) is a connected graph with at least one edge that is not a cycle, then \(Z(G) \leq p(G) + 2 \varphi(G) - 1\), the extremal graphs whose zero forcing number attain the upper bound are characterized. For a connected graph \(G\) with at least one edge that is not a path, we prove \(Z(G) \geq p(G) - \omega(G)\), where \(\omega(G)\) is the number of similar equivalency classes of the set of all pendant vertices of \(G\) and two pendant vertices are said to be similar if they are the terminal vertices of a common major vertex. The extremal graphs with zero forcing number \(p(G) - \omega(G)\) are also characterized.On a question of Haemers regarding vectors in the nullspace of Seidel matrices.https://zbmath.org/1459.051542021-05-28T16:06:00+00:00"Akbari, S."https://zbmath.org/authors/?q=ai:akbari.saeeid|akbari.samira|akbari.saieed|akbari.samin|akbari.shahabeddin|akbari.soheil"Cioabă, S. M."https://zbmath.org/authors/?q=ai:cioaba.sebastian-m"Goudarzi, S."https://zbmath.org/authors/?q=ai:goudarzi.sobhan"Niaparast, Aidan"https://zbmath.org/authors/?q=ai:niaparast.aidan"Tajdini, Artin"https://zbmath.org/authors/?q=ai:tajdini.artinSummary: \textit{W. H. Haemers} [MATCH Commun. Math. Comput. Chem. 68, No. 3, 653--659 (2012; Zbl 1289.05290)] asked the following question: If \(S\) is the Seidel matrix of a graph of order \(n\) and \(S\) is singular, does there exist an eigenvector of \(S\) corresponding to 0 which has only \(\pm 1\) elements? In this paper, we construct infinite families of graphs which give a negative answer to this question. One of our constructions implies that for every natural number \(N\), there exists a graph whose Seidel matrix \(S\) is singular such that for any integer vector in the nullspace of \(S\), the absolute value of any entry in this vector is more than \(N\). We also derive some characteristics of vectors in the nullspace of Seidel matrices, which lead to some necessary conditions for the singularity of Seidel matrices. Finally, we obtain some properties of the graphs which affirm the above question.No signed graph with the nullity \(\eta(G,\sigma)=|V(G)|-2m(G)+2c(G)-1\).https://zbmath.org/1459.051122021-05-28T16:06:00+00:00"Lu, Yong"https://zbmath.org/authors/?q=ai:lu.yong.2|lu.yong.3|lu.yong|lu.yong.1"Wu, Jingwen"https://zbmath.org/authors/?q=ai:wu.jingwenSummary: Let \(G^\sigma=(G,\sigma)\) be a signed graph and \(A(G,\sigma)\) be its adjacency matrix. Denote by \(m(G)\) the matching number of \(G\). Let \(\eta(G,\sigma)\) be the nullity of \((G,\sigma)\). \textit{S. He} et al. [ibid. 572, 273--291 (2019; Zbl 1411.05107)] proved that
\[
|V(G)|-2m(G)-c(G)\leq\eta(G,\sigma)\leq |V(G)|-2m(G)+2c(G),
\]
where \(c(G)\) is the dimension of cycle space of \(G\). Signed graphs reaching the lower bound or the upper bound are respectively characterized by the same paper. In this paper, we will prove that there are no signed graphs with nullity \(|V(G)|-2m(G)+2c(G)-1\). We also prove that there are infinitely many signed graphs with nullity \(|V(G)|-2m(G)+2c(G)-s\), \((0\leq s\leq 3c(G)\), \(s\neq 1)\) for a given \(c(G)\).Graphs with few trivial characteristic ideals.https://zbmath.org/1459.051152021-05-28T16:06:00+00:00"Alfaro, Carlos A."https://zbmath.org/authors/?q=ai:alfaro.carlos-a"Barrus, Michael D."https://zbmath.org/authors/?q=ai:barrus.michael-d"Sinkovic, John"https://zbmath.org/authors/?q=ai:sinkovic.john-henry"Villagrán, Ralihe R."https://zbmath.org/authors/?q=ai:villagran.ralihe-rSummary: We give a characterization of the graphs with at most three trivial characteristic ideals. This implies the complete characterization of the regular graphs whose critical groups have at most three invariant factors equal to 1 and the characterization of the graphs whose Smith groups have at most 3 invariant factors equal to 1. We also give an alternative and simpler way to obtain the characterization of the graphs whose Smith groups have at most 3 invariant factors equal to 1, and a list of minimal forbidden graphs for the family of graphs with Smith group having at most 4 invariant factors equal to 1.The first two maximum ABC spectral radii of bicyclic graphs.https://zbmath.org/1459.051962021-05-28T16:06:00+00:00"Yuan, Yan"https://zbmath.org/authors/?q=ai:yuan.yan"Du, Zhibin"https://zbmath.org/authors/?q=ai:du.zhibinSummary: The ABC matrix of a graph \(G\), proposed by \textit{E. Estrada} [J. Math. Chem. 55, No. 4, 1021--1033 (2017; Zbl 1380.92097)], can be regarded as a weighed version of adjacency matrices of graphs, in which the \((u, v)\)-entry is equal to \(\sqrt{\frac{du+d_v-2}{d_ud_v}}\) if \(uv\) is an edge of a graph \(G\), and 0 otherwise, where \(d_u\) represents the degree of \(u\) in \(G\). The research about ABC spectral radius (largest eigenvalue of ABC matrix) of graphs is rather active in recent years. In this paper, we characterize the bicyclic graphs with the first two maximum ABC spectral radii, which are just the unique two bicyclic graphs of order \(n\) with maximum degree \(n-1\) if \(n\geq 7\).Constraints on Brouwer's Laplacian spectrum conjecture.https://zbmath.org/1459.051672021-05-28T16:06:00+00:00"Cooper, Joshua N."https://zbmath.org/authors/?q=ai:cooper.joshua-nSummary: Brouwer's Conjecture states that, for any graph \(G\), the sum of the \(k\) largest (combinatorial) Laplacian eigenvalues of \(G\) is at most \(|E(G)|+\binom{k+1}{2}\), \(1\leq k\leq n\). We present several interrelated results establishing Brouwer's conjecture for a wide range of graphs \(G\) and parameters \(k\). In particular, we show that (1) is true for low-arboricity graphs, and in particular for planar \(G\) when \(k\geq 11\); (2) is true whenever the variance of the degree sequence is not very high, generalizing previous results for \(G\) regular or random; (3) is true if \(G\) belongs to a hereditarily spectrally-bounded class and \(k\) is sufficiently large as a function of \(k\), in particular \(k\geq\sqrt{32n}\) for bipartite graphs; (4) holds unless \(G\) has edge-edit distance \(<k\sqrt{2n}=O(n^{3/2})\) from a split graph; (5) no \(G\) violates the conjectured upper bound by more than \(O(n^{5/4})\), and bipartite \(G\) by no more than \(O(n)\); and (6) holds for all \(k\) outside an interval of length \(O( n^{3/4})\). Furthermore, we show that a natural generalization of Brouwer's conjecture surprisingly is quite false: asymptotically almost surely, a uniform random signed complete graph violates the conjectured bound by \(\Omega(n)\).The distance energy of clique trees.https://zbmath.org/1459.051762021-05-28T16:06:00+00:00"Jin, Ya-Lei"https://zbmath.org/authors/?q=ai:jin.yalei"Gu, Rui"https://zbmath.org/authors/?q=ai:gu.rui"Zhang, Xiao-Dong"https://zbmath.org/authors/?q=ai:zhang.xiaodongSummary: The distance energy of a simple connected graph \(G\) is defined as the sum of absolute values of its distance eigenvalues. In this paper, we mainly give a positive answer to a conjecture of distance energy of clique trees proposed by \textit{H. Lin} et al. [ibid. 467, 29--39 (2015; Zbl 1304.05093)].Sedentary quantum walks.https://zbmath.org/1459.051712021-05-28T16:06:00+00:00"Godsil, Chris"https://zbmath.org/authors/?q=ai:godsil.christopher-davidSummary: Let \(X\) be a graph with adjacency matrix \(A\). The continuous quantum walk on \(X\) is determined by the unitary matrices \(U(t)=\exp(itA)\) (for \(t\in\mathbb{R})\). If \(X\) is the complete graph \(K_n\) and \(a\in V(X)\), then
\[
1-|U(t)_{a,a}|\leq 2/n.
\]
Roughly speaking, this means that a quantum walk on a complete graph stays home with high probability. We say that a family of graphs is sedentary if there is a constant \(c\) such that \(1-|U(t)_{a,a}|\leq c/|V(X)|\) for all \(t\). In this paper we investigate this condition, and produce further examples of sedentary graphs.
A cone over a graph \(X\) is the graph we get by adjoining a new vertex and making it adjacent to each vertex of \(X\). We prove that if \(X\) is the cone over an \(\ell\)-regular graph on \(n\) vertices, then \(|U(t)_{a,a}|\leq\ell^2/(\ell^2+4n)\). It follows that if we choose \(\ell\) and \(n\) such that \(n/\ell^2\to 0\), then a continuous quantum walk starting on the ``conical'' vertex will remain there with probability close to 1. On the other hand, if \(\ell\leq 2\), we show there is a time \(t\) such that all entries in the \(a\)-column of \(U(t)e_a\) have absolute value \(1/\sqrt{n}\). We show that there are large classes of strongly regular graphs such that \(1-|U(t)_{a,a}|\leq c/V(X)\) for some constant \(c\). On the other hand, for Paley graphs on \(n\) vertices, we prove that if \(t=\pi/\sqrt{n}\), then \(|U(t)_{a,a}|\leq 1/n\).Spectral radius and matchings in graphs.https://zbmath.org/1459.051842021-05-28T16:06:00+00:00"O, Suil"https://zbmath.org/authors/?q=ai:o.suilSummary: A perfect matching in a graph \(G\) is a set of disjoint edges covering all vertices of \(G\). Let \(\rho(G)\) be the spectral radius of a graph \(G\), and let \(\theta(n)\) be the largest root of \(x^3-(n-4)x^2-(n-1)x+2(n-4)=0\). In this paper, we prove that for a positive even integer \(n\geq 8\) or \(n=4\), if \(G\) is an \(n\)-vertex graph with \(\rho(G)>\theta(n)\), then \(G\) has a perfect matching; for \(n=6\), if \(\rho(G)>\frac{1+\sqrt{33}}{2}\), then \(G\) has a perfect matching. It is sharp for every positive even integer \(n \geq 4\) in the sense that there are graphs \(H\) with \(\rho(H)=\theta^\prime(n)\) and no perfect matching, where \(\theta^\prime(n)=\theta(n)\) if \(n=4\) or \(n\geq 8\) and \(\theta^\prime(6)=\frac{1+\sqrt{33}}{2}\).On graphs with adjacency and signless Laplacian matrices eigenvectors entries in \(\{-1,+1\}\).https://zbmath.org/1459.051562021-05-28T16:06:00+00:00"Alencar, Jorge"https://zbmath.org/authors/?q=ai:alencar.jorge"de Lima, Leonardo"https://zbmath.org/authors/?q=ai:de-lima.leonardo-sSummary: \textit{H. S. Wilf} [J. Comb. Theory, Ser. B 40, 113--117 (1986; Zbl 0598.05047)] asked what kind of graphs have an eigenvector with entries formed only by \(\pm 1\). In this paper, we answer this question for the adjacency, Laplacian, and signless Laplacian matrices of a graph. To this end, we generalize the concept of an exact graph to the adjacency and signless Laplacian matrices. Infinite families of exact graphs for all those matrices are presented.Integral unicyclic graphs.https://zbmath.org/1459.051622021-05-28T16:06:00+00:00"Braga, Rodrigo O."https://zbmath.org/authors/?q=ai:braga.rodrigo-o"Del-Vecchio, Renata R."https://zbmath.org/authors/?q=ai:del-vecchio.renata-raposo"Rodrigues, Virgínia M."https://zbmath.org/authors/?q=ai:rodrigues.virginia-mSummary: A graph is integral if the spectrum of its adjacency matrix consists entirely of integers. The question about which unicyclic graphs are integral remains open. We contribute to this problem by presenting three infinite families of integral unicyclic graphs. These families are generated by distinct particular solutions of a Diophantine equation. We also show that two integral unicyclic graphs found through a computer search that do not belong to the families we present are unique with their shapes. Necessary conditions for certain unicyclic graphs to be integral are also given.The main eigenvalues of signed graphs.https://zbmath.org/1459.051552021-05-28T16:06:00+00:00"Akbari, S."https://zbmath.org/authors/?q=ai:akbari.saieed"França, Franscisca A. M."https://zbmath.org/authors/?q=ai:franca.franscisca-a-m"Ghasemian, E."https://zbmath.org/authors/?q=ai:ghasemian.ebrahim"Javarsineh, M."https://zbmath.org/authors/?q=ai:javarsineh.mehrnoosh"de Lima, Leonardo S."https://zbmath.org/authors/?q=ai:de-lima.leonardo-sSummary: A signed graph \(G^\sigma\) is an ordered pair \((V(G), E(G))\), where \(V(G)\) and \(E(G)\) are the set of vertices and edges of \(G\), respectively, along with a map \(\sigma\) that signs every edge of \(G\) with +1 or \(-1\). An eigenvalue of the associated adjacency matrix of \(G^\sigma\), denoted by \(A(G^\sigma)\), is a main eigenvalue if the corresponding eigenspace has a non-orthogonal eigenvector to the all-one vector \(j\). We conjectured that for every graph \(G\neq K_2\), \(K_4\backslash\{e\}\), there is a switching \(\sigma\) such that all eigenvalues of \(G^\sigma\) are main. We show that this conjecture holds for every Cayley graphs, distance-regular graphs, vertex and edge-transitive graphs as well as double stars and paths.Connectivity and eigenvalues of graphs with given girth or clique number.https://zbmath.org/1459.051732021-05-28T16:06:00+00:00"Hong, Zhen-Mu"https://zbmath.org/authors/?q=ai:hong.zhenmu"Lai, Hong-Jian"https://zbmath.org/authors/?q=ai:lai.hong-jian"Xia, Zheng-Jiang"https://zbmath.org/authors/?q=ai:xia.zhengjiangSummary: Let \(\kappa^\prime(G)\), \(\mu_{n-1}(G)\) and \(\mu_1(G)\) denote the edge-connectivity, the algebraic connectivity and the Laplacian spectral radius of \(G\), respectively. In this paper, we prove that for integers \(k\geq 2\) and \(r\geq 2\), and any simple graph \(G\) of order \(n\) with minimum degree \(\delta\geq k\), girth \(g\geq 3\) and clique number \(\omega(G)\leq r\), the edge-connectivity \(\kappa^\prime(G)\geq k\) if \(\mu_{n-1}(G)\geq\frac{(k-1)n}{N(\delta,g)(n-N(\delta,g))}\) or if \(\mu_{n-1}(G)\geq\frac{(k-1)n}{\varphi (\delta,r)(n-\varphi(\delta,r))}\), where \(N(\delta,g)\) is the Moore bound on the smallest possible number of vertices such that there exists a \(\delta\)-regular simple graph with girth \(g\), and \(\varphi(\delta,r)=\max\{\delta+1,\lfloor\frac{r\delta}{r-1} \rfloor\}\). Analogue results involving \(\mu_{n-1}(G)\) and \(\frac{\mu_1(G)}{\mu_{n-1}(G)}\) to characterize vertex-connectivity of graphs with fixed girth and clique number are also presented. Former results in [\textit{H. Liu} et al., ibid. 439, No. 12, 3777--3784 (2013; Zbl 1282.05134); \textit{R. Liu} et al., ibid. 578, 411--424 (2019; Zbl 1419.05130); \textit{Z.-M. Hong} et al., ibid. 579, 72--88 (2019; Zbl 1419.05126); \textit{R. Liu} et al., Appl. Math. Comput. 344--345, 141--149 (2019; Zbl 1428.05198); \textit{A. Abiad} et al., Electron. J. Linear Algebra 34, 428--443 (2018; Zbl 1396.05063)] are improved or extended.On \((k+1)\)-line graphs of \(k\)-trees and their nullities.https://zbmath.org/1459.051832021-05-28T16:06:00+00:00"Oliveira, Allana S. S."https://zbmath.org/authors/?q=ai:oliveira.allana-s-s"de Freitas, Maria Aguieiras A."https://zbmath.org/authors/?q=ai:de-freitas.maria-aguieiras-a"Vinagre, Cybele T. M."https://zbmath.org/authors/?q=ai:vinagre.cybele-t-m"Markenzon, Lilian"https://zbmath.org/authors/?q=ai:markenzon.lilianSummary: The nullity of a graph, denoted by \(\eta(G)\), is the multiplicity of zero as an eigenvalue of the adjacency matrix of \(G\). \textit{S. Fiorini} et al. [ibid. 397, 245--251 (2005; Zbl 1068.05015)] presented an upper bound for the nullities of trees in terms of the order \(n\) and the maximum degree \(\Delta\). In our work we show that, under the same conditions, all possible nullities below that bound are attained. This result allows us to obtain an upper bound for the nullities of the \((k+1)\)-line graphs of a particular family of \(k\)-trees, generalizing a known result about the nullities of line graphs of trees. We also present a general characterization for the \((k+1)\)-line graphs of \(k\)-trees.Null decomposition of bipartite graphs without cycles of length 0 modulo 4.https://zbmath.org/1459.052622021-05-28T16:06:00+00:00"Jaume, Daniel A."https://zbmath.org/authors/?q=ai:jaume.daniel-a"Molina, Gonzalo"https://zbmath.org/authors/?q=ai:molina.gonzalo"Pastine, Adrián"https://zbmath.org/authors/?q=ai:pastine.adrianSummary: In this work we study the null space of bipartite graphs without cycles of length 0 modulo 4 (denoted as \(C_{4k}\)-free bipartite graphs), and its relation to structural properties. We extend the Null Decomposition of trees, introduced by \textit{D. A. Jaume} and \textit{G. Molina} [Discrete Math. 341, No. 3, 836--850 (2018; Zbl 1378.05026)], to \(C_{4k}\)-free bipartite graphs. This decomposition uses the null space of the adjacency matrix of a graph \(G\) to decompose it into two different types of graphs: \(C_N(G)\) and \(C_S(G)\). \(C_N\) has perfect matching number. \(C_S(G)\) has a unique maximum independent set. We obtain formulas for the independence number and the matching number of a \(C_{4k}\)-free bipartite graph using this decomposition. We also show how the number of maximum matchings and the number of maximum independent sets in a \(C_{4k}\)-free bipartite graph are related to its null decomposition.Symmetric completions of cycles and bipartite graphs.https://zbmath.org/1459.051642021-05-28T16:06:00+00:00"Cohen, Nir"https://zbmath.org/authors/?q=ai:cohen.nir"Pereira, Edgar"https://zbmath.org/authors/?q=ai:pereira.edgarSummary: The analysis of symmetric completions of partial matrices associated with a simple graph \(G\), in terms of inertias and minimal rank, simplifies dramatically when \(G\) is bipartite. Essentially, it is equivalent to the analysis of an associated non-symmetric completion problem. All the inertias down to the minimum rank can be obtained, but the minimal rank itself remains NP-hard for general graphs in this class. The class of bipartite graphs includes even cycles but excludes odd cycles. By the above reduction we provide relatively sharp minimal rank estimates for even cycles and discuss some counter-examples raised by odd cycles.\(Q\)-integral graphs with at most two vertices of degree greater than or equal to three.https://zbmath.org/1459.051822021-05-28T16:06:00+00:00"Novanta, Anderson Fernandes"https://zbmath.org/authors/?q=ai:novanta.anderson-fernandes"de Lima, Leonardo"https://zbmath.org/authors/?q=ai:de-lima.leonardo-s"Oliveira, Carla Silva"https://zbmath.org/authors/?q=ai:silva-oliveira.carlaSummary: Let \(G\) a graph on \(n\) vertices. The signless Laplacian matrix of \(G\), denoted by \(Q(G)\), is defined as \(Q(G)=D(G)+A(G)\), where \(A(G)\) is the adjacency matrix of \(G\) and \(D(G)\) is the diagonal matrix of the degrees of \(G\). A graph \(G\) is said to be \(Q\)-integral if all eigenvalues of the matrix \(Q(G)\) are integers. In this paper, we characterize all \(Q\)-integral graphs among all connected graphs with at most two vertices of degree greater than or equal to three.Stick number of non-paneled knotless spatial graphs.https://zbmath.org/1459.570262021-05-28T16:06:00+00:00"Flapan, Erica"https://zbmath.org/authors/?q=ai:flapan.erica"Kozai, Kenji"https://zbmath.org/authors/?q=ai:kozai.kenji"Nikkuni, Ryo"https://zbmath.org/authors/?q=ai:nikkuni.ryoA graph embedded in \(\mathbb{R}^3\) is said to be paneled if every cycle in the graph bounds a disk whose interior is disjoint from the graph. The non-paneled knotless stick number of a graph is the minimum number of sticks required to create an embedding of the graph which is not paneled and yet contains no knots.
In this paper the authors are interested in the non-paneled knotless stick number of complete graphs. It is known that \(K_3\) is paneled if and only if it is knotless, and every embedding of \(K_n\) with \(n \ge 7\) contains a non-trivial knot. Moreover the non-paneled knotless stick number of \(K_6\) is 15. So the only complete graphs whose non-paneled knotless stick number is unknown are \(K_4\) and \(K_5\).
The authors show that the minimum number of sticks required to construct a non-paneled knotless embedding of \(K_4\) is 8 and of \(K_5\) is 12 or 13.
Using the results about \(K_4\) the authors show that the probability that a random linear embedding of \(K_{3,3}\) in a cube is in the form of a Möbius ladder is \(0.97380 \pm 0.00003\), and offer this as a possible explanation for why \(K_{3,3}\) subgraphs of metalloproteins occur primarily in this form.
Reviewer: Claus Ernst (Bowling Green)On the spectral radius of block graphs with prescribed independence number \(\alpha\).https://zbmath.org/1459.051662021-05-28T16:06:00+00:00"Conde, Cristian M."https://zbmath.org/authors/?q=ai:conde.cristian-m"Dratman, Ezequiel"https://zbmath.org/authors/?q=ai:dratman.ezequiel"Grippo, Luciano N."https://zbmath.org/authors/?q=ai:grippo.luciano-norbertoSummary: Let \(\mathcal{G}(n,\alpha)\) be the class of block graphs on \(n\) vertices and prescribed independence number \(\alpha\). In this article we prove that the maximum spectral radius \(\rho(G)\), among all graphs \(G\in\mathcal{G}(n,\alpha)\), is reached at a unique graph. As a byproduct we obtain an upper for \(\rho(G)\), when \(G\in\mathcal{G}(n,\alpha)\).On the spectrum of hypergraphs.https://zbmath.org/1459.051582021-05-28T16:06:00+00:00"Banerjee, Anirban"https://zbmath.org/authors/?q=ai:banerjee.anirbanSummary: Here, we introduce different connectivity matrices and study their eigenvalues to explore various structural properties of a general hypergraph. We investigate how the diameter, connectivity and vertex chromatic number of a hypergraph are related to the spectrum of these matrices. Different properties of a regular hypergraph are also characterized by the same. Cheeger constant on a hypergraph is defined and its spectral bounds have been derived for a connected general hypergraph. Random walk on a general hypergraph can also be well studied by analyzing the spectrum of the transition probability operator defined on the hypergraph. We also introduce Ricci curvature on a general hypergraph and study its relation with the hypergraph spectra.Integer Laplacian eigenvalues of chordal graphs.https://zbmath.org/1459.051532021-05-28T16:06:00+00:00"Abreu, Nair"https://zbmath.org/authors/?q=ai:abreu.nair-maria-maia-de"Justel, Claudia Marcela"https://zbmath.org/authors/?q=ai:justel.claudia-marcela"Markenzon, Lilian"https://zbmath.org/authors/?q=ai:markenzon.lilianSummary: In this paper, structural properties of chordal graphs are studied, establishing a relationship between these structures and integer Laplacian eigenvalues. We present the characterization of chordal graphs with equal vertex and algebraic connectivities, by means of the vertices that compose the minimal vertex separators of the graph; we stablish a sufficient condition for the cardinality of a maximal clique to appear as an integer Laplacian eigenvalue. Finally, we review two subclasses of chordal graphs, showing particular properties.On the one edge algorithm for the orthogonal double covers.https://zbmath.org/1459.052552021-05-28T16:06:00+00:00"El-Shanawany, R."https://zbmath.org/authors/?q=ai:el-shanawany.r-a"El-Mesady, A."https://zbmath.org/authors/?q=ai:el-mesady.ahmedSummary: The existing problem of the orthogonal double covers of the graphs is well-known in the theory of combinatorial designs. In this paper, a new technique called the one edge algorithm for constructing the orthogonal double covers of the complete bipartite graphs by copies of a graph is introduced. The advantage of this algorithm is that it is accessible to discrete mathematicians not intimately familiar with the theory of the orthogonal double covers.Fourier analysis of a delayed Rulkov neuron network.https://zbmath.org/1459.920072021-05-28T16:06:00+00:00"Lozano, Roberto"https://zbmath.org/authors/?q=ai:lozano.roberto"Used, Javier"https://zbmath.org/authors/?q=ai:used.javier"Sanjuán, Miguel A. F."https://zbmath.org/authors/?q=ai:sanjuan.miguel-a-fOne refers to the map-based neuron model, the chaotic Rulkov model, defined by \(x_n=\frac {\alpha}{1+x^2_{n-1}}+y_{n-1}\), \(y_n= y_{n-1}-\beta(x_{n-1} -\sigma)\) where \(\alpha\), \(\beta\) and \(\sigma\) are constant parameters, \(x_n\) represents the transmembrane voltage of a single neuron and \(y_n\) designs the slow gating process.
The main goal is to study the global behavior of coupled neurons in a neural network. In this paper two neural networks of chaotic Rulkov neurons are considered, the small-world and the Erdös-Rényi network models. An algorithm that improves the synchronization of the neuron network is developed. The algorithm computes a delay that included into the electrical coupling improves the synchronization of the neural network. The steps of the delay algorithm are: i) Compute the fundamental frequency (non-zero frequency with the largest amplitude) of each neuron, ii) Calculate the period that corresponds to the fundamental frequency of the previous step, iii) Use this period as the delay and compute again. Frequency analysis of a single Rulkov neuron has been performed and the behavior of neurons in small-world networks and in Erdös-Rényi networks have been analyzed.
Reviewer: Claudia Simionescu-Badea (Wien)On cardinality of complementarity spectra of connected graphs.https://zbmath.org/1459.051862021-05-28T16:06:00+00:00"Seeger, Alberto"https://zbmath.org/authors/?q=ai:seeger.alberto"Sossa, David"https://zbmath.org/authors/?q=ai:sossa.davidSummary: This work deals with complementarity spectra of connected graphs and, specifically, with the associated concept of spectral capacity of a finite set of connected graphs. The cardinality of the complementarity spectrum of a connected graph \(G\) serves as lower bound for the number of connected induced subgraphs of \(G\). Motivated by this observation, we establish various results on cardinality of complementarity spectra. Special attention is paid to the asymptotic behavior of spectral capacities as the number of vertices goes to infinity.Generalized power sum and Newton-Girard identities.https://zbmath.org/1459.050142021-05-28T16:06:00+00:00"Bera, Sudip"https://zbmath.org/authors/?q=ai:bera.sudip"Mukherjee, Sajal Kumar"https://zbmath.org/authors/?q=ai:mukherjee.sajal-kumarSummary: In this article we prove an algebraic identity which significantly generalizes the formula for sum of powers of consecutive integers involving Stirling numbers of the second kind. Also we have obtained a generalization of Newton-Girard power sum identity.Comparison of sufficient degree based conditions for Hamiltonian graph.https://zbmath.org/1459.051462021-05-28T16:06:00+00:00"Abrosimov, M. B."https://zbmath.org/authors/?q=ai:abrosimov.mikhail-borisovichSummary: A graph \(G\) is said to be Hamiltonian if it contains a spanning cycle, i.e. a cycle that passes through all of its vertices. The Hamiltonian cycle problem is NP-complete, and many sufficient conditions have been found after the first sufficient condition proposed by \textit{G. A. Dirac} [Proc. Lond. Math. Soc. (3) 2, 69--81 (1952; Zbl 0047.17001)]. In this paper for all graphs with a number of vertices up to 12, the most popular sufficient degree based conditions for Hamiltonian graph are compared: theorems by Dirac, Ore, Posa, Chvatal and Bondy-Chvatal. The number of graphs which satisfy each condition is counted. With the number of vertices from 3 to 12, the number of graphs satisfying the Dirac condition is 1, 3, 3, 19, 29, 424, 1165, 108376, 868311, 495369040; the number of graphs satisfying the Ore condition is 1, 3, 5, 21, 68, 503, 4942, 128361, 5315783, 575886211; the number of graphs satisfying the Posha condition is 1, 3, 6, 31, 190, 2484, 53492, 2683649, 216082075, 40913881116; the number of graphs satisfying the Chvatal condition is 1, 3, 6, 34, 194, 2733, 54435, 2914167, 218674224, 43257613552 and the number of graphs satisfying the Bondy-Chvatal condition is 1, 3, 7, 45, 352, 5540, 157016, 8298805, 802944311, 141613919605. This result is the best one: about 90 \% of the Hamiltonian graphs satisfy condition proposed by \textit{J. A. Bondy} and \textit{V. Chvatal} [Discrete Math. 15, 111--135 (1976; Zbl 0331.05138)]. The FHCP Challenge Set is a collection of 1001 instances of the Hamiltonian Cycle Problem, ranging in size from 66 vertices up to 9528. All graphs from the FHCP Challenge Set were checked whether they satisfy considered conditions. It turned out that 11 graphs satisfy the Bondy-Chvatal condition: no. 59 (with 400 vertices), no. 72 (460), no. 79 (480), no. 84 (500), no. 90 (510), no. 96 (540), no. 128 (677), no. 134 (724), no. 150 (823), no. 162 (909), and no. 188 (with 1123 vertices). For these graphs we can check and find Hamiltonian cycle using Bondy-Chvatal's theorem with computational complexity \(O(n^4)\) where \(n\) is the number of graph vertices.Ear-slicing for matchings in hypergraphs.https://zbmath.org/1459.052732021-05-28T16:06:00+00:00"Sebő, András"https://zbmath.org/authors/?q=ai:sebo.andrasSummary: We study when a given edge of a factor-critical graph is contained in a matching avoiding exactly one, pregiven vertex of the graph. We then apply the results to always partition the vertex-set of a 3-regular, 3-uniform hypergraph into at most one triangle (hyperedge of size 3) and edges (subsets of size 2 of hyperedges), corresponding to the intuition, and providing new insight to triangle and edge packings of \textit{G. Cornuéjols} and \textit{W. R. Pulleyblank}'s [Combinatorica 3, 35--52 (1983; Zbl 0535.05038)]. The existence of such a packing can be considered to be a hypergraph variant of \textit{J. Petersen}'s theorem on perfect matchings [Acta Math. 15, 193--220 (1891; JFM 23.0115.03)], and leads to a simple proof for a sharpening of Lu's theorem on antifactors of graphs [\textit{H. Lu} et al., ``Antifactor of regular bipartite graphs'', Preprint, \url{arXiv:1511.09277}].On degree sum conditions and vertex-disjoint chorded cycles.https://zbmath.org/1459.050462021-05-28T16:06:00+00:00"Elliott, Bradley"https://zbmath.org/authors/?q=ai:elliott.bradley"Gould, Ronald J."https://zbmath.org/authors/?q=ai:gould.ronald-j"Hirohata, Kazuhide"https://zbmath.org/authors/?q=ai:hirohata.kazuhideSummary: In this paper, we consider a general degree sum condition sufficient to imply the existence of \(k\) vertex-disjoint chorded cycles in a graph \(G\). Let \(\sigma_t(G)\) be the minimum degree sum of \(t\) independent vertices of \(G\). We prove that if \(G\) is a graph of sufficiently large order and \(\sigma_t(G)\geq 3kt-t+1\) with \(k\geq 1\), then \(G\) contains \(k\) vertex-disjoint chorded cycles. We also show that the degree sum condition on \(\sigma_t(G)\) is sharp. To do this, we also investigate graphs without chorded cycles.\(\lambda\)-core distance partitions.https://zbmath.org/1459.051812021-05-28T16:06:00+00:00"Mifsud, Xandru"https://zbmath.org/authors/?q=ai:mifsud.xandruSummary: The \(\lambda\)-core vertices of a graph correspond to the non-zero entries of some eigenvector of \(\lambda\) for a universal adjacency matrix \(\mathfrak{U}\) of the graph. We define a partition of the vertex set \(V\) based on the \(\lambda\)-core vertex set and its neighbourhoods at a distance \(r\), and give a number of results relating the structure of the graph to this partition. For such partitions, we also define an entropic measure for the information content of a graph, related to every distinct eigenvalue \(\lambda\) of \(\mathfrak{U}\), and discuss its properties and potential applications.Separability of Schur rings over abelian groups of odd order.https://zbmath.org/1459.053522021-05-28T16:06:00+00:00"Ryabov, Grigory"https://zbmath.org/authors/?q=ai:ryabov.grigorii-konstantinovichSummary: An \(S\)-ring (a Schur ring) is said to be separable with respect to a class of groups \(\mathcal{K}\) if every algebraic isomorphism from the \(S\)-ring in question to an \(S\)-ring over a group from \(\mathcal{K}\) is induced by a combinatorial isomorphism. A finite group \(G\) is said to be separable with respect to \(\mathcal{K}\) if every \(S\)-ring over \(G\) is separable with respect to \(\mathcal{K}\). We prove that every abelian group \(G\) of order \(9p\), where \(p\) is a prime, is separable with respect to the class of all finite abelian groups. Modulo previously obtained results, this completes a classification of noncyclic abelian groups of odd order that are separable with respect to the class of all finite abelian groups. This also implies that the relative Weisfeiler-Leman dimension of a Cayley graph over \(G\) with respect to the class of all Cayley graphs over abelian groups is at most 2.Decomposing degenerate graphs into locally irregular subgraphs.https://zbmath.org/1459.052522021-05-28T16:06:00+00:00"Bensmail, Julien"https://zbmath.org/authors/?q=ai:bensmail.julien"Dross, François"https://zbmath.org/authors/?q=ai:dross.francois"Nisse, Nicolas"https://zbmath.org/authors/?q=ai:nisse.nicolasSummary: A (undirected) graph is locally irregular if no two of its adjacent vertices have the same degree. A decomposition of a graph \(G\) into \(k\) locally irregular subgraphs is a partition \(E_1,\dots ,E_k\) of \(E(G)\) into \(k\) parts each of which induces a locally irregular subgraph. Not all graphs decompose into locally irregular subgraphs; however, it was conjectured that, whenever a graph does, it should admit such a decomposition into at most three locally irregular subgraphs. This conjecture was verified for a few graph classes in recent years. This work is dedicated to the decomposability of degenerate graphs with low degeneracy. Our main result is that decomposable \(k\)-degenerate graphs decompose into at most \(3k+1\) locally irregular subgraphs, which improves on previous results whenever \(k \le 9\). We improve this result further for some specific classes of degenerate graphs, such as bipartite cacti, \(k\)-trees, and planar graphs. Although our results provide only little progress towards the leading conjecture above, the main contribution of this work is rather the decomposition schemes and methods we introduce to prove these results.Donaghey's transformation: carousel effects and tame components.https://zbmath.org/1459.050442021-05-28T16:06:00+00:00"Pushkarev, I. A."https://zbmath.org/authors/?q=ai:pushkarev.i-a"Byzov, V. A."https://zbmath.org/authors/?q=ai:byzov.v-aSummary: In this paper, Donaghey's transformation is investigated. It is a combinatorial dynamical system, based on the combinatorial interpretations of Catalan numbers, with the transition function of it being the composition of the direct and the inverse bijections between cubic and non-cubic trees. The dynamical system under investigation operates on a finite set and is inversible; therefore it is a mere permutation of trees. The properties of the cycles of this permutation, called the orbits, are studied in terms of permutation of structural elements of trees. In the course of studies, the systematic initiation of particular effects is indicated. These particular effects are referred to as ``carousel'': it is the movement of objects from one classification category to another, typical of natural classifications. In this form, they look as an indicator of system complexity. Two new carousel effects for structural elements, referred to as triads and germs, are described. The carousel effect for triads is used for the detection of two families of trees; the lengths of all tree arcs in the first family are equal to one; in the second family, they are equal to two. Here, the term ``the arcs of the orbit'' is used to denote its fragments between two trees, which have no left subtrees. Therefore, the properties of the arcs are the global properties of the orbits, and the carousel effects are local. The corresponding enumeration problems are solved: it is demonstrated that the number of trees of the first family increases as \(\dfrac{C}{n^{3/2}}\left(\dfrac{5}{2^{4/3}-2^{2/3}+1}\right)^n\), the number of trees of the second family as \(\dfrac{3^{n+1/2}}{n^{3/2}\sqrt{\pi}}\) (\(n\) is the number of triads). The paper presents the family of cycles with the length 6, which are different from the ones discovered by \textit{L. W. Shapiro} [Fibonacci Q. 17, 253--259 (1979; Zbl 0451.05020)], the number of which increases as \(\Theta(n^2)\), and the family of cycles with length 9, the number of which increases as \(\Theta(2^{n/2})\). A class of orbits with the lengths growing up as \(\Theta(n^2)\) is detected.On improved universal estimation of exponents of digraphs.https://zbmath.org/1459.051032021-05-28T16:06:00+00:00"Fomichev, V. M."https://zbmath.org/authors/?q=ai:fomichev.v-mSummary: An improved formula for universal estimation of exponent is obtained for \(n\)-vertex primitive digraphs. A previous formula by \textit{A. L. Dulmage} and \textit{N. S. Mendelsohn} [Ill. J. Math. 8, 642--656 (1964; Zbl 0125.00706)] is based on a system \(\hat{C}\) of directed circuits \(C_1,\ldots,C_m\), which are held in a graph and have lengths \(l_1,\ldots,l_m\) with \(\gcd(l_1,\ldots,l_m)=1\). A new formula is based on a similar circuit system \(\hat{C} \), where \(\gcd(l_1,\ldots,l_m)=d\geq 1\). Also, the new formula uses \(r_{i,j}^{s/d}(\hat{C})\), that is the length of the shortest path from \(i\) to \(j\) going through the circuit system \(\hat{C}\) and having the length which is comparable to \(s\) modulo \(d, s=0,\ldots,d-1\). It is shown, that \(\text{exp}\,\Gamma\leq 1+\hat{F}(L(\hat{C}))+R(\hat{C})\), where \(\hat{F}(L)=d\cdot F(l_1/d,\ldots, l_m/d)\) and \(F(a_1,\ldots,a_m)\) is the Frobenius number, \(R(\hat{C})=\max_{(i,j)}\max_s\{r_{i,j}^{s/d}(\hat{C})\} \). For some class of \(2k\)-vertex primitive digraphs, it is proved, that the improved formula gives the value of estimation \(2k\), and the previous formula gives the value of estimation \(3k-2\).On properties of primitive sets of digraphs with common cycles.https://zbmath.org/1459.050982021-05-28T16:06:00+00:00"Avezova, Y. E."https://zbmath.org/authors/?q=ai:avezova.ya-eSummary: Let \(\hat{\Gamma}=\{\Gamma_1,\ldots,\Gamma_p\}\) be a set of digraphs with vertex set \(V\), \(p>1\), and \(U^{(p)}\) be the union of digraphs \(\Gamma_1\cup\ldots\cup\Gamma_p\) with no multiple arcs. The smallest number such that the union of \(\mu\) digraphs of the set \(\hat{\Gamma}\) contains all arcs of \(U^{(p)}\) is denoted by \(\mu \). Suppose \(\hat{C}=\{C_1,\ldots,C_m\}\) is a set of elementary cycles. This set is called common for \(\hat{\Gamma}\) if every digraph of the set \(\hat{\Gamma}\) contains all cycles of the set \(\hat{C} \). Assume that \(C_1^\ast\cup\ldots\cup C_m^\ast=V\) where \(C_i^\ast\) denotes the vertex set of \(C_i\), \(i=1,\ldots,m\). For a given digraph \(\Gamma \), the loop-character index in the semigroup \(\langle \Gamma \rangle\) is the smallest integer \(h\) for which there is a loop on every vertex of \(\Gamma^h\). In this paper, we study conditions for the set of digraphs with common cycles to be primitive. For \(m\geq 1\), the set \(\hat{\Gamma}\) with common cycles set \(\hat{C}\) is primitive if and only if the digraph \(U^{(p)}\) is primitive. If \(\hat{\Gamma}\) is primitive, then \(\operatorname{exp}\hat{\Gamma} \leq \bigl((\mu-1)h+1\bigr)\operatorname{exp}U^{(p)} \), where \(h\) is the loop-character index in the semigroup \(\langle \Gamma(\hat{C})\rangle\), \(\Gamma(\hat{C})=C_1\cup\ldots\cup C_m\). For \(m=1\), we establish an improved bound on the exponent. Let all digraphs of the primitive set \(\hat{\Gamma}\) have a common Hamiltonian cycle, then \(\operatorname{exp}\hat{\Gamma} \leq(2n-1)\mu+\sum\limits_{\tau=1}^\mu{\bigl(F(l_1^\tau,\ldots,l_{m(\tau)}^\tau)+d_\tau-l_1^\tau\bigr)} \), where \(l_1^\tau,\ldots,l_{m(\tau)}^\tau\) are all cycle lengths in \(\Gamma_\tau \), ordered so that \(l_1^\tau<\ldots<l_{m(\tau)}^\tau=n\), \(d_\tau=\operatorname{gcd}(l_1^\tau,\ldots,l_{m(\tau)}^\tau)\), \(F(l_1^\tau,\ldots,l_{m(\tau)}^\tau)=d_\tau\Phi(l_1^\tau/ d_\tau,\ldots,l_{m(\tau)}^\tau/ d_\tau)\), \(\Phi(l_1^\tau/ d_\tau,\ldots,l_{m(\tau)}^\tau/d_\tau)\) denotes the Frobenius number, \( \tau=1,\ldots,\mu \). Finally, if \(n=q^{\alpha}\), \(q\) is prime, \( \alpha\in \mathbb{N}\), \(m=n^2\), then the number of primitive sets of \(n\)-vertex digraphs with a common Hamiltonian cycle equals \(2^\sigma-2^\varepsilon \), where \(\sigma=2^{m-n}\), \(\varepsilon=2^{m/q-n} \).Spanning triangle-trees and flows of graphs.https://zbmath.org/1459.051082021-05-28T16:06:00+00:00"Li, Jiaao"https://zbmath.org/authors/?q=ai:li.jiaao"Li, Xueliang"https://zbmath.org/authors/?q=ai:li.xueliang"Wang, Meiling"https://zbmath.org/authors/?q=ai:wang.meilingSummary: In this paper we study the flow properties of graphs containing a spanning triangle-tree. Our main results provide a structure characterization of graphs with a spanning triangle-tree admitting a nowhere-zero 3-flow. All these graphs without nowhere-zero 3-flows are constructed from \(K_4\) by a so-called bull-growth operation. This generalizes a result of \textit{G. Fan} et al. [J. Comb. Theory, Ser. B 98, No. 6, 1325--1336 (2008; Zbl 1171.05026)] on triangularly-connected graphs and particularly shows that every 4-edge-connected graph with a spanning triangle-tree has a nowhere-zero 3-flow. A well-known classical theorem of \textit{F. Jaeger} et al. [ibid. 56, No. 2, 165--182 (1992; Zbl 0824.05043)] shows that every graph with two edge-disjoint spanning trees admits a nowhere-zero 4-flow. We prove that every graph with two edge-disjoint spanning triangle-trees has a flow strictly less than 3.Two-coloring triples such that in each color class every element is missed at least once.https://zbmath.org/1459.050822021-05-28T16:06:00+00:00"Keszegh, Balázs"https://zbmath.org/authors/?q=ai:keszegh.balazsSummary: We give a characterization of finite sets of triples of elements (e.g., positive integers) that can be colored with two colors such that for every element \(i\) in each color class there exists a triple which does not contain \(i\). We give a linear (in the number of triples) time algorithm to decide if such a coloring exists and find one if it does. We also consider generalizations of this result and an application to a matching problem, which motivated this study. Finally, we show how these results translate to results about colorings of hypergraphs in which the degree of every vertex is \(k\) less than the number of hyperedges.Partitioning claw-free subcubic graphs into two dominating sets.https://zbmath.org/1459.052352021-05-28T16:06:00+00:00"Cui, Qing"https://zbmath.org/authors/?q=ai:cui.qingSummary: A dominating set in a graph \(G\) is a set \(S\subseteq V(G)\) such that every vertex in \(V(G)\setminus S\) has at least one neighbor in \(S\). Let \(G\) be an arbitrary claw-free graph containing only vertices of degree two or three. In this paper, we prove that the vertex set of \(G\) can be partitioned into two dominating sets \(V_1\) and \(V_2\) such that for \(i=1,2\), the subgraph of \(G\) induced by \(V_i\) is triangle-free and every vertex \(v\in V_i\) also has at least one neighbor in \(V_i\) if \(v\) has degree three in \(G\). This gives an affirmative answer to a problem of \textit{G. Bacsó} et al. [ibid. 35, No. 5, 1129--1138 (2019; Zbl 1426.05124)] and generalizes a result of \textit{W. J. Desormeaux} et al. [``Partitioning the vertices of a cubic graph into two total dominating sets'', Discret. Appl. Math. 223, 52--63 (2017; \url{doi:10.1016/j.dam.2017.01.032})].There are no cubic graphs on 26 vertices with crossing number 10 or 11.https://zbmath.org/1459.050552021-05-28T16:06:00+00:00"Clancy, Kieran"https://zbmath.org/authors/?q=ai:clancy.kieran"Haythorpe, Michael"https://zbmath.org/authors/?q=ai:haythorpe.michael"Newcombe, Alex"https://zbmath.org/authors/?q=ai:newcombe.alex"Pegg, Ed jun."https://zbmath.org/authors/?q=ai:pegg.ed-junSummary: We show that no cubic graphs of order 26 have crossing number larger than 9, which proves a conjecture of \textit{E. Pegg} and \textit{G. Exoo} [``Crossing number graphs'', Math. J. 11, No. 2, 161--170 (2009), \url{https://www.mathematica-journal.com/2009/11/23/crossing-number-graphs/}] that the smallest cubic graphs with crossing number 11 have 28 vertices. This result is achieved by first eliminating all girth 3 graphs from consideration, and then using the recently developed QuickCross heuristic to find drawings with few crossings for each remaining graph. We provide a minimal example of a cubic graph on 28 vertices with crossing number 10, and also exhibit for the first time a cubic graph on 30 vertices with crossing number 12, which we conjecture is minimal.Sub-Ramsey numbers for matchings.https://zbmath.org/1459.052062021-05-28T16:06:00+00:00"Wu, Fangfang"https://zbmath.org/authors/?q=ai:wu.fangfang"Zhang, Shenggui"https://zbmath.org/authors/?q=ai:zhang.shenggui"Li, Binlong"https://zbmath.org/authors/?q=ai:li.binlongSummary: Given a graph \(G\) and a positive integer \(k\), the sub-Ramsey number \(\operatorname{sr}(G, k)\) is defined to be the minimum number \(m\) such that every \(K_m\) whose edges are colored using every color at most \(k\) times contains a subgraph isomorphic to \(G\) all of whose edges have distinct colors. In this paper, we will concentrate on \(\operatorname{sr}(nK_2,k)\) with \(nK_2\) denoting a matching of size \(n\). We first give upper and lower bounds for \(sr(nK_2,k)\) and exact values of \(\operatorname{sr}(nK_2,k)\) for some \(n\) and \(k\). Afterwards, we show that \(\operatorname{sr}(nK_2,k)=2n\) when \(n\) is sufficiently large and \(k<\frac{n}{8}\) by applying the Local Lemma.Bipartite independent number and Hamilton-biconnectedness of bipartite graphs.https://zbmath.org/1459.051502021-05-28T16:06:00+00:00"Li, Binlong"https://zbmath.org/authors/?q=ai:li.binlongSummary: Let \(G\) be a balanced bipartite graph with bipartite sets \(X\), \(Y\). We say that \(G\) is Hamilton-biconnected if there is a Hamilton path connecting any vertex in \(X\) and any vertex in \(Y\). We define the bipartite independent number \(\alpha^o_B(G)\) to be the maximum integer \(\alpha\) such that for any integer partition \(\alpha =s+t\), \(G\) has an independent set formed by \(s\) vertices in \(X\) and \(t\) vertices in \(Y\). In this paper we show that if \(\alpha^o_B(G)\leq\delta (G)\) then \(G\) is Hamilton-biconnected, unless \(G\) has a special construction.On colorful edge triples in edge-colored complete graphs.https://zbmath.org/1459.052042021-05-28T16:06:00+00:00"Simonyi, Gábor"https://zbmath.org/authors/?q=ai:simonyi.gaborSummary: An edge-coloring of the complete graph \(K_n\) we call \(F\)-caring if it leaves no \(F\)-subgraph of \(K_n\) monochromatic and at the same time every subset of |\(V(F)\)| vertices contains in it at least one completely multicolored version of \(F\). For the first two meaningful cases, when \(F=K_{1,3}\) and \(F=P_4\) we determine for infinitely many \(n\) the minimum number of colors needed for an \(F\)-caring edge-coloring of \(K_n\). An explicit family of \(2\lceil\log_2 n\rceil 3\)-edge-colorings of \(K_n\) so that every quadruple of its vertices contains a totally multicolored \(P_4\) in at least one of them is also presented. Investigating related Ramsey-type problems we also show that the Shannon (OR-)capacity of the Grötzsch graph is strictly larger than that of the cycle of length 5.Gallai-Ramsey numbers for a class of graphs with five vertices.https://zbmath.org/1459.052022021-05-28T16:06:00+00:00"Li, Xihe"https://zbmath.org/authors/?q=ai:li.xihe"Wang, Ligong"https://zbmath.org/authors/?q=ai:wang.ligongSummary: Given two graphs \(G\) and \(H\), the \(k\)-colored Gallai-Ramsey number \(\operatorname{gr}_k(G : H)\) is defined to be the minimum integer \(n\) such that every \(k\)-coloring of the complete graph on \(n\) vertices contains either a rainbow copy of \(G\) or a monochromatic copy of \(H\). In this paper, we consider \(\operatorname{gr}_k(K_3:H)\), where \(H\) is a connected graph with five vertices and at most six edges. There are in total thirteen graphs in this graph class, and the Gallai-Ramsey numbers for eight of them have been studied step by step in several papers. We determine all the Gallai-Ramsey numbers for the remaining five graphs, and we also obtain some related results for a class of unicyclic graphs. As applications, we find the mixed Ramsey spectra \(S(n;H,K_3)\) for these graphs by using the Gallai-Ramsey numbers.Vertex distinction with subgraph centrality: a proof of Estrada's conjecture and some generalizations.https://zbmath.org/1459.053102021-05-28T16:06:00+00:00"Ballini, Francesco"https://zbmath.org/authors/?q=ai:ballini.francesco"Deniskin, Nikita"https://zbmath.org/authors/?q=ai:deniskin.nikitaSummary: Centrality measures are used in network science to identify the most important vertices for transmission of information and dynamics on a graph. One of these measures, introduced by \textit{E. Estrada} and \textit{J. A. Rodríguez-Velázquez} [``Subgraph centrality in complex networks'', Phys. Rev. E 71, Article ID 056103, 9 p. (2005; \url{doi:10.1103/PhysRevE.71.056103})], is the \(\beta\)-subgraph centrality, which is based on the exponential of the matrix \(\beta A\), where \(A\) is the adjacency matrix of the graph and \(\beta\) is a real parameter (``inverse temperature''). We prove that for algebraic \(\beta\), two vertices with equal \(\beta\)-subgraph centrality are necessarily cospectral. We further show that two such vertices must have the same degree and eigenvector centralities. Our results settle a conjecture of Estrada and a generalization of it due to \textit{K. Kloster} et al. [Linear Algebra Appl. 546, 115--121 (2018; Zbl 1391.05169)]. We also discuss possible extensions of our results.Asymptotic values of four Laplacian-type energies for matrices with degree-distance-based entries of random graphs.https://zbmath.org/1459.051792021-05-28T16:06:00+00:00"Li, Xueliang"https://zbmath.org/authors/?q=ai:li.xueliang"Li, Yiyang"https://zbmath.org/authors/?q=ai:li.yiyang"Wang, Zhiqian"https://zbmath.org/authors/?q=ai:wang.zhiqianThis paper studies asymptotic values of Laplacian-type energies for matrices with degree-distance-based entries of the Erdös-Rényi random graph \(G(n,p)\). Let a real symmetric function \(f\) be given over \(G\), let \(\operatorname{LEL}_f(G)\) be the associated weighted Laplacian-energy like invariant and \(\operatorname{IE}_f(G)\) be the weighted incidence energy. Let \(f_1(d_i,d_j)\) and \(f_2(d_i,d_j)\) be two symmetric functions satisfying \(f_1((1+o(1))np,(1+o(1))np)=(1+o(1))f_1(np,np)\) and \(f_s((1+o(1))np,(1+o(1))np)=(1+o(1))f_2(np,np)\), then it is shown that almost surely
\begin{itemize}
\item[(i)] if \(f_1(np,np)/f_2(np,np)\rightarrow\infty\) or \(f_1(np,np)/f_2(np,np)\rightarrow-\infty\), then \(\operatorname{LEL}_f(G(n,p))=\sqrt{|f_1(np,np)|}(\sqrt{p}+o(1))n^{3/2}\) and \(\operatorname{IE}_f(G(n,p))=\sqrt{|f_1(np,np)|}(\sqrt{p}+o(1))n^{3/2}\);
\item[(ii)] if \(f_1(np,np)/f_2(np,np)\rightarrow C\) for some constant \(C\), then \(\operatorname{LEL}_f(G(n,p))=\sqrt{|f_2(np,np)|}\) \((\sqrt{1+(C-1)p}+o(1))\) \(n^{3/2}\) and \(\operatorname{LEL}_f(G(n,p))=\sqrt{|f_2(np,np)|}\) \((\sqrt{1+(C-1)p}+o(1))\) \(n^{3/2}\).
\end{itemize}
Reviewer: Yilun Shang (Newcastle)Edge-matching graph contractions and their interlacing properties.https://zbmath.org/1459.051782021-05-28T16:06:00+00:00"Leiter, Noam"https://zbmath.org/authors/?q=ai:leiter.noam"Zelazo, Daniel"https://zbmath.org/authors/?q=ai:zelazo.danielSummary: For a given graph \(\mathcal{G}\) of order \(n\) with \(m\) edges, and a real symmetric matrix associated to the graph, \(M(\mathcal{G})\in\mathbb{R}^{n\times n}\), the interlacing graph reduction problem is to find a graph \(\mathcal{G}_r\) of order \(r<n\) such that the eigenvalues of \(M(\mathcal{G}_r)\) interlace the eigenvalues of \(M(\mathcal{G})\). Graph contractions over partitions of the vertices are widely used as a combinatorial graph reduction tool. In this study, we derive a graph reduction interlacing theorem based on subspace mappings and the minmax theory. We then define a class of edge-matching graph contractions and show how two types of edge-matching contractions provide Laplacian and normalized Laplacian interlacing. An \(\mathcal{O}(mn)\) algorithm is provided for finding a normalized Laplacian interlacing contraction and an \(\mathcal{O}(n^2+nm)\) algorithm is provided for finding a Laplacian interlacing contraction.A Coxeter spectral classification of positive edge-bipartite graphs. II: Dynkin type \(\mathbb{D}_n\).https://zbmath.org/1459.051132021-05-28T16:06:00+00:00"Simson, Daniel"https://zbmath.org/authors/?q=ai:simson.danielSummary: We continue the Coxeter spectral study of finite positive edge-bipartite signed (multi)graphs \(\Delta\) (bigraphs, for short), with \(n\geq 2\) vertices started in [the author, SIAM J. Discrete Math. 27, No. 2, 827--854 (2013; Zbl 1272.05072)] and developed in [the author, Linear Algebra Appl. 557, 105--133 (2018; Zbl 1396.05049)]. We do it by means of the non-symmetric Gram matrix \(\check{G}_{\Delta}\in \mathbb{M}_n(\mathbb{Z})\) defining \(\Delta\), its Gram quadratic form \(q_{\Delta}:\mathbb{Z}^n\to\mathbb{Z}\), \(v\mapsto v\cdot\check{G}_{\Delta}\cdot v^{tr}\) (that is positive definite, by definition), the complex spectrum \(\operatorname{specc}_{\Delta}\subset\mathcal{S}^1:=\{z\in\mathbb{C},|z|=1\}\) of the Coxeter matrix \(\operatorname{Cox}_{\Delta}:=-\check{G}_{\Delta}\cdot \check{G}_{\Delta}^{-tr}\in\mathbb{M}_n(\mathbb{Z})\), called the Coxeter spectrum of \(\Delta\), and the Coxeter polynomial \(\operatorname{cox}_{\Delta}(t):=\det(t\cdot E-\operatorname{Cox}_{\Delta})\in\mathbb{Z} [t]\). One of the aims of the Coxeter spectral analysis is to classify the connected bigraphs \(\Delta\) with \(n\geq 2\) vertices up to the \(\ell\)-weak Gram \(\mathbb{Z}\)-congruence \(\Delta\sim_{\ell\mathbb{Z}} \Delta^\prime\) and up to the strong Gram \(\mathbb{Z}\)-congruence \(\Delta\approx_{\mathbb{Z}}\Delta^\prime\), where \(\Delta\sim_{\ell\mathbb{Z}}\Delta^\prime\) (resp. \(\Delta\approx_{\mathbb{Z}}\Delta^\prime)\) means that \(\det \check{G}_{\Delta}=\det \check{G}_{\Delta^\prime}\) and \(G_{\Delta^\prime}=B^{tr}\cdot G_{\Delta}\cdot B\) (resp. \(\check{G}_{\Delta^\prime}=B^{tr}\cdot \check{G}_{\Delta}\cdot B)\), for some \(B\in\mathbb{M}_n(\mathbb{Z})\) with \(\det B=\pm 1\), where \(G_{\Delta}:=\frac{1}{2}[\check{G}_{\Delta}+\check{G}_{\Delta}^{tr}]\in \mathbb{M}_n(\frac{1}{2}\mathbb{Z})\).
Here we study connected signed simple graphs \(\Delta\), with \(n\geq 2\) vertices, that are positive, i.e., the symmetric Gram matrix \(G_{\Delta}\in\mathbb{M}_n(\frac{1}{2}\mathbb{Z})\) of \(\Delta\) is positive definite. It is known that every such a signed graph is \(\ell\)-weak Gram \(\mathbb{Z}\)-congruent with a unique simply laced Dynkin graph \(\operatorname{Dyn}_{\Delta} \in\{\mathbb{A}_n,\mathbb{D}_n,n\geq 4,\mathbb{E}_6, \mathbb{E}_7,\mathbb{E}_8\}\), called the Dynkin type of \(\Delta\). A classification up to the strong Gram \(\mathbb{Z}\)-congruence \(\Delta\approx_{\mathbb{Z}}\Delta^\prime\) is still an open problem and only partial results are known. In this paper, we obtain such a classification for the positive signed simple graphs \(\Delta\) of Dynkin type \(\mathbb{D}_n\) by means of the family of the signed graphs \(\mathcal{D}_n^{(1)}=\mathbb{D}_n, \mathcal{D}_n^{(2)},\dots,\mathcal{D}_n^{(r_n)}\) constructed in Section 2, for any \(n\geq 4\), where \(r_n=\llcorner n/2\lrcorner\). More precisely, we prove that any connected signed simple graph of Dynkin type \(\mathbb{D}_n\), with \(n\geq 4\) vertices, is strongly \(\mathbb{Z}\)-congruent with a signed graph \(\mathcal{D}_n^{(s)}\), for some \(s\leq r_n\), and its Coxeter polynomial \(\operatorname{cox}_{\Delta}(t)\) is of the form \((t^s+1)(t^{n-s}+1)\). We do it by a matrix morsification type reduction to the classification of the conjugacy classes in the integral orthogonal group \(\operatorname{O}(n, \mathbb{Z})\) of the integer orthogonal matrices \(C\), with \(\det(E-C)=4\).Classes of nonbipartite graphs with reciprocal eigenvalue property.https://zbmath.org/1459.051592021-05-28T16:06:00+00:00"Barik, Sasmita"https://zbmath.org/authors/?q=ai:barik.sasmita"Pati, Sukanta"https://zbmath.org/authors/?q=ai:pati.sukantaSummary: Let \(G\) be a simple connected graph and \(A(G)\) be the adjacency matrix of \(G\). The graph \(G\) is said to have the reciprocal eigenvalue property (R) if \(A(G)\) is nonsingular and \(\frac{1}{\lambda}\) is an eigenvalue of \(A(G)\) whenever \(\lambda\) is an eigenvalue of \(A(G)\). Further, if \(\lambda\) and \(\frac{1}{\lambda}\) have the same multiplicity, for each eigenvalue \(\lambda\), then \(G\) is said to have the strong reciprocal eigenvalue property (SR). Till date, all the classes of bipartite graphs that are found to have property (R), are found to have property (SR) and it is not known whether these two properties are equivalent even for the bipartite graphs with a unique perfect matching. Among nonbipartite graphs, there is only one known graph class for which these two properties are not equivalent. In this article, we construct some more classes of nonbipartite graphs with property (R) but not (SR).On the Smith normal form of walk matrices.https://zbmath.org/1459.051912021-05-28T16:06:00+00:00"Wang, Wei"https://zbmath.org/authors/?q=ai:wang.wei.17|wang.wei.25|wang.wei.8|wang.wei.21|wang.wei.26|wang.wei.18|wang.wei.29|wang.wei.15|wang.wei.24|wang.wei.28|wang.wei.30|wang.wei.12|wang.wei.19|wang.wei.3|wang.wei.13|wang.wei.16|wang.wei.5|wang.wei.27|wang.wei.1|wang.wei.2|wang.wei.9|wang.wei.20|wang.wei.23Summary: Let \(G\) be a graph with \(n\) vertices. The walk matrix \(W(G)\) of \(G\) is the matrix \([e,Ae,\dots,A^{n-1}e]\), where \(A\) is the adjacency matrix of \(G\) and \(e\) is the all-one vector. Let \(W\) be a walk matrix of order \(n\). We show that at most \(\lfloor \frac{n}{2}\rfloor\) invariant factors of \(W\) are congruent to 2 modulo 4. As a consequence, it is proved that, for any \(n \times n\) walk matrix \(W\) with 2-rank \(r\), the determinant of \(W\) is always a multiple of \(2^{\lceil\frac{3n-4r}{2}\rceil}\). Moreover, if \(2^{-\lceil\frac{3n-4r}{2}\rceil}\det W\) is odd and square-free, then the Smith normal form of \(W\) can be recovered uniquely from the triple \((n,r,\det W)\).Bounds for the energy of a complex unit gain graph.https://zbmath.org/1459.051852021-05-28T16:06:00+00:00"Samanta, Aniruddha"https://zbmath.org/authors/?q=ai:samanta.aniruddha"Kannan, M. Rajesh"https://zbmath.org/authors/?q=ai:rajesh-kannan.mSummary: A \(\mathbb{T}\)-gain graph, \(\Phi=(G,\varphi)\), is a graph in which the function \(\varphi\) assigns a unit complex number to each orientation of an edge of \(G\), and its inverse is assigned to the opposite orientation. The associated adjacency matrix \(A(\Phi)\) is defined canonically. The energy \(\mathcal{E}(\Phi)\) of a \(\mathbb{T}\)-gain graph \(\Phi\) is the sum of the absolute values of all eigenvalues of \(A(\Phi)\). We study the notion of energy of a vertex of a \(\mathbb{T}\)-gain graph, and establish bounds for it. For any \(\mathbb{T}\)-gain graph \(\Phi\), we prove that \(2\tau(G)-2c(G)\leq\mathcal{E}(\Phi)\leq 2\tau(G) \sqrt{\Delta(G)}\), where \(\tau(G)\), \(c(G)\) and \(\Delta(G)\) are the vertex cover number, the number of odd cycles and the largest vertex degree of \(G\), respectively. Furthermore, using the properties of vertex energy, we characterize the class of \(\mathbb{T}\)-gain graphs for which \(\mathcal{E}(\Phi)=2\tau(G)-2c(G)\) holds. Also, we characterize the \(\mathbb{T}\)-gain graphs for which \(\mathcal{E}(\Phi)=2\tau(G) \sqrt{\Delta(G)}\) holds. This characterization solves a general version of an open problem. In addition, we establish bounds for the energy in terms of the spectral radius of the associated adjacency matrix.The \(\alpha\)-normal labeling for generalized directed uniform hypergraphs.https://zbmath.org/1459.052902021-05-28T16:06:00+00:00"Li, Wei"https://zbmath.org/authors/?q=ai:li.wei.11|li.wei.2|li.wei.10|li.wei.9|li.wei.3|li.wei.7|li.wei.4|li.wei|li.wei.8|li.wei-wayne"Liu, Lele"https://zbmath.org/authors/?q=ai:liu.leleSummary: Let \(\pi=(\nu_1,\dots,\nu_d)\) be an ordered partition of the integer \(m\geq 1\), where \(\sum_{i=1}^d\nu_i=m\) and \(\nu_i\in\mathbb{Z}^+\) for all \(i\in [d]\). A \(\pi\)-directed \(m\)-uniform hypergraph \(G\) consists of a finite vertex set \(V(G)\) and a collection of edges \(E(G)\). Each edge is an ordered tuple, \(e=(S_1(e),S_2(e),\dots,S_d(e))\), of disjoint subsets of vertices such that \(|S_i|=\nu_i\), for all \(i\in [d]\). For each edge \(e,|\bigcup_{i=1}^dS_i(e)|=m\). Moreover, \(\bigcup_{e\in E(G)} S_i(e)=S_i(G)\) and \(|S_i(G)|=n_i\). Given \(\mathfrak{p}=(p_1,p_2,\dots,p_d)\in(1,\infty)^d\), the \(\mathfrak{p}\)-spectral radius of \(G\) is defined as
\[
\lambda_{\mathfrak{p}}(G):=\max\limits_{\|\mathfrak{x}_i\|_{p_i}=1,i\in [d]}\sum\limits_{e\in E(G)}\prod\limits_{i=1}^d\prod\limits_{v\in S_i(e)}x_{i,v},
\]
where \(\mathfrak{x}_i=(x_{i,1},x_{i,2},\dots,x_{i,n_i})\in \mathbb{R}^{n_i}\). In this paper, we develop the \(\alpha\)-normal labeling method for calculating \(\lambda_{\mathfrak{p}}(G)\) and some related properties are given. Moreover, we give a new lower bound of the \(\mathfrak{p}\)-spectral radius of \(\pi\)-directed \(m\)-uniform hypergraphs for the case \(\sum_{i=1}^d \frac{\nu_i}{p_i}>1\) by using the inverse Hölder's inequality.Gain-line graphs via \(G\)-phases and group representations.https://zbmath.org/1459.051632021-05-28T16:06:00+00:00"Cavaleri, Matteo"https://zbmath.org/authors/?q=ai:cavaleri.matteo"D'Angeli, Daniele"https://zbmath.org/authors/?q=ai:dangeli.daniele"Donno, Alfredo"https://zbmath.org/authors/?q=ai:donno.alfredoSummary: Let \(G\) be an arbitrary group. We define a gain-line graph for a gain graph \((\Gamma,\psi)\) through the choice of an incidence \(G\)-phase matrix inducing \(\psi\). We prove that the switching equivalence class of the gain function on the line graph \(L(\Gamma)\) does not change if one chooses a different \(G\)-phase inducing \(\psi\) or a different representative of the switching equivalence class of \(\psi\). In this way, we generalize to any group some results proven by \textit{N. Reff} [ibid. 436, No. 9, 3165--3176 (2012; Zbl 1241.05085)] in the abelian case. The investigation of the orbits of some natural actions of \(G\) on the set \(\mathcal{H}_{\Gamma}\) of \(G\)-phases of \(\Gamma\) allows us to characterize gain functions on \(\Gamma\), gain functions on \(L(\Gamma)\), their switching equivalence classes and their balance property. The use of group algebra valued matrices plays a fundamental role and, together with the matrix Fourier transform, allows us to represent a gain graph with Hermitian matrices and to perform spectral computations. Our spectral results also provide some necessary conditions for a gain graph to be a gain-line graph.Some new bounds of weighted graph entropies with GA and Gaurava indices edge weights.https://zbmath.org/1459.941782021-05-28T16:06:00+00:00"Wu, Tiejun"https://zbmath.org/authors/?q=ai:wu.tiejun"ur Rehman, Hafiz Mutee"https://zbmath.org/authors/?q=ai:ur-rehman.hafiz-mutee"Chu, Yu-Ming"https://zbmath.org/authors/?q=ai:chu.yuming"Afzal, Deeba"https://zbmath.org/authors/?q=ai:afzal.deeba"Yu, Jianfeng"https://zbmath.org/authors/?q=ai:yu.jianfengSummary: Motivated by the concept of Shannon's entropy, the degree-dependent weighted graph entropy was defined which is now become a tool for measurement of structural information of complex graph networks. The aim of this paper is to study weighted graph entropy. We used GA and Gaurava indices as edge weights to define weighted graph entropy and establish some bounds for different families of graphs. Moreover, we compute the defined weighted entropies for molecular graphs of some dendrimer structures.Proof of a conjecture on distance energy change of complete multipartite graph due to edge deletion.https://zbmath.org/1459.051882021-05-28T16:06:00+00:00"Sun, Shaowei"https://zbmath.org/authors/?q=ai:sun.shaowei"Das, Kinkar Chandra"https://zbmath.org/authors/?q=ai:das.kinkar-chandraSummary: Let \(K_{n_1,n_2,\dots,n_k}\) be a complete \(k\)-partite graph with \(k\geq 2\) and \(n_i\geq 2\) for \(i=1,2,\dots,k\). The Turán graph \(T(n, k)\) is a complete \(k\)-partite graph of \(n\) vertices with sizes of partitions as equal as possible. The distance energy \(E_D(G)\) of a graph \(G\) is defined as the sum of absolute values of distance eigenvalues of the graph \(G\). \textit{A. Varghese} et al. [ibid. 553, 211--222 (2018; Zbl 1391.05179)] conjectured that
\[
E_D(K_{n_1,n_2,\dots,n_k})<E_D(K_{n_1,n_2,\dots,n_k}-e),
\]
where \(e\) is any edge of \(K_{n_1,n_2,\dots,n_k}\) and proved that the above relation holds for \(k=2\). Very recently, \textit{G.-X. Tian} et al. [ibid. 584, 438--457 (2020; Zbl 1426.05111)] confirmed that the above conjecture holds for \(T(n,k)\) with \(n\equiv 0\pmod{k}\) and \(T(n,3)\). They also mentioned a weaker conjecture as follows:
\[
E_D(T(n, k))<E_D(T(n, k)-e),
\]
where \(e\) is any edge of \(T(n, k)\) and \(k\geq 2\), \(n\geq 2k\). In this paper, we confirm that the former conjecture is true for \(k\geq 3\) and then the latter conjecture follows immediately.The lattice of cycles of an undirected graph.https://zbmath.org/1459.051572021-05-28T16:06:00+00:00"Averkov, Gennadiy"https://zbmath.org/authors/?q=ai:averkov.gennadiy"Chavez, A."https://zbmath.org/authors/?q=ai:chavez.a-t|chavez.anastasia|chavez.a-e|chavez.alan|chavez.angel"De Loera, J. A."https://zbmath.org/authors/?q=ai:de-loera.jesus-a"Gillespie, Bryan"https://zbmath.org/authors/?q=ai:gillespie.bryan-rSummary: We study bases of the lattice generated by the cycles of an undirected graph, defined as the integer linear combinations of the 0/1-incidence vectors of cycles. We prove structural results for this lattice, including explicit formulas for its dimension and determinant, and we present efficient algorithms to construct lattice bases, using only cycles as generators, in quadratic time. By algebraic considerations, we relate these results to the more general setting with coefficients from an arbitrary abelian group. Our results generalize classical results for the vector space of cycles of a graph over the binary field to the case of an arbitrary field.A reduction formula for the characteristic polynomial of hypergraph with pendant edges.https://zbmath.org/1459.052222021-05-28T16:06:00+00:00"Chen, Lixiang"https://zbmath.org/authors/?q=ai:chen.lixiang"Bu, Changjiang"https://zbmath.org/authors/?q=ai:bu.changjiangSummary: In this paper, we give a reduction formula for the characteristic polynomial of \(k\)-uniform hypergraphs with pendant edges, and use the reduction formula to derive the explicit expression for the characteristic polynomial and all distinct eigenvalues of \(k\)-uniform loose hyperpaths.Computing edge weights of magic labeling on rooted products of graphs.https://zbmath.org/1459.052882021-05-28T16:06:00+00:00"Liu, Jia-Bao"https://zbmath.org/authors/?q=ai:liu.jia-bao"Afzal, Hafiz Usman"https://zbmath.org/authors/?q=ai:afzal.hafiz-usman"Javaid, Muhammad"https://zbmath.org/authors/?q=ai:javaid.muhammadSummary: Labeling of graphs with numbers is being explored nowadays due to its diverse range of applications in the fields of civil, software, electrical, and network engineering. For example, in network engineering, any systems interconnected in a network can be converted into a graph and specific numeric labels assigned to the converted graph under certain rules help us in the regulation of data traffic, connectivity, and bandwidth as well as in coding/decoding of signals. Especially, both antimagic and magic graphs serve as models for surveillance or security systems in urban planning. In [SUT J. Math. 34, No. 2, 105--109 (1998; Zbl 0918.05090)], \textit{H. Enomoto} et al. introduced the notion of super \(\left(a, 0\right)\) edge-antimagic labeling of graphs. In this article, we shall compute super \(\left(a, 0\right)\) edge-antimagic labeling of the rooted product of \(P_n\) and the complete bipartite graph \(\left( K_{2, m}\right)\) combined with the union of path, copies of paths, and the star. We shall also compute a super \(\left(a, 0\right)\) edge-antimagic labeling of rooted product of \(P_n\) with a special type of pancyclic graphs. The labeling provided here will also serve as super \(\left( a^\prime, 2\right)\) edge-antimagic labeling of the aforesaid graphs. All the structures discussed in this article are planar. Moreover, our findings have also been illustrated with examples and summarized in the form of a table and \(3D\) plots.Fractional arboricity, strength and eigenvalues of graphs with fixed girth or clique number.https://zbmath.org/1459.051742021-05-28T16:06:00+00:00"Hong, Zhen-Mu"https://zbmath.org/authors/?q=ai:hong.zhenmu"Xia, Zheng-Jiang"https://zbmath.org/authors/?q=ai:xia.zhengjiang"Lai, Hong-Jian"https://zbmath.org/authors/?q=ai:lai.hong-jian"Liu, Ruifang"https://zbmath.org/authors/?q=ai:liu.ruifangSummary: Let \(c(G)\), \(g(G)\), \(\omega(G)\) and \(\mu_{n-1}(G)\) denote the number of components, the girth, the clique number and the second smallest Laplacian eigenvalue of the graph \(G\), respectively. The strength \(\eta(G)\) and the fractional arboricity \(\gamma(G)\) are defined by
\begin{align*}
(\eta(G) & =\min_{F\subseteq E(G)} \frac{|F|}{c(G-F)-c(G)} \text{ and}\\
\gamma(G) & = \max_{H\subseteq G}\frac{|E(H)|}{|V(H)|-1},
\end{align*}
where the optima are taken over all edge subsets \(F\) and all subgraphs \(H\) whenever the denominator is non-zero, respectively. Nash-Williams and Tutte proved that \(G\) has \(k\) edge-disjoint spanning trees if and only if \(\eta(G)\geq k\); and Nash-Williams showed that \(G\) can be covered by at most \(k\) forests if and only if \(\gamma(G)\leq k\). In this paper, for integers \(r\geq 2\), \(s\) and \(t\), and any simple graph \(G\) of order \(n\) with minimum degree \(\delta\geq\frac{2s}{t}\) and either clique number \(\omega(G)\leq r\) or girth \(g\geq 3\), we prove that if \(\mu_{n-1}(G)>\frac{2s-1}{t\varphi (\delta,r)}\) or \(\mu_{n-1}(G)>\frac{2s-1}{tN(\delta,g)}\), then \(\eta(G)\geq\frac{s}{t}\), where \(\varphi(\delta, r)=\max\{\delta+1,\lfloor\frac{r\delta}{r-1}\rfloor\}\) and \(N(\delta, g)\) is the Moore bound on the smallest possible number of vertices such that there exists a \(\delta\)-regular simple graph with girth \(g\). As corollaries, sufficient conditions on \(\mu_{n-1}(G)\) such that \(G\) has \(k\) edge-disjoint spanning trees are obtained. Analogous result involving \(\mu_{n-1}(G)\) to characterize fractional arboricity of graphs with given clique number is also presented. Former results in [\textit{Q. Liu} et al., ibid. 458, 128--133 (2014; Zbl 1295.05146); \textit{Y. Hong} et al., Discrete Appl. Math. 213, 219--223 (2016; Zbl 1344.05088)] are extended, and the result in [\textit{R. Liu} et al., Linear Algebra Appl. 578, 411--424 (2019; Zbl 1419.05130)] is improved.\(P_3\)-factors in the square of a tree.https://zbmath.org/1459.051362021-05-28T16:06:00+00:00"Dai, Guowei"https://zbmath.org/authors/?q=ai:dai.guowei"Hu, Zhiquan"https://zbmath.org/authors/?q=ai:hu.zhiquanSummary: A spanning subgraph \(H\) of a graph \(G\) is a \(P_3\)-factor of \(G\) if every component of \(H\) is a path of order three. The square of a graph \(G\), denoted by \(G^2\), is the graph with vertex set \(V(G)\) such that two vertices are adjacent in \(G^2\) if and only if their distance in \(G\) is at most 2. A graph \(G\) is subcubic if it has maximum degree at most three. In this paper, we give a sharp necessary condition for the existence of \(P_3\)-factors in the square of a tree, which improves a result of \textit{X. Li} and \textit{Z. Zhang} [ibid. 24, No. 2, 107--111 (2008; Zbl 1155.05053)]. In addition, we will also present a sufficient condition for the existence of \(P_3 \)-factors in the square of a tree, which has the following interesting application to subcubic trees: if \(T\) is a subcubic tree of order \(3n\) such that \(|L(T-L(T))|\le 7\), then \(T^2\) has a \(P_3\)-factor, where \(L(T)\) denotes the set of leaves in \(T\). Examples show that the upper bound 7 on \(|L(T-L(T))|\) is sharp.The influence of network structural preference on link prediction.https://zbmath.org/1459.911522021-05-28T16:06:00+00:00"Wang, Yongcheng"https://zbmath.org/authors/?q=ai:wang.yongcheng"Wang, Yu"https://zbmath.org/authors/?q=ai:wang.yu.4|wang.yu|wang.yu.1|wang.yu.3|wang.yu.2|wang.yu.5|wang.yu.8|wang.yu.9"Lin, Xinye"https://zbmath.org/authors/?q=ai:lin.xinye"Wang, Wei"https://zbmath.org/authors/?q=ai:wang.wei.21|wang.wei.26|wang.wei.1|wang.wei.8|wang.wei.25|wang.wei.29|wang.wei.5|wang.wei.15|wang.wei.28|wang.wei.19|wang.wei.2|wang.wei.23|wang.wei.16|wang.wei.13|wang.wei.18|wang.wei.20|wang.wei.3|wang.wei.24|wang.wei.27|wang.wei.30|wang.wei.17|wang.wei.9|wang.wei.12Summary: Link prediction in complex networks predicts the possibility of link generation between two nodes that have not been linked yet in the network, based on known network structure and attributes. It can be applied in various fields, such as friend recommendation in social networks and prediction of protein-protein interaction in biology. However, in the social network, link prediction may raise concerns about privacy and security, because, through link prediction algorithms, criminals can predict the friends of an account user and may even further discover private information such as the address and bank accounts. Therefore, it is urgent to develop a strategy to prevent being identified by link prediction algorithms and protect privacy, utilizing perturbation on network structure at a low cost, including changing and adding edges. This article mainly focuses on the influence of network structural preference perturbation through deletion on link prediction. According to a large number of experiments on the various real networks, edges between large-small degree nodes and medium-medium degree nodes have the most significant impact on the quality of link prediction.Sequential metric dimension.https://zbmath.org/1459.052092021-05-28T16:06:00+00:00"Bensmail, Julien"https://zbmath.org/authors/?q=ai:bensmail.julien"Mazauric, Dorian"https://zbmath.org/authors/?q=ai:mazauric.dorian"Mc Inerney, Fionn"https://zbmath.org/authors/?q=ai:mc-inerney.fionn"Nisse, Nicolas"https://zbmath.org/authors/?q=ai:nisse.nicolas"Pérennes, Stéphane"https://zbmath.org/authors/?q=ai:perennes.stephaneSummary: In the localization game, introduced by \textit{S. M. Seager} [Ars Comb. 110, 45--54 (2013; Zbl 1313.05099)], an invisible and immobile target is hidden at some vertex of a graph \(G\). At every step, one vertex \(v\) of \(G\) can be probed which results in the knowledge of the distance between \(v\) and the secret location of the target. The objective of the game is to minimize the number of steps needed to locate the target whatever be its location. We address the generalization of this game where \(k\ge 1\) vertices can be probed at every step. Our game also generalizes the notion of the metric dimension of a graph. Precisely, given a graph \(G\) and two integers \(k,\ell \ge 1\), the Localization problem asks whether there exists a strategy to locate a target hidden in \(G\) in at most \(\ell\) steps and probing at most \(k\) vertices per step. We first show that, in general, this problem is NP-complete for every fixed \(k \ge 1\) (resp., \( \ell \ge 1)\). We then focus on the class of trees. On the negative side, we prove that the Localization problem is NP-complete in trees when \(k\) and \(\ell\) are part of the input. On the positive side, we design a \((+\,1)\)-approximation algorithm for the problem in \(n\)-node trees, i.e., an algorithm that computes in time \(O(n \log n)\) (independent of \(k)\) a strategy to locate the target in at most one more step than an optimal strategy. This algorithm can be used to solve the Localization problem in trees in polynomial time if \(k\) is fixed. We also consider some of these questions in the context where, upon probing the vertices, the relative distances to the target are retrieved. This variant of the problem generalizes the notion of the centroidal dimension of a graph.Resolvability in subdivision of circulant networks \(C_n[1, k]\).https://zbmath.org/1459.050692021-05-28T16:06:00+00:00"Wei, Jianxin"https://zbmath.org/authors/?q=ai:wei.jianxin"Bokhary, Syed Ahtsham Ul Haq"https://zbmath.org/authors/?q=ai:bokhary.syed-ahtsham-ul-haq"Abbas, Ghulam"https://zbmath.org/authors/?q=ai:abbas.ghulam"Imran, Muhammad"https://zbmath.org/authors/?q=ai:imran.muhammadSummary: Circulant networks form a very important and widely explored class of graphs due to their interesting and wide-range applications in networking, facility location problems, and their symmetric properties. A resolving set is a subset of vertices of a connected graph such that each vertex of the graph is determined uniquely by its distances to that set. A resolving set of the graph that has the minimum cardinality is called the basis of the graph, and the number of elements in the basis is called the metric dimension of the graph. In this paper, the metric dimension is computed for the graph \(G_n[1, k]\) constructed from the circulant graph \(C_n[1, k]\) by subdividing its edges. We have shown that, for \(k=2\), \(G_n[1, k]\) has an unbounded metric dimension, and for \(k=3\) and 4, \(G_n[1, k]\) has a bounded metric dimension.Some bounds of weighted entropies with augmented Zagreb index edge weights.https://zbmath.org/1459.050622021-05-28T16:06:00+00:00"Huang, Yujie"https://zbmath.org/authors/?q=ai:huang.yujie"Mutee-ur-Rehman, Hafiz"https://zbmath.org/authors/?q=ai:ur-rehman.hafiz-mutee"Nazeer, Saima"https://zbmath.org/authors/?q=ai:nazeer.saima"Afzal, Deeba"https://zbmath.org/authors/?q=ai:afzal.deeba"Qiang, Xiaoli"https://zbmath.org/authors/?q=ai:qiang.xiaoliSummary: The graph entropy was proposed by \textit{J. Körner} [in: Transactions of the 6th Prague conference on information theory, statistical decision functions, random processes, held at Prague, from September 19 to 25, 1971. Prague: Academia, Publishing House of the Czechoslovak Academy of Sciences. 411--425 (1973; Zbl 0298.94022)] when he was studying the problem of coding in information theory. The foundation of graph entropy is in information theory, but it was demonstrated to be firmly identified with some established and often examined graph-theoretic ideas. For instance, it gives an equal definition to a graph to be flawless, and it can likewise be connected to acquire lower bounds in graph covering problems. The objective of this study is to solve the open problem suggested by Kwun et al. in 2018. In this paper, we study the weighted graph entropy by taking augmented Zagreb edge weight and give bounds of it for regular, connected, bipartite, chemical, unicyclic, etc., graphs. Moreover, we compute the weighted graph entropy of certain nanotubes and plot our results to see dependence of weighted entropy on involved parameters.Multi-color forcing in graphs.https://zbmath.org/1459.050732021-05-28T16:06:00+00:00"Bozeman, Chassidy"https://zbmath.org/authors/?q=ai:bozeman.chassidy"Harris, Pamela E."https://zbmath.org/authors/?q=ai:harris.pamela-e"Jain, Neel"https://zbmath.org/authors/?q=ai:jain.neel"Young, Ben"https://zbmath.org/authors/?q=ai:young.ben"Yu, Teresa"https://zbmath.org/authors/?q=ai:yu.teresaSummary: Let \(G=(V,E)\) be a finite connected graph along with a coloring of the vertices of \(G\) using the colors in a given set \(X\). In this paper, we introduce multi-color forcing, a generalization of zero-forcing on graphs, and give conditions in which the multi-color forcing process terminates regardless of the number of colors used. We give an upper bound on the number of steps required to terminate a forcing procedure in terms of the number of vertices in the graph on which the procedure is being applied. We then focus on multi-color forcing with three colors and analyze the end states of certain families of graphs, including complete graphs, complete bipartite graphs, and paths, based on various initial colorings. We end with a few directions for future research.On variable sum exdeg indices of quasi-tree graphs and unicyclic graphs.https://zbmath.org/1459.050682021-05-28T16:06:00+00:00"Sun, Xiaoling"https://zbmath.org/authors/?q=ai:sun.xiaoling"Du, Jianwei"https://zbmath.org/authors/?q=ai:du.jianweiSummary: In this work, by using the properties of the variable sum exdeg indices and analyzing the structure of the quasi-tree graphs and unicyclic graphs, the minimum and maximum variable sum exdeg indices of quasi-tree graphs and quasi-tree graphs with perfect matchings were presented; the minimum and maximum variable sum exdeg indices of unicyclic graphs with given pendant vertices and cycle length were determined.Uniform length dominating sequence graphs.https://zbmath.org/1459.052372021-05-28T16:06:00+00:00"Erey, Aysel"https://zbmath.org/authors/?q=ai:erey.ayselSummary: A sequence of vertices \((v_1,\dots,v_k)\) of a graph \(G\) is called a dominating closed neighborhood sequence if \(\{v_1,\dots,v_k\}\) is a dominating set of \(G\) and \(N[v_i]\nsubseteq\cup_{j=1}^{i-1}N[v_j]\) for every \(i\). A graph \(G\) is said to be \(k\)-uniform if all dominating closed neighborhood sequences in the graph have equal length \(k\). \textit{B. Brešar} et al. [Discrete Math. 336, 22--36 (2014; Zbl 1300.05210)] characterized \(k\)-uniform graphs with \(k\leq 3\). In this article we extend their work by giving a complete characterization of all \(k\)-uniform graphs with \(k\geq 4\).The surface of a sufficiently large sphere has chromatic number at most 7.https://zbmath.org/1459.050932021-05-28T16:06:00+00:00"Sirgedas, Tomas"https://zbmath.org/authors/?q=ai:sirgedas.tomasSummary: We present a method to assign, for any radius \(r\) greater than about 12.44, one of seven colors to each point in \(\mathbb{R}^3\) lying at distance \(r\) from the origin, such that no two points at unit distance from each other are assigned the same color. The existence of such a construction contrasts with the recent demonstration that, for any positive value \(\varepsilon\), if no two points assigned the same color lie at any distance in \([1,1+\varepsilon]\) (and with certain other restrictions that are also satisfied with our coloring), then eight colors are needed for any finite \(r\ge 18\), even though seven colors suffice in the plane when \(\varepsilon\le\frac{\sqrt 7}{2}-1\).Fault-tolerant resolvability in some classes of line graphs.https://zbmath.org/1459.052792021-05-28T16:06:00+00:00"Guo, Xuan"https://zbmath.org/authors/?q=ai:guo.xuan"Faheem, Muhammad"https://zbmath.org/authors/?q=ai:faheem.muhammad"Zahid, Zohaib"https://zbmath.org/authors/?q=ai:zahid.zohaib"Nazeer, Waqas"https://zbmath.org/authors/?q=ai:nazeer.waqas"Li, Jingjng"https://zbmath.org/authors/?q=ai:li.jingjngSummary: Fault tolerance is the characteristic of a system that permits it to carry on its intended operations in case of the failure of one of its units. Such a system is known as the fault-tolerant self-stable system. In graph theory, if we remove any vertex in a resolving set, then the resulting set is also a resolving set, called the fault-tolerant resolving set, and its minimum cardinality is called the fault-tolerant metric dimension. In this paper, we determine the fault-tolerant resolvability in line graphs. As a main result, we computed the fault-tolerant metric dimension of line graphs of necklace and prism graphs.Complete bipartite graphs deleted in Ramsey graphs.https://zbmath.org/1459.052032021-05-28T16:06:00+00:00"Li, Yan"https://zbmath.org/authors/?q=ai:li.yan.4|li.yan.2|li.yan.6|li.yan.10|li.yan.5|li.yan.1|li.yan.3|li.yan.8|li.yan.9|li.yan.7|li.yan"Li, Yusheng"https://zbmath.org/authors/?q=ai:li.yusheng"Wang, Ye"https://zbmath.org/authors/?q=ai:wang.yeSummary: For graphs \(F, G\) and \(H\), let \(F \to (G, H)\) signify that any red/blue edge coloring of \(F\) contains either a red \(G\) or a blue \(H\). The Ramsey number \(R(G, H)\) is defined as \(\min \{r \mid K_r \to (G, H)\}\). In this note, we consider an optimization problem to bound the complete bipartite-critical Ramsey number \(R_{\Lambda} (G, H)\) defined as \(\max \{t \mid K_r \setminus K_{t, t} \to (G, H)\}\) where \(r = R (G, H)\) and \(\Lambda\) is a set of \(K_{t, t}\), and determine \(R_{\Lambda}(G, H)\) for some pairs \((G, H)\).The structure of hypergraphs without long Berge cycles.https://zbmath.org/1459.052242021-05-28T16:06:00+00:00"Győri, Ervin"https://zbmath.org/authors/?q=ai:gyori.ervin"Lemons, Nathan"https://zbmath.org/authors/?q=ai:lemons.nathan"Salia, Nika"https://zbmath.org/authors/?q=ai:salia.nika"Zamora, Oscar"https://zbmath.org/authors/?q=ai:zamora.oscarSummary: We study the structure of \(r\)-uniform hypergraphs containing no Berge cycles of length at least \(k\) for \(k \leq r\), and determine that such hypergraphs have some special substructure. In particular we determine the extremal number of such hypergraphs, giving an affirmative answer to the conjectured value when \(k = r\) and giving a simple solution to a recent result of \textit{A. Kostochka} and \textit{R. Luo} [Discrete Appl. Math. 276, 69--91 (2020; Zbl 1435.05153)] when \(k < r\).A localization method in Hamiltonian graph theory.https://zbmath.org/1459.051482021-05-28T16:06:00+00:00"Asratian, Armen S."https://zbmath.org/authors/?q=ai:asratian.armen-s"Granholm, Jonas B."https://zbmath.org/authors/?q=ai:granholm.jonas-b"Khachatryan, Nikolay K."https://zbmath.org/authors/?q=ai:khachatryan.nikolay-kSummary: The classical global criteria for the existence of Hamilton cycles only apply to graphs with large edge density and small diameter. In a series of papers \textit{A. S. Asratian} and \textit{N. K. Khachatrian} [Dokl., Akad. Nauk Arm. SSR 81, 103--106 (1985; Zbl 0657.05052); J. Comb. Theory, Ser. B 49, No. 2, 287--294 (1990; Zbl 0708.05038); Australas. J. Comb. 38, 77--86 (2007; Zbl 1141.05045)] developed local criteria for the existence of Hamilton cycles in finite connected graphs, which are analogues of the classical global criteria due to \textit{G. A. Dirac} [Proc. Lond. Math. Soc. (3) 2, 69--81 (1952; Zbl 0047.17001)], \textit{Ø. Ore} [Am. Math. Mon. 67, 55 (1960; Zbl 0089.39505)], \textit{H. A. Jung} [Ann. Discrete Math. 3, 129--144 (1978; Zbl 0399.05039)], and \textit{C. St. J. A. Nash-Williams} [in: Studies in pure mathematics. Papers in combinatorial theory, analysis, geometry, algebra, and the theory of numbers presented to Richard Rado on the occasion of his sixty-fifth birthday. London-New York: Academic Press. 157--183 (1971; Zbl 0223.05123)]. The idea was to show that the global concept of Hamiltonicity can, under rather general conditions, be captured by local phenomena, using the structure of balls of small radii. (The ball of radius \(r\) centered at a vertex \(u\) is a subgraph of \(G\) induced by the set of vertices whose distances from \(u\) do not exceed \(r\).) Such results are called localization theorems and present a possibility to extend known classes of finite Hamiltonian graphs. In this paper we formulate a general approach for finding localization theorems and use this approach to formulate local analogues of well-known results of \textit{D. Bauer} et al. [J. Comb. Theory, Ser. B 47, No. 2, 237--243 (1989; Zbl 0634.05053)], \textit{J. A. Bondy} [``Longest paths and cycles in graphs of high degree'', Res. Rep. CORR 80-16, University of Waterloo (1980)], \textit{R. Häggkvist} and \textit{G. G. Nicoghossian} [J. Comb. Theory, Ser. B 30, 118--120 (1981; Zbl 0462.05046)], and \textit{J. Moon} and \textit{L. Moser} [Isr. J. Math. 1, 163--165 (1963; Zbl 0119.38806)]. Finally we extend two of our results to infinite locally finite graphs and show that they guarantee the existence of Hamiltonian curves, introduced by \textit{A. Kündgen} et al. [Eur. J. Comb. 65, 259--275 (2017; Zbl 1369.05123)].A scaling limit for the length of the longest cycle in a sparse random graph.https://zbmath.org/1459.050592021-05-28T16:06:00+00:00"Anastos, Michael"https://zbmath.org/authors/?q=ai:anastos.michael"Frieze, Alan"https://zbmath.org/authors/?q=ai:frieze.alan-mSummary: We discuss the length \(L_{c , n}\) of the longest cycle in a sparse random graph \(G_{n , p}\), \(p = c / n\), \(c\) constant. We show that for large \(c\) there exists a function \(f(c)\) such that \(L_{c , n} / n \to f(c)\) a.s. The function \(f(c) = 1 - \sum_{k = 1}^\infty p_k(c) e^{- k c}\) where \(p_k(c)\) is a polynomial in \(c\). We are only able to explicitly give the values \(p_1\), \(p_2\), although we could in principle compute any \(p_k\). We see immediately that the length of the longest path is also asymptotic to \(f(c) n\) w.h.p..Approximating infinite graphs by normal trees.https://zbmath.org/1459.052202021-05-28T16:06:00+00:00"Kurkofka, Jan"https://zbmath.org/authors/?q=ai:kurkofka.jan"Melcher, Ruben"https://zbmath.org/authors/?q=ai:melcher.ruben"Pitz, Max"https://zbmath.org/authors/?q=ai:pitz.max-fSummary: We show that every connected graph can be approximated by a normal tree, up to some arbitrarily small error phrased in terms of neighbourhoods around its ends. The existence of such approximate normal trees has consequences of both combinatorial and topological nature. On the combinatorial side, we show that a graph has a normal spanning tree as soon as it has normal spanning trees locally at each end; i.e., the only obstruction for a graph to having a normal spanning tree is an end for which none of its neighbourhoods has a normal spanning tree. On the topological side, we show that the end space \(\Omega(G)\), as well as the spaces \(| G | = G \cup \Omega(G)\) naturally associated with a graph \(G\), are always paracompact. This gives unified and short proofs for a number of results by \textit{R. Diestel} and \textit{I. Leader} [J. Lond. Math. Soc., II. Ser. 63, No. 1, 16--32 (2001; Zbl 1012.05051)], \textit{P. Sprüssel} [J. Comb. Theory, Ser. B 98, No. 4, 798--804 (2008; Zbl 1198.05032)] and \textit{N. Polat} [ibid., No. 1, 86--110 (1996; Zbl 0855.05051)], and answers an open question about metrizability of end spaces by Polat.On the rational Turán exponents conjecture.https://zbmath.org/1459.051332021-05-28T16:06:00+00:00"Kang, Dong Yeap"https://zbmath.org/authors/?q=ai:kang.dong-yeap"Kim, Jaehoon"https://zbmath.org/authors/?q=ai:kim.jaehoon"Liu, Hong"https://zbmath.org/authors/?q=ai:liu.hong.1Summary: The extremal number \(\operatorname{ex}(n, F)\) of a graph \(F\) is the maximum number of edges in an \(n\)-vertex graph not containing \(F\) as a subgraph. A real number \(r \in [1, 2]\) is realisable if there exists a graph \(F\) with \(\operatorname{ex}(n, F) = \Theta( n^r)\). Several decades ago, Erdős and Simonovits (see [\textit{P. Erdős}, Combinatorica 1, 25--42 (1981; Zbl 0486.05001)]) conjectured that every rational number in \([1, 2]\) is realisable. Despite decades of effort, the only known realisable numbers are \(0, 1, \frac{ 7}{ 5}, 2\), and the numbers of the form \(1 + \frac{ 1}{ m}, 2 - \frac{ 1}{ m}, 2 - \frac{ 2}{ m}\) for integers \(m \geq 1\). In particular, it is not even known whether the set of all realisable numbers contains a single limit point other than the two numbers 1 and 2. In this paper, we make progress on the conjecture of Erdős and Simonovits. First, we show that \(2 - \frac{ a}{ b}\) is realisable for any integers \(a, b \geq 1\) with \(b > a\) and \(b \equiv \pm 1\pmod a\). This includes all previously known ones, and gives infinitely many limit points \(2 - \frac{ 1}{ m}\) in the set of all realisable numbers as a consequence. Secondly, we propose a conjecture on subdivisions of bipartite graphs. Apart from being interesting on its own, we show that, somewhat surprisingly, this subdivision conjecture in fact implies that every rational number between 1 and 2 is realisable.Regular Cayley maps for dihedral groups.https://zbmath.org/1459.051192021-05-28T16:06:00+00:00"Kovács, István"https://zbmath.org/authors/?q=ai:kovacs.istvan.1|kovacs.istvan|kovacs.istvan.2|kovacs.istvan-a"Kwon, Young Soo"https://zbmath.org/authors/?q=ai:kwon.young-sooSummary: An orientably-regular map \(\mathcal{M}\) is a 2-cell embedding of a finite connected graph in a closed orientable surface such that the group \(\mathrm{Aut}^\circ M\) of orientation-preserving automorphisms of \(\mathcal{M}\) acts transitively on the set of arcs. Such a map \(\mathcal{M}\) is called a Cayley map for the finite group \(G\) if \(\mathrm{Aut}^\circ \mathcal{M}\) contains a subgroup, which is isomorphic to \(G\) and acts regularly on the set of vertices. \textit{M. D. E. Conder} and \textit{T. W. Tucker} [Trans. Am. Math. Soc. 366, No. 7, 3585--3609 (2014; Zbl 1290.05160)] classified the regular Cayley maps for finite cyclic groups, and obtain two two-parameter families \(\mathcal{M}(n, r)\), one for odd \(n\) and one for even \(n\), where \(n\) is the order of the regular cyclic group and \(r\) is a positive integer satisfying certain arithmetical conditions. In this paper, we classify the regular Cayley maps for dihedral groups in the same fashion. Five two-parameter families \(\mathcal{M}_i(n, r)\), \(1 \leq i \leq 5\), are derived, where \(2n\) is the order of the regular dihedral group and \(r\) is an integer satisfying certain arithmetical conditions. For each map \(\mathcal{M}_i(n, r)\), we determine its valence and covalence, and also describe the structure of the group \(\mathrm{Aut}^\circ \mathcal{M}_i(n, r)\). Unlike the approach of Conder and Tucker, which is entirely algebraic, we follow the traditional combinatorial representation of Cayley maps, and use a combination of permutation group theoretical techniques, the method of quotient Cayley maps, and computations with skew morphisms.\(K_r\)-factors in graphs with low independence number.https://zbmath.org/1459.053292021-05-28T16:06:00+00:00"Knierim, Charlotte"https://zbmath.org/authors/?q=ai:knierim.charlotte"Su, Pascal"https://zbmath.org/authors/?q=ai:su.pascalSummary: A classical result by \textit{A. Hajnal} and \textit{E. Szemerédi} [in: Combinatorial theory and its applications. Vols. I--III. Proceedings of a colloqium, Balatonfüred, 1969. Budapest: Bolyai János Matematikai Társulat; Amsterdam-London; North Holland Publishing Company. 601--623 (1970; Zbl 0217.02601)] determines the minimal degree conditions necessary to guarantee for a graph to contain a \(K_r\)-factor. Namely, any graph on \(n\) vertices, with minimum degree \(\delta(G) \geq ( 1 - \frac{ 1}{ r} ) n\) and \(r\) dividing \(n\) has a \(K_r\)-factor. This result is tight but the extremal examples are unique in that they all have a large independent set which is the bottleneck. \textit{R. Nenadov} and \textit{Y. Pehova} [SIAM J. Discrete Math. 34, No. 2, 1001--1010 (2020; Zbl 1436.05087)] showed that by requiring a sub-linear independence number the minimum degree condition in the Hajnal-Szemerédi theorem can be improved. We show that, with the same minimum degree and sub-linear independence number, we can find a clique-factor with double the clique size. More formally, we show for every \(r \in \mathbb{N}\) and constant \(\mu > 0\) there is a positive constant \(\gamma\) such that every graph \(G\) on \(n\) vertices with \(\delta(G) \geq ( 1 - \frac{ 2}{ r} + \mu ) n\) and \(\alpha(G) < \gamma n\) has a \(K_r\)-factor. We also give examples showing the minimum degree condition is asymptotically best possible.The feasible region of hypergraphs.https://zbmath.org/1459.052262021-05-28T16:06:00+00:00"Liu, Xizhi"https://zbmath.org/authors/?q=ai:liu.xizhi"Mubayi, Dhruv"https://zbmath.org/authors/?q=ai:mubayi.dhruvSummary: Let \(\mathcal{F}\) be a family of \(r\)-uniform hypergraphs. The feasible region \(\Omega(\mathcal{F})\) of \(\mathcal{F}\) is the set of points \((x, y)\) in the unit square such that there exists a sequence of \(\mathcal{F} \)-free \(r\)-uniform hypergraphs whose shadow density approaches \(x\) and whose edge density approaches \(y\). The feasible region provides a lot of combinatorial information, for example, the supremum of \(y\) over all \((x, y) \in \Omega(\mathcal{F})\) is the Turán density \(\pi(\mathcal{F})\), and \(\Omega(\emptyset)\) gives the Kruskal-Katona theorem. We undertake a systematic study of \(\Omega(\mathcal{F})\), and prove that \(\Omega(\mathcal{F})\) is completely determined by a left-continuous almost everywhere differentiable function; and moreover, there exists an \(\mathcal{F}\) for which this function is not continuous. We also extend some old related theorems. For example, we generalize a result of \textit{D. C. Fisher} and \textit{J. Ryan} [Discrete Math. 103, No. 3, 313--320 (1992; Zbl 0817.05035)] to hypergraphs and extend a classical result of \textit{B. Bollobas} [ibid. 8, 21--24 (1974; Zbl 0291.05114)] by almost completely determining the feasible region for cancellative triple systems.A Menger-like property of tree-cut width.https://zbmath.org/1459.052602021-05-28T16:06:00+00:00"Giannopoulou, Archontia C."https://zbmath.org/authors/?q=ai:giannopoulou.archontia-c"Kwon, O-joung"https://zbmath.org/authors/?q=ai:kwon.ojoung"Raymond, Jean-Florent"https://zbmath.org/authors/?q=ai:raymond.jean-florent"Thilikos, Dimitrios M."https://zbmath.org/authors/?q=ai:thilikos.dimitrios-mSummary: \textit{R. Thomas} [J. Comb. Theory, Ser. B 48, No. 1, 67--76 (1990; Zbl 0636.05022)] proved that every graph admits a tree decomposition of minimum width that additionally satisfies a certain vertex-connectivity condition called leanness. This result had many uses and has been extended to several other decompositions. In this paper, we consider tree-cut decompositions, that have been introduced by \textit{P. Wollan} [ibid. 110, 47--66 (2015; Zbl 1302.05148)] as a possible edge-version of tree decompositions. We show that every graph admits a tree-cut decomposition of minimum width that additionally satisfies an edge-connectivity condition analogous to Thomas' leanness.On density-critical matroids.https://zbmath.org/1459.050352021-05-28T16:06:00+00:00"Campbell, Rutger"https://zbmath.org/authors/?q=ai:campbell.rutger"Grace, Kevin"https://zbmath.org/authors/?q=ai:grace.kevin"Oxley, James"https://zbmath.org/authors/?q=ai:oxley.james-g"Whittle, Geoff"https://zbmath.org/authors/?q=ai:whittle.geoffrey-pFor a matroid $M$ having $m$ rank-one flats, the density $d(M)$ is $m/r(M)$ for all $m>0$. A matroid is density-critical if every proper minors of non-zero rank has lower density. It is shown that there are exactly ten minor-minimal simple obstructions to a matroid being able to be covered by two independent sets. These ten matroids are precisely the density critical matroids $M$ such that $d(M) > 2$ but $d(N) \leq 2$ for all proper minors $N$ of $M$. All density-critical matroids with $d(M)<2$ are series-parallel networks. The authors have found all density-critical matroids of density at most 9/4.
Reviewer: Wei Yao (Nanjing)The 3-sline graph of a given graph.https://zbmath.org/1459.052822021-05-28T16:06:00+00:00"Johns, Garry"https://zbmath.org/authors/?q=ai:johns.garry-l"Wang, Bianka"https://zbmath.org/authors/?q=ai:wang.bianka"Zayed, Mohra"https://zbmath.org/authors/?q=ai:zayed.mohraSummary: For a given graph \(G\), a variation of its line graph is the 3-xline graph, where two 3-paths \(P\) and \(Q\) are adjacent in \(G\) if \(V(P)\cap V(Q)=\{v\}\) when \(v\) is the interior vertex of both \(P\) and \(Q\). The vertices of the 3-xline graph \(XL_3(G)\) correspond to the 3-paths in \(G\), and two distinct vertices of the 3-xline graph are adjacent if and only if they are adjacent 3-paths in \(G\). In this paper, we study 3-xline graphs for several classes of graphs, and show that for a connected graph \(G\), the 3-xline graph is never isomorphic to \(G\) and is connected only when \(G\) is the star \(K_{1,n}\), for \(n=2\) or \(n\ge 5\). We also investigate cycles in 3-xline graphs and characterize those graphs \(G\) where \(XL_3(G)\) is Eulerian.Decomposition of complete graphs into unicyclic bipartite graphs with eight edges.https://zbmath.org/1459.052582021-05-28T16:06:00+00:00"Freyberg, Bryan"https://zbmath.org/authors/?q=ai:freyberg.bryan"Tran, Nhan"https://zbmath.org/authors/?q=ai:tran.nhanSummary: We introduce a variation of \(\sigma\)-labeling to prove that every disconnected unicyclic bipartite graph with eight edges decomposes the complete graph \(K_n\) whenever the necessary conditions are satisfied. We combine this result with known results in the connected case to prove that every unicyclic bipartite graph with eight edges other than \(C_s\) decomposes \(K_n\) if and only if \(n\equiv 0,1\pmod{16}\) and \(n\ge 16\).Solving the weighted \(k\)-separator problem in graphs with specific modules.https://zbmath.org/1459.052662021-05-28T16:06:00+00:00"Lepin, V. V."https://zbmath.org/authors/?q=ai:lepin.v-vSummary: Given a graph \(G\) with a vertex weight function \(\omega_V: V(G)\to\mathbb{R}^+\) and a positive integer \(k,\) we consider the \(k\)-separator problem: it consists in finding a minimum-weight subset of vertices whose removal leads to a graph where the size of each connected component is less than or equal to \(k.\) Using the notion of modular decomposition we extend the class of graphs on which this problem can be solved in polynomial time. For a graph \(G\) that is modular decomposable into \(\pi(G)\subseteq\{P_4,\dots,P_m\}\cup\{C_5,\dots,C_m\}\) we give an \(O(n^2)\) algorithm for finding the minimum weight of \(k\)-separators. The algorithm solves this problem for cographs in time \(O(n)\). Moreover, we give an \(O(n)\) time algorithm solving this problem for the series-parallel graphs.Three coloring of pure children drawings of snarks and the problem ``The hunting of the snark''.https://zbmath.org/1459.050832021-05-28T16:06:00+00:00"Krenkel', T. E."https://zbmath.org/authors/?q=ai:krenkel.t-e"Kulikova, T. A."https://zbmath.org/authors/?q=ai:kulikova.t-aSummary: The Tait theorem as a consequence of the Four Color Theorem states that planar cubic graphs are edge three-colorable. The first graph which is a counterexample for the Tait theorem was a nontrivial cubic (trivalent) Petersen graph \(P,\) which is the only and minimal graph with chromatic index 4. The integer sequence OEIS A130315 describes (as defined by Martin Gardner) the number of (with girth \(\ge 5\)) snarks, that are nontrivial cubic graphs with \(2n\) verticies. The conjecture is presented that via transition from category \textit{Snarks} to category \textit{SnarksPureDessins} the derived two-colored graphs (pure children drawings of snarks) can be three-colored at half-edges. The embedding of the Petersen graph in double torus \(\Sigma_2\) is presented. The \(RGB\) theorem about the cycle double cover of the Peterson-Belyi graph \(PB\) is proved.Control of multiple viruses interacting and propagating in multilayer networks.https://zbmath.org/1459.921462021-05-28T16:06:00+00:00"Tu, Xiao"https://zbmath.org/authors/?q=ai:tu.xiao"Jiang, Guo-Ping"https://zbmath.org/authors/?q=ai:jiang.guoping"Song, Yurong"https://zbmath.org/authors/?q=ai:song.yurong"Wang, Xiaoling"https://zbmath.org/authors/?q=ai:wang.xiaolingSummary: Experimental studies involving control against virus propagation have attracted the interest of scientists. However, most accomplishments have been constrained by the simple assumption of a single virus in various networks, but this assumption apparently conflicts with recent developments in complex network theory, which details that each node might play multiple roles in different topological connections. Multiple viruses propagate through individuals via different routes, and thus, each individual component could be located in various positions of differing importance in each virus propagation process in each network. Therefore, we propose several control strategies for establishing a multiple-virus interaction and propagation model involving multiplex networks, including a novel Multiplex PageRank target control model and a multiplex random control model. Using computer experiments and simulations derived from actual examples, we exploit several actual cases to determine the relationship of the relative infection probability with the immunization probability. The results demonstrate the differences between our multiple-virus interaction and propagation model and the single-virus propagation model and verify the effectiveness of our novel Multiplex PageRank target control strategy. Moreover, we use parallel computing for simulating and identifying the relationships of the immunization thresholds with both interaction coefficients, which is beneficial for further practical applications because it can reduce the multiple interactions between viruses and allows achieving a greater effect through the immunization of fewer nodes in the multilayer networks.Reappraising the distribution of the number of edge crossings of graphs on a sphere.https://zbmath.org/1459.052932021-05-28T16:06:00+00:00"Alemany-Puig, Lluís"https://zbmath.org/authors/?q=ai:alemany-puig.lluis"Mora, Mercè"https://zbmath.org/authors/?q=ai:mora.merce"Ferrer-i-Cancho, Ramon"https://zbmath.org/authors/?q=ai:ferrer-i-cancho.ramonPerfect state transfer on Cayley graphs over dihedral groups.https://zbmath.org/1459.051172021-05-28T16:06:00+00:00"Cao, Xiwang"https://zbmath.org/authors/?q=ai:cao.xiwang"Feng, Keqin"https://zbmath.org/authors/?q=ai:feng.keqinSummary: Recently, there are extensive studies on perfect state transfer on graphs due to their significant applications in quantum information processing and quantum computations. However, most of the graphs previously investigated are abelian Cayley graphs. In this paper, we study perfect state transfer on Cayley graphs over dihedral groups. Using the representations of the dihedral group \(D_n\), we present some necessary and sufficient conditions for the Cayley graph \(\operatorname{Cay}(D_n,S)\) to have a perfect state transfer between two distinct vertices for some connection set \(S\). Based on these conditions, we show that \(\operatorname{Cay}(D_n,S)\) cannot have PST if \(n\) is odd and \(S\) is conjugation-closed. For some even integers \(n\), it is possible for \(\operatorname{Cay}(D_n,S)\) to have PST, some concrete constructions are provided.The size of the giant joint component in a binomial random double graph.https://zbmath.org/1459.053032021-05-28T16:06:00+00:00"Jerrum, Mark"https://zbmath.org/authors/?q=ai:jerrum.mark-r"Makai, Tamás"https://zbmath.org/authors/?q=ai:makai.tamasThis paper studies the size of the giant joint component in a binomial random double graph. A double graph is formed by superposing two graphs \(G_1=(V, E_1)\) and \(G_2=(V,E_2)\) over the same vertex set \(V\). The binomial random double graph \(G(n,p_1,p_2)\) is a graph where \(G_1\) and \(G_2\) are independent binomial random graph over \([n]\) and the edge probabilities are \(p_1\) and \(p_2\). For
\(p_1=\lambda_1/n\) and \(p_2=\lambda_2/n\), let the solution of the equation \(\beta=\mathbb{P}(Po(\lambda_1\beta)>0)\mathbb{P}(Po(\lambda_2\beta)>0)\) be denoted by \(\beta(\lambda_1,\lambda_2)\). Let \(C=\partial\{(\lambda_1,\lambda_2):\beta(\lambda_1,\lambda_2)=0\}\). For \((\lambda_1,\lambda_2)\in\mathbb{R}^+\backslash C\) the number of
vertices in the giant joint component of \(G(n,\lambda_1/n,\lambda_2/n)\) is \(\beta(\lambda_1,\lambda_2)n+o_p(n)\) as \(n\) goes to infinity.
Moreover, it is shown that when \(\beta(\lambda_1,\lambda_2)\) is positive, the giant joint component is unique.
Reviewer: Yilun Shang (Newcastle)The signless Laplacian state transfer in coronas.https://zbmath.org/1459.051902021-05-28T16:06:00+00:00"Tian, Gui-Xian"https://zbmath.org/authors/?q=ai:tian.guixian"Yu, Ping-Kang"https://zbmath.org/authors/?q=ai:yu.ping-kang"Cui, Shu-Yu"https://zbmath.org/authors/?q=ai:cui.shuyuSummary: For two graphs \(G\) and \(H\), the corona product \(G \circ H\) is the graph obtained by taking one copy of \(G\) and \(|V_G|\) copies of \(H\), and joining the \(i\)th vertex of \(G\) with every vertex of the \(i\)th copy of \(H\). In this paper, we study the state transfer of corona relative to the signless Laplacian matrix. We explore some conditions that guarantee the signless Laplacian perfect state transfer in \(G \circ H\). We prove that \(G \circ K_m\) has no signless Laplacian perfect state transfer for some special \(m\). We also show that \(K_2 \circ H\) has pretty good state transfer but no perfect state transfer relative to the signless Laplacian matrix for a regular graph \(H\). Furthermore, we show that \(\overline{nK_2} \circ K_1\) has signless Laplacian pretty good state transfer, where \(\overline{nK_2}\) is the cocktail party graph.Complex adjacency spectra of digraphs.https://zbmath.org/1459.051062021-05-28T16:06:00+00:00"Sahoo, Gopinath"https://zbmath.org/authors/?q=ai:sahoo.gopinathSummary: In this article, we consider only those (simple) digraphs which satisfy the property that if \((u,v)\) is an edge of a digraph, then \((v,u)\) is not an edge of it. A new matrix representation of a digraph is considered and the matrix is named as the complex adjacency matrix. The eigenvalues and the eigenvectors of the complex adjacency matrices of cycle digraphs and directed trees are obtained and it is shown that not only the eigenvalues of these matrices but also the eigenvectors provide a lot of information about the structure of these digraphs.Matching number, connectivity and eigenvalues of distance signless Laplacians.https://zbmath.org/1459.052702021-05-28T16:06:00+00:00"Li, Shuchao"https://zbmath.org/authors/?q=ai:li.shuchao"Sun, Wanting"https://zbmath.org/authors/?q=ai:sun.wantingSummary: Let \(G\) be a connected graph. The first and the second largest distance signless Laplacian eigenvalues of \(G\) are denoted by \(q_1(G)\) and \(q_2(G)\). In this paper, we determine the graphs with the minimum \(q_1(G)\) among \(n\)-vertex graphs with given matching number. We also establish sharp lower bounds on \(q_2(G)\) in terms of the order and the matching number of \(G\). Moreover, the unique graph with the minimum \(q_2(G)\) among the \(n\)-vertex connected graphs with fixed connectivity is identified.The cyclic rank completion problem with general blocks.https://zbmath.org/1459.051652021-05-28T16:06:00+00:00"Cohen, Nir"https://zbmath.org/authors/?q=ai:cohen.nir"Pereira, Edgar"https://zbmath.org/authors/?q=ai:pereira.edgarSummary: We present an upper bound for the minimal completion rank of a partial matrix \(P\) whose block pattern is a single cycle of size \(2k\) with specified blocks \(A_1,\dots,A_{2k}\). Under certain conditions, the bound becomes quite sharp when \(k\) increases. This extends previous results in which the blocks are regular. The upper bound is constructed from invariants associated with the canonical form of the partial matrix, under row and column operations. These invariants can be expressed in terms of ranks of certain matrices constructed directly from the data blocks, independent of \(P\) being in canonical form.Optimal network topologies: expanders, cages, Ramanujan graphs, entangled networks and all that.https://zbmath.org/1459.051692021-05-28T16:06:00+00:00"Donetti, Luca"https://zbmath.org/authors/?q=ai:donetti.luca"Neri, Franco"https://zbmath.org/authors/?q=ai:neri.franco-maria"Muñoz, Miguel A."https://zbmath.org/authors/?q=ai:munoz.miguel-aOn the zeros of the partial Hosoya polynomial of graphs.https://zbmath.org/1459.051292021-05-28T16:06:00+00:00"Ghorbani, Modjtaba"https://zbmath.org/authors/?q=ai:ghorbani.modjtaba"Dehmer, Matthias"https://zbmath.org/authors/?q=ai:dehmer.matthias"Cao, Shujuan"https://zbmath.org/authors/?q=ai:cao.shujuan"Feng, Lihua"https://zbmath.org/authors/?q=ai:feng.lihua"Tao, Jin"https://zbmath.org/authors/?q=ai:tao.jin"Emmert-Streib, Frank"https://zbmath.org/authors/?q=ai:emmert-streib.frankSummary: The partial Hosoya polynomial (or briefly the partial \(H\)-polynomial) can be used to construct the well-known Hosoya polynomial. The \(i\) th coefficient of this polynomial, defined for an arbitrary vertex \(u\) of a graph \(G\), is the number of vertices at distance \(i\) from \(u\). The aim of this paper is to determine the partial \(H\)-polynomial of several well-known graphs and, then, to investigate the location of their zeros. To pursue, we characterize the structure of graphs with the minimum and the maximum modulus of the zeros of partial \(H\)-polynomial. Finally, we define another graph polynomial of the partial \(H\)-polynomial, see [\textit{M. Dehmer} et al., ``The orbit-polynomial: a novel measure of symmetry in graphs'', submitted]. Also, we determine the unique positive root of this polynomial for particular graphs.Neighbor sum distinguishing total choice number of planar graphs without 6-cycles.https://zbmath.org/1459.050942021-05-28T16:06:00+00:00"Zhang, Dong Han"https://zbmath.org/authors/?q=ai:zhang.donghan"Lu, You"https://zbmath.org/authors/?q=ai:lu.you"Zhang, Sheng Gui"https://zbmath.org/authors/?q=ai:zhang.shengguiSummary: \textit{M. Pilśniak} and \textit{M. Woźniak} [Graphs Comb. 31, No. 3, 771--782 (2015; Zbl 1312.05054)] put forward the concept of neighbor sum distinguishing (NSD) total coloring and conjectured that any graph with maximum degree \(\Delta\) admits an NSD total \((\Delta+3)\)-coloring. \textit{C. Qu} et al. [J. Comb. Optim. 32, No. 3, 906--916 (2016; Zbl 1348.05082)] showed that the list version of the conjecture holds for any planar graph with \(\Delta\geq 13\). In this paper, we prove that any planar graph with \(\Delta\geq 7\) but without 6-cycles satisfies the list version of the conjecture.Phase transitions in optimized network models.https://zbmath.org/1459.820352021-05-28T16:06:00+00:00"Cheng, An-Liang"https://zbmath.org/authors/?q=ai:cheng.an-liang"Lai, Pik-Yin"https://zbmath.org/authors/?q=ai:lai.pik-yinNon-Euclidean braced grids.https://zbmath.org/1459.520172021-05-28T16:06:00+00:00"Power, Stephen C."https://zbmath.org/authors/?q=ai:power.stephen-cSummary: Necessary and sufficient conditions are obtained for the infinitesimal rigidity of braced grids in the plane with respect to non-Euclidean norms. Component rectangles of the grid may carry 0, 1 or 2 diagonal braces, and the combinatorial part of the conditions is given in terms of a matroid for the bicoloured bipartite multigraph defined by the braces.Competing synchronization on random networks.https://zbmath.org/1459.340922021-05-28T16:06:00+00:00"Park, Jinha"https://zbmath.org/authors/?q=ai:park.jinha"Kahng, B."https://zbmath.org/authors/?q=ai:kahng.byungik|kahng.byeong-hoon|kahng.byung-jay|kahng.byungnamCommunity detection via a triangle and edge combination conductance partitioning.https://zbmath.org/1459.911552021-05-28T16:06:00+00:00"Zhang, Teng"https://zbmath.org/authors/?q=ai:zhang.teng"Sun, Lizhu"https://zbmath.org/authors/?q=ai:sun.lizhu"Bu, Changjiang"https://zbmath.org/authors/?q=ai:bu.changjiangSelf-organization in many-body systems with short-range interactions: clustering, correlations and topology.https://zbmath.org/1459.811362021-05-28T16:06:00+00:00"Kleftogiannis, Ioannis"https://zbmath.org/authors/?q=ai:kleftogiannis.ioannis"Amanatidis, Ilias"https://zbmath.org/authors/?q=ai:amanatidis.iliasLearning performance in inverse Ising problems with sparse teacher couplings.https://zbmath.org/1459.820222021-05-28T16:06:00+00:00"Abbara, Alia"https://zbmath.org/authors/?q=ai:abbara.alia"Kabashima, Yoshiyuki"https://zbmath.org/authors/?q=ai:kabashima.yoshiyuki"Obuchi, Tomoyuki"https://zbmath.org/authors/?q=ai:obuchi.tomoyuki"Xu, Yingying"https://zbmath.org/authors/?q=ai:xu.yingyingAnti-Ramsey numbers in complete \(k\)-partite graphs.https://zbmath.org/1459.051982021-05-28T16:06:00+00:00"Ding, Jili"https://zbmath.org/authors/?q=ai:ding.jili"Bian, Hong"https://zbmath.org/authors/?q=ai:bian.hong"Yu, Haizheng"https://zbmath.org/authors/?q=ai:yu.haizhengSummary: The anti-Ramsey number \(AR\left( G, H\right)\) is the maximum number of colors in an edge-coloring of \(G\) such that \(G\) contains no rainbow subgraphs isomorphic to \(H\). In this paper, we discuss the anti-Ramsey numbers \(AR\left( K_{p_1, p_2, \ldots, p_k}, \mathcal{T}_n\right)\), \(AR\left( K_{p_1, p_2, \ldots, p_k}, \mathcal{M}\right)\), and \(AR\left( K_{p_1, p_2, \ldots, p_k}, \mathcal{C}\right)\) of \(K_{p_1, p_2, \ldots, p_k} \), where \(\mathcal{T}_n\), \(\mathcal{M}\), and \(\mathcal{C}\) denote the family of all spanning trees, the family of all perfect matchings, and the family of all Hamilton cycles in \(K_{p_1, p_2, \ldots, p_k}\), respectively.A Pfaffian formula for the Ising partition function of surface graphs.https://zbmath.org/1459.820552021-05-28T16:06:00+00:00"Pham, Anh Minh"https://zbmath.org/authors/?q=ai:pham.anh-minhThe cavity master equation: average and fixed point of the ferromagnetic model in random graphs.https://zbmath.org/1459.821612021-05-28T16:06:00+00:00"Domínguez, E."https://zbmath.org/authors/?q=ai:dominguez.eduardo"Machado, D."https://zbmath.org/authors/?q=ai:machado.deiwison-s"Mulet, R."https://zbmath.org/authors/?q=ai:mulet.robertoOn a conjecture of Karasev.https://zbmath.org/1459.520062021-05-28T16:06:00+00:00"Lee, Seunghun"https://zbmath.org/authors/?q=ai:lee.seunghun"Yoo, Kangmin"https://zbmath.org/authors/?q=ai:yoo.kangminAuthors' abstract: Karasev conjectured that for any set of \(3k\) lines in general position in the plane, which is partitioned into 3 color classes of equal size \(k\), the set can be partitioned into \(k\) colorful 3-subsets such that all the triangles formed by the subsets have a point in common. Although the general conjecture is false, we show that Karasev's conjecture is true for lines in convex position. We also discuss possible generalizations of this result.
Reviewer: Mircea Balaj (Oradea)Geometry and volume product of finite dimensional Lipschitz-free spaces.https://zbmath.org/1459.520072021-05-28T16:06:00+00:00"Alexander, Matthew"https://zbmath.org/authors/?q=ai:alexander.matthew"Fradelizi, Matthieu"https://zbmath.org/authors/?q=ai:fradelizi.matthieu"García-Lirola, Luis C."https://zbmath.org/authors/?q=ai:garcia-lirola.luis-carlos"Zvavitch, Artem"https://zbmath.org/authors/?q=ai:zvavitch.artemLet \((M,d)\) be a finite, pointed metric space, where the special designated point is denoted by \(a_0\). The family of Lipschitz functions \(f : M \to \mathbb{R}\) with the property that \(f(a_0)=0\) is a Banach space with respect to a norm defined by \(d\), and is called the Lipschitz dual \(\mathrm{Lip}_0(M)\) of \(M\). The canonical predual \(\mathcal{F}(M)\) of this space is called the Lipschitz-free space over \(M\). The authors study the geometric properties of Lipschitz-free spaces over finite, pointed metric spaces.
The main topics and results in the paper are as follows. The authors characterize the weighted graphs induced by finite, pointed metric spaces, and describe the face structure of the unit ball \(B(\mathcal{F}(M))\) of the Lipschitz-free space over \(M\) in terms of the properties of this graph. They characterize the Lipschitz-free spaces that can be decomposed into an \(\ell_1\)- or \(\ell_{\infty}\)-sum of other Lipschitz-free spaces, and those that are zonotopes or Hanner polytopes. Furthermore, they give equivalent reformulations of the property that two Lipschitz-free spaces \(\mathcal{F}(M)\) and \(\mathcal{F}(M')\) are isomorphic. Finally, they investigate the volume products of the unit balls of Lipschitz-free spaces over pointed metric spaces with a fixed number of points, and prove some partial results regarding the minimal and the maximal values of these quantities.
Reviewer: Zsolt Lángi (Budapest)Polynomial configurations in sets of positive upper density over local fields.https://zbmath.org/1459.050712021-05-28T16:06:00+00:00"Bardestani, Mohammad"https://zbmath.org/authors/?q=ai:bardestani.mohammad"Mallahi-Karai, Keivan"https://zbmath.org/authors/?q=ai:karai.keivan-mallahiSummary: Let \(F(x)=(f_1(x),\dots,f_m(x))\) be such that \(1,f_1,\dots,f_m\) are linearly independent polynomials with real coefficients. Based on ideas of \textit{C. Bachoc} et al. [Isr. J. Math. 202, 227--254 (2014; Zbl 1302.05047)] in combination with estimating certain oscillatory integrals with polynomial phase we will show that the independence ratio of the Cayley graph of \(\mathbb{R}^m\) with respect to the portion of the graph of \(F\) defined by \(a\leq\log|s|\leq T\) is at most \(O(1/T-a))\). We conclude that if \(I \subseteq\mathbb{R}^m\) has positive upper density, then the difference set \(I-I\) contains vectors of the form \(F(s)\) for an unbounded set of values \(s\in\mathbb{R}\). It follows that the Borel chromatic number of the Cayley graph of \(\mathbb{R}^m\) with respect to the set \(\{\pm F(s):s\in\mathbb{R}\}\) is infinite. Analogous results are also proven when \(\mathbb{R}\) is replaced by the field of \(p\)-adic numbers \(\mathbb{Q}_p\). At the end, we will also show the existence of real analytic functions \(f_1,\dots,f_m\), for which the analogous statements no longer hold.Computation of irregularity indices of certain computer networks.https://zbmath.org/1459.053132021-05-28T16:06:00+00:00"Liu, Jiangnan"https://zbmath.org/authors/?q=ai:liu.jiangnan"Cai, Lulu"https://zbmath.org/authors/?q=ai:cai.lulu"Virk, Abaid ur Rehman"https://zbmath.org/authors/?q=ai:virk.abaid-ur-rehman"Akhtar, Waheed"https://zbmath.org/authors/?q=ai:akhtar.waheed"Maitla, Shahzad Ahmed"https://zbmath.org/authors/?q=ai:maitla.shahzad-ahmed"Wei, Yang"https://zbmath.org/authors/?q=ai:wei.yangSummary: A graph is said to be a regular graph if all its vertices have the same degree; otherwise, it is irregular. In general, irregularity indices are used for computational analysis of nonregular graph topological composition. The creation of irregular indices is based on the conversion of a structural graph into a total count describing the irregularity of the molecular design on the map. It is important to be notified how unusual a molecular structure is in various situations and problems in structural science and chemistry. In this paper, we will compute irregularity indices of certain networks.Irregularity measures for benzene ring embedded in P-type surface.https://zbmath.org/1459.050512021-05-28T16:06:00+00:00"Liu, Yun"https://zbmath.org/authors/?q=ai:liu.yun"Siddiqa, Aysha"https://zbmath.org/authors/?q=ai:siddiqa.aysha"Chu, Yu-Ming"https://zbmath.org/authors/?q=ai:chu.yuming"Azam, Muhammad"https://zbmath.org/authors/?q=ai:azam.muhammad-umer"Basra, Muhammad Asim Raza"https://zbmath.org/authors/?q=ai:basra.muhammad-asim-raza"Virk, Abaid ur Rehman"https://zbmath.org/authors/?q=ai:virk.abaid-ur-rehmanSummary: A topological index is an important tool in predicting physicochemical properties of a chemical compound. Topological indices help us to assign a single number to a chemical compound. Drugs and other chemical compounds are frequently demonstrated as different polygonal shapes, trees, graphs, etc. In this paper, we will compute irregularity indices for the benzene ring embedded in a P-type surface \(\left( B R_p\right)\) and the simple bounded dual of the benzene ring embedded in a P-type surface \(\left( S B R_p\right)\).Computing vertex-based eccentric topological descriptors of zero-divisor graph associated with commutative rings.https://zbmath.org/1459.051142021-05-28T16:06:00+00:00"Ahmadini, Abdullah Ali H."https://zbmath.org/authors/?q=ai:ahmadini.abdullah-ali-h"Koam, Ali N. A."https://zbmath.org/authors/?q=ai:koam.ali-n-a"Ahmad, Ali"https://zbmath.org/authors/?q=ai:ahmad.ali"Bača, Martin"https://zbmath.org/authors/?q=ai:baca.martin"Semaničová-Feňovčíková, Andrea"https://zbmath.org/authors/?q=ai:semanicova-fenovcikova.andreaSummary: The applications of finite commutative ring are useful substances in robotics and programmed geometric, communication theory, and cryptography. In this paper, we study the vertex-based eccentric topological indices of a zero-divisor graphs of commutative ring \(\mathbb{Z}_{p^2}\times \mathbb{Z}_q\), where \(p\) and \(q\) are primes.On the geodesic identification of vertices in convex plane graphs.https://zbmath.org/1459.050582021-05-28T16:06:00+00:00"Alsaadi, Fawaz E."https://zbmath.org/authors/?q=ai:alsaadi.fawaz-e"Salman, Muhammad"https://zbmath.org/authors/?q=ai:salman.muhammad"Rehman, Masood Ur"https://zbmath.org/authors/?q=ai:rehman.masood-ur"Khan, Abdul Rauf"https://zbmath.org/authors/?q=ai:khan.abdul-rauf-khan|rauf-khan.abdul"Cao, Jinde"https://zbmath.org/authors/?q=ai:cao.jinde.1|cao.jinde"Alassafi, Madini Obad"https://zbmath.org/authors/?q=ai:alassafi.madini-obadSummary: A shortest path between two vertices \(u\) and \(v\) in a connected graph \(G\) is a \(u\)-\(v\) geodesic. A vertex \(w\) of \(G\) performs the geodesic identification for the vertices in a pair \((u,v)\) if either \(v\) belongs to a \(u-w\) geodesic or \(u\) belongs to a \(v-w\) geodesic. The minimum number of vertices performing the geodesic identification for each pair of vertices in \(G\) is called the strong metric dimension of \(G\). In this paper, we solve the strong metric dimension problem for three convex plane graphs by performing the geodesic identification of their vertices.Metric dimension of heptagonal circular ladder.https://zbmath.org/1459.050672021-05-28T16:06:00+00:00"Sharma, Sunny Kumar"https://zbmath.org/authors/?q=ai:sharma.sunny-kumar"Bhat, Vijay Kumar"https://zbmath.org/authors/?q=ai:bhat.vijay-kumarGraphons, permutons and the Thoma simplex: three mod-Gaussian moduli spaces.https://zbmath.org/1459.600072021-05-28T16:06:00+00:00"Féray, Valentin"https://zbmath.org/authors/?q=ai:feray.valentin"Méliot, Pierre-Loïc"https://zbmath.org/authors/?q=ai:meliot.pierre-loic"Nikeghbali, Ashkan"https://zbmath.org/authors/?q=ai:nikeghbali.ashkanThe paper under review is devoted to the theory of graphons, which are limit objects for sequences of graphs \((G_{n})\) on increasing numbers of vertices. Given a graphon \(\gamma\), one can construct a model of random graphs \((G_{n}(\gamma))\) which converges to \(\gamma\). Convergence of a sequence of graphs to a graphon is known, through, e.g., work of Lovász and co-authors, to be closely linked to convergence of (various definitions of) subgraph densities \(t(F,G_{n})\), informally the ratio of the number of occurrences of the fixed graph \(F\) in \(G_{n}\) to the maximum possible such number.
The main concern of the paper under review is to show that for a graphon \(\gamma\) and a fixed graph \(F\) the sequence \(t(F,G_{n}(\gamma))\) has the property that generically it is mod-Gaussian after a suitable renormalisation. Being mod-Gaussian is a property of a sequence \((X_{n})\) of random variables which can be defined in terms of the Laplace transforms of the random variables. If a sequence is mod-Gaussian, desirable consequences include a central limit theorem, moderate deviation principle, estimate of speed of convergence, local limit theorem and concentration inequality. One of the methods for proving that a sequence is mod-Gaussian is through considering estimates of cumulants. One of the estimates needed to get this approach to work can be approached through dependency graphs. The word ``generically'' above is a technical limitation of the method: for example, it transpires that the standard Erdős-Rényi random graphs are singular so this theory gives less insight there, the theory is best for models where there is less symmetry.
In addition to the results on graphons proved in the paper, there is a theory of permutons (limit objects for permutations) and limit objects for random integer partitions. The authors also prove several results based on the mod-Gaussian property for these random combinatorial objects. An application to the Ising model in statistical physics is also given.
Reviewer: David B. Penman (Colchester)An explicit formula for the number of labeled series-parallel \(k\)-cyclic blocks.https://zbmath.org/1459.052922021-05-28T16:06:00+00:00"Voblyi, V. A."https://zbmath.org/authors/?q=ai:voblyi.vitaliy-a(no abstract)The salesman's improved tours for fundamental classes.https://zbmath.org/1459.901752021-05-28T16:06:00+00:00"Boyd, Sylvia"https://zbmath.org/authors/?q=ai:boyd.sylvia-c"Sebő, András"https://zbmath.org/authors/?q=ai:sebo.andrasSummary: Finding the exact integrality gap \(\alpha\) for the LP relaxation of the metric travelling salesman problem (TSP) has been an open problem for over 30 years, with little progress made. It is known that \(4/3 \le \alpha \le 3/2\), and a famous conjecture states \(\alpha = 4/3\). It has also been conjectured that the integrality gap is achieved for half-integer basic solutions of the linear program. For this problem, essentially two ``fundamental'' classes of instances have been proposed. This fundamental property means that in order to show that the integrality gap is at most \(\rho\) for all instances of the metric TSP, it is sufficient to show it only for the instances in the fundamental class. However, despite the importance and the simplicity of such classes, no apparent effort has been deployed for improving the integrality gap bounds for them. In this paper we take a natural first step in this endeavour, and consider the 1/2-integer points of one such class. We successfully improve the upper bound for the integrality gap from 3/2 to 10/7 for a superclass of these points for which a lower bound of 4/3 is proved. A key role in the proof of this result is played by finding Hamiltonian cycles whose existence is equivalent to Kotzig's result on ``compatible Eulerian tours'', and which lead us to delta-matroids for developing the related algorithms. Our arguments also involve other innovative tools from combinatorial optimization with the potential of a broader use.Algorithms and hardness for signed domination.https://zbmath.org/1459.681612021-05-28T16:06:00+00:00"Lin, Jin-Yong"https://zbmath.org/authors/?q=ai:lin.jin-yong"Poon, Sheung-Hung"https://zbmath.org/authors/?q=ai:poon.sheung-hungSummary: A signed dominating function for a graph \(G=(V, E)\) is a function \(f\): \(V \rightarrow \{ +1, -1\}\) such that for all \(v \in V\), the sum of the function values over the closed neighborhood of \(v\) is at least one. The weight \(w(f(V))\) of signed dominating function \(f\) for vertex set \(V\) is the sum of \(f(v)\) for \(v \in V\). The \textit{signed domination number}\(\gamma_s\) of \(G\) is the minimum weight of a signed dominating function for \(G\). The \textit{signed domination (SD) problem} asks for a signed dominating function which contributes the signed domination number. First we show that the SD problem is W[2]-hard. Next we show that the SD problem on graphs of maximum degree six is APX-hard. Then we present constant-factor approximation algorithms for the SD problem on subcubic graphs, graphs of maximum degree four, and graphs of maximum degree five, respectively. In addition, we present an alternative and more direct proof for the NP-completeness of the SD problem on subcubic
planar bipartite graphs. Lastly, we obtain an \(O^{*}(5.1957^k)\)-time FPT-algorithm for the SD problem on subcubic graphs \(G\), where \(k\) is the signed domination number of \(G\).
For the entire collection see [Zbl 1320.68020].Completion of the mixed unit interval graphs hierarchy.https://zbmath.org/1459.052182021-05-28T16:06:00+00:00"Talon, Alexandre"https://zbmath.org/authors/?q=ai:talon.alexandre"Kratochvil, Jan"https://zbmath.org/authors/?q=ai:kratochvil.janSummary: We describe the missing class of the hierarchy of mixed unit interval graphs, generated by the intersection graphs of closed, open and one type of half-open intervals of the real line. This class lies strictly between unit interval graphs and mixed unit interval graphs. We give a complete characterization of this new class, as well as a polynomial time algorithm to recognize graphs from this class and to produce a corresponding interval representation if one exists.
For the entire collection see [Zbl 1320.68020].An improved exact algorithm for maximum induced matching.https://zbmath.org/1459.053182021-05-28T16:06:00+00:00"Xiao, Mingyu"https://zbmath.org/authors/?q=ai:xiao.mingyu"Tan, Huan"https://zbmath.org/authors/?q=ai:tan.huanSummary: This paper studies exact algorithms for the Maximum Induced Matching problem, in which an \(n\)-vertex graph is given and we are asked to find a set of maximum number of edges in the graph such that no pair of edges in the set have a common endpoint or are adjacent by another edge. This problem has applications in many different areas. We will give several structural properties of the problem and present an \(O^*(1.4391^n)\)-time algorithm, which improves previous exact algorithms for this problem.
For the entire collection see [Zbl 1320.68020].The complexity of degree anonymization by graph contractions.https://zbmath.org/1459.681582021-05-28T16:06:00+00:00"Hartung, Sepp"https://zbmath.org/authors/?q=ai:hartung.sepp"Talmon, Nimrod"https://zbmath.org/authors/?q=ai:talmon.nimrodSummary: We study the computational complexity of \(k\)-anonymizing a given graph by as few graph contractions as possible. A graph is said to be \(k\)-anonymous if for every vertex in it, there are at least \(k-1\) other vertices with exactly the same degree. The general degree anonymization problem is motivated by applications in privacy-preserving data publishing, and was studied to some extent for various graph operations (most notable operations being edge addition, edge deletion, vertex addition, and vertex deletion). We complement this line of research by studying several variants of graph contractions, which are operations of interest, for example, in the contexts of social networks and clustering algorithms. We show that the problem of degree anonymization by graph contractions is \({\mathsf {NP}}\)-hard even for some very restricted inputs, and identify some fixed-parameter tractable cases.
For the entire collection see [Zbl 1320.68020].Finding connected dense \(k\)-subgraphs.https://zbmath.org/1459.681552021-05-28T16:06:00+00:00"Chen, Xujin"https://zbmath.org/authors/?q=ai:chen.xujin"Hu, Xiaodong"https://zbmath.org/authors/?q=ai:hu.xiaodong"Wang, Changjun"https://zbmath.org/authors/?q=ai:wang.changjunSummary: Given a connected graph \(G\) on \(n\) vertices and a positive integer \(k\leq n\), a subgraph of \(G\) on \(k\) vertices is called a \(k\)-subgraph in \(G\). We design combinatorial approximation algorithms for finding a connected \(k\)-subgraph in \(G\) such that its density is at least a factor \(\varOmega (\max \{n^{-2/5},k^2/n^2\})\) of the density of the densest \(k\)-subgraph in \(G\) (which is not necessarily connected). These particularly provide the first non-trivial approximations for the densest connected \(k\)-subgraph problem on general graphs.
For the entire collection see [Zbl 1320.68020].Scale-free percolation in continuous space: quenched degree and clustering coefficient.https://zbmath.org/1459.052982021-05-28T16:06:00+00:00"Dalmau, Joseba"https://zbmath.org/authors/?q=ai:dalmau.joseba"Salvi, Michele"https://zbmath.org/authors/?q=ai:salvi.micheleSummary: Spatial random graphs capture several important properties of real-world networks. We prove quenched results for the continuous-space version of scale-free percolation introduced in [\textit{P. Deprez} and \textit{M. V. Wüthrich}, Commun. Math. Stat. 7, No. 3, 269--308 (2019; Zbl 1436.60079)]. This is an undirected inhomogeneous random graph whose vertices are given by a Poisson point process in \(\mathbb{R}^d\). Each vertex is equipped with a random weight, and the probability that two vertices are connected by an edge depends on their weights and on their distance. Under suitable conditions on the parameters of the model, we show that, for almost all realizations of the point process, the degree distributions of all the nodes of the graph follow a power law with the same tail at infinity. We also show that the averaged clustering coefficient of the graph is self-averaging. In particular, it is almost surely equal to the annealed clustering coefficient of one point, which is a strictly positive quantity.Construction for the sequences of \(Q\)-borderenergetic graphs.https://zbmath.org/1459.051682021-05-28T16:06:00+00:00"Deng, Bo"https://zbmath.org/authors/?q=ai:deng.bo"Chang, Caibing"https://zbmath.org/authors/?q=ai:chang.caibing"Zhao, Haixing"https://zbmath.org/authors/?q=ai:zhao.haixing"Das, Kinkar Chandra"https://zbmath.org/authors/?q=ai:das.kinkar-chandraSummary: This research intends to construct a signless Laplacian spectrum of the complement of any \(k\)-regular graph \(G\) with order \(n\). Through application of the join of two arbitrary graphs, a new class of \(Q\)-borderenergetic graphs is determined with proof. As indicated in the research, with a regular \(Q\)-borderenergetic graph, sequences of regular \(Q\)-borderenergetic graphs can be constructed. The procedures for such a construction are determined and demonstrated. Significantly, all the possible regular \(Q\)-borderenergetic graphs of order \(7<n\leq10\) are determined.Nordhaus-Gaddum-type relations for arithmetic-geometric spectral radius and energy.https://zbmath.org/1459.051922021-05-28T16:06:00+00:00"Wang, Yajing"https://zbmath.org/authors/?q=ai:wang.yajing"Gao, Yubin"https://zbmath.org/authors/?q=ai:gao.yubinSummary: Spectral graph theory plays an important role in engineering. Let \(G\) be a simple graph of order \(n\) with vertex set \(V=\left\{ v_1, v_2, \ldots, v_n\right\}\). For \(v_i\in V \), the degree of the vertex \(v_i\), denoted by \(d_i\), is the number of the vertices adjacent to \(v_i\). The arithmetic-geometric adjacency matrix \(A_{a g}\left( G\right)\) of \(G\) is defined as the \(n\times n\) matrix whose \(\left( i, j\right)\) entry is equal to \(\left( \left( d_i + d_j\right)/2 \sqrt{ d_i d_j}\right)\) 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 spectral radius and energy of its arithmetic-geometric adjacency matrix, respectively. In this paper, some new upper bounds on arithmetic-geometric energy are obtained. In addition, we present the Nordhaus-Gaddum-type relations for arithmetic-geometric spectral radius and arithmetic-geometric energy and characterize corresponding extremal graphs.On the Duffin-Schaeffer conjecture.https://zbmath.org/1459.111542021-05-28T16:06:00+00:00"Koukoulopoulos, Dimitris"https://zbmath.org/authors/?q=ai:koukoulopoulos.dimitris"Maynard, James"https://zbmath.org/authors/?q=ai:maynard.jamesThis very well written and important paper settles the long standing Duffin-Schaeffer conjecture along with some related conjectures in metric Diophantine approximation. The conjecture is concerned with the set
\[
\Bigg\{ \alpha \in [0,1] : \bigg\vert \alpha - \frac{a}{q}\bigg\vert \le \frac{\psi(q)}{q} \; \text{ for infinitely many coprime } \; p,q \in {\mathbb Z}, q > 0 \Bigg\},
\]
where \(\psi: {\mathbb N} \rightarrow [0,\infty)\) is some function. It is an easy consequence of the Borel-Cantelli lemma that if \(\sum_{q=1}^\infty \psi(q) < \infty\), the Lebesgue measure of this set is equal to \(0\). Conversely, it was famously shown by [\textit{A. Khintchine}, Math. Ann. 92, 115--125 (1924; JFM 50.0125.01)] that if \(q \psi(q)\) is decreasing, the condition that \(\sum_{q=1}^\infty \psi(q) = \infty\) implies that the Lebesgue measure of the set is equal to \(1\), thus providing a complete description under the assumption of monotonicity.
\textit{R. J. Duffin} and \textit{A. C. Schaeffer} [Duke Math. J. 8, 243--255 (1941; Zbl 0025.11002)] proved that the monotonicity condition on \(\psi\) is in fact necessary for the validity of Khintchine's theorem. Letting \(\phi\) denote the Euler totient function, they conjectured that the Lebesgue measure of the set should instead be governed by the series \(\sum_{q=1}^\infty \psi(q)\phi(q)/q\) in the same manner: convergence should imply measure \(0\) and divergence should imply measure \(1\). This long standing conjecture is settled in the affirmative in the present paper.
The proof starts with a series of reductions. Via a mean-and-variance argument, it is shown that a certain second moment bound is sufficient for the conclusion. This bound is subsequently interpreted as a statement on a bipartite graph with a lot of additional arithmetic structure. With arithmetic methods, it is then shown that the existence of a certain highly structured subgraph is sufficient for the conclusion. To conclude, the authors perform a clever iterative procedure on the original graph to deduce the existence of such a subgraph and hence the Duffin-Schaeffer conjecture.
As consequences of the main result, a conjecture of \textit{P. A. Catlin} [J. Number Theory 8, 282--288, 289--297 (1976; Zbl 0337.10038)], which provides a zero-one law for the corresponding set without the assumption of coprimality, is deduced. Additionally, results on the Hausdorff dimension of the exceptional set when the series is convergent are deduced by appealing to results of \textit{V. Beresnevich} and \textit{S. Velani} [Ann. Math. (2) 164, No. 3, 971--992 (2006; Zbl 1148.11033)].
Reviewer: Simon Kristensen (Aarhus)Hierarchical graph Laplacian eigen transforms.https://zbmath.org/1459.420562021-05-28T16:06:00+00:00"Irion, Jeff"https://zbmath.org/authors/?q=ai:irion.jeff"Saito, Naoki"https://zbmath.org/authors/?q=ai:saito.naokiSummary: We describe a new transform that generates a dictionary of bases for handling data on a graph by combining recursive partitioning of the graph and the Laplacian eigenvectors of each subgraph. Similar to the wavelet packet and local cosine dictionaries for regularly sampled signals, this dictionary of bases on the graph allows one to select an orthonormal basis that is most suitable to one's task at hand using a best-basis type algorithm. We also describe a few related transforms including a version of the Haar wavelet transform on a graph, each of which may be useful in its own right.The smallest symmetric cubic graphs with given type.https://zbmath.org/1459.053492021-05-28T16:06:00+00:00"Conder, Marston D. E."https://zbmath.org/authors/?q=ai:conder.marston-d-eThis paper concerns highly symmetric cubic graphs. The basic setup is as follows: we say that a graph is symmetric if its automorphism group acts transitively on pairs consisting of a vertex and an incident edge. This paper is devoted to symmetric finite cubic graphs, for which there exists a good structure theory: given such a graph, its automorphism group is known to act regularly on \(k\)-arcs for some \(1\leqslant k\leqslant5\). This naturally divides the set of symmetric cubic graphs into five types. The types for \(k=2,4\) are further subdivided according to the action of an edge-stabiliser. So there are seven classes, labelled \(1\), \(2^1\), \(2^2\), \(3\), \(4^1\), \(4^2\), \(5\). In [\textit{M. Conder} and \textit{R. Nedela}, J. Algebra 322, No. 3, 722--740 (2009; Zbl 1183.05034)], the classes are further subdivided according to the existence of subgroups of the automorphism group which act in each of these seven ways. They showed that there are seventeen possible action types (each of which corresponds to a non-empty subset of \(\{1, 2^1, 2^2, 3, 4^1, 4^2, 5\}\)). They found an example of a graph with each action type, and in fourteen of the cases showed that their example is the smallest. The object of the paper under review is to find the smallest example for each of the other three action types.
The method is via computational group theory: it is known that for each of the seven basic types there is a finitely-presented group which is universal in the sense that the automorphism group of any finite cubic graph of that type is a quotient. So the object is to find suitable quotients of these groups. This is done efficiently using MAGMA.
The paper is well written, but should not be used as an introduction to the subject; it should be read as a sequel to [loc. cit.].
Reviewer: Matthew Fayers (London)Bounds on the \(\alpha \)-distance energy and \(\alpha \)-distance Estrada index of graphs.https://zbmath.org/1459.051952021-05-28T16:06:00+00:00"Yang, Yang"https://zbmath.org/authors/?q=ai:yang.yang|yang.yang.3|yang.yang.4|yang.yang.2|yang.yang.1|yang.yang.5"Sun, Lizhu"https://zbmath.org/authors/?q=ai:sun.lizhu"Bu, Changjiang"https://zbmath.org/authors/?q=ai:bu.changjiangSummary: Let \(G\) be a simple undirected connected graph, then \(D_\alpha\left( G\right)=\alpha Tr\left( G\right)+\left( 1 - \alpha\right)D\left( G\right)\) is called the \(\alpha \)-distance matrix of \(G\), where \(\alpha\in\left[ 0,1\right]\), \(D\left( G\right)\) is the distance matrix of \(G\), and \(Tr\left( G\right)\) is the vertex transmission diagonal matrix of \(G\). In this paper, we study some bounds on the \(\alpha \)-distance energy and \(\alpha \)-distance Estrada index of \(G\). Furthermore, we establish the relation between \(\alpha \)-distance Estrada index and \(\alpha \)-distance energy.Lower bounds on the entire Zagreb indices of trees.https://zbmath.org/1459.053202021-05-28T16:06:00+00:00"Luo, Liang"https://zbmath.org/authors/?q=ai:luo.liang"Dehgardi, Nasrin"https://zbmath.org/authors/?q=ai:dehgardi.nasrin"Fahad, Asfand"https://zbmath.org/authors/?q=ai:fahad.asfandSummary: For a (molecular) graph \(G\), the first and the second entire Zagreb indices are defined by the formulas \(M_1^\varepsilon\left( G\right)=\sum_{x \in V \left( G\right) \cup E \left( G\right)} d \left( x\right)^2\) and \(M_2^\varepsilon\left( G\right)= \sum_{x \text{ is either adjacent or incident to } y} d \left( x\right) d \left( y\right)\) in which \(d\left( x\right)\) represents the degree of a vertex or an edge \(x\). In the current manuscript, we establish some lower bounds on the first and the second entire Zagreb indices and determine the extremal trees which achieve these bounds.The bondage number of generalized Petersen graphs \(P(n,2)\).https://zbmath.org/1459.052452021-05-28T16:06:00+00:00"Pei, Lidan"https://zbmath.org/authors/?q=ai:pei.lidan"Pan, Xiangfeng"https://zbmath.org/authors/?q=ai:pan.xiangfengSummary: The domination number \(\gamma\left( G\right)\) of a nonempty graph \(G\) is the minimum cardinality among all subsets \(D\subseteq V\left( G\right)\) such that \(N_G\left[ D\right]=V\left( G\right)\). The bondage number \(b\left( G\right)\) of a graph \(G\) is the smallest number of edges whose removal from \(G\) results in a graph with larger domination number. The exact value of \(b\left( P \left( n , 2\right)\right)\) for \(n=0,3,4\pmod{5}\) and the bounds of \(b\left( P \left( n , 2\right)\right)\) for \(n=1,2\pmod{5}\) are determined.The ordered covering problem.https://zbmath.org/1459.052562021-05-28T16:06:00+00:00"Feige, Uriel"https://zbmath.org/authors/?q=ai:feige.uriel"Hitron, Yael"https://zbmath.org/authors/?q=ai:hitron.yaelSummary: We study the ordered covering (OC) problem. The input is a finite set of \(n\) elements \(X\), a color function \(c:X \rightarrow \{0,1\}\) and a collection \(\mathcal S\) of subsets of \(X\). A solution consists of an ordered tuple \(T=(S_1,\dots ,S_{\ell})\) of sets from \(\mathcal {S}\) which covers \(X\), and a coloring \(g:\{S_i\}_{i=1}^\ell \rightarrow \{0,1\}\) such that for all \(x \in X\), the first set covering \(x\) in the tuple, namely \(S_j\) with \(j=\min \{i : x \in S_i\}\), has color \(g(S_j)=c(x)\). The minimization version is to find a solution using the minimum number of sets. Variants of OC include OC\((\alpha _0,\alpha _1)\) in which each element of color \(i \in \{0,1\}\) appears in at most \(\alpha _i\) sets of \(\mathcal S\), and \(k\)-OC in which the first set of the solution \(S_1\) is required to have color~0, and there are at most \(k-1\) alternations of colors in the solution. Among other results we show:
(i) there is a polynomial time approximation algorithm for Min-OC\((2, 2)\) with approximation ratio~2. (This is best possible unless vertex cover can be approximated within a ratio better than 2.) Moreover, Min-OC\((2,2)\) can be solved optimally in polynomial time if the underlying instance is bipartite.
(ii) For every \(\alpha _0, \alpha _1 \geq 2\), there is a polynomial time approximation algorithm for Min-3-OC\((\alpha _0,\alpha _1)\) with approximation \(\alpha _1(\alpha _0 - 1)\). Unless the unique games conjecture is false, this is best possible.On cycle transversals and their connected variants in the absence of a small linear forest.https://zbmath.org/1459.053322021-05-28T16:06:00+00:00"Dabrowski, Konrad K."https://zbmath.org/authors/?q=ai:dabrowski.konrad-kazimierz"Feghali, Carl"https://zbmath.org/authors/?q=ai:feghali.carl"Johnson, Matthew"https://zbmath.org/authors/?q=ai:johnson.matthew"Paesani, Giacomo"https://zbmath.org/authors/?q=ai:paesani.giacomo"Paulusma, Daniël"https://zbmath.org/authors/?q=ai:paulusma.daniel"Rzążewski, Paweł"https://zbmath.org/authors/?q=ai:rzazewski.pawelSummary: A graph is \(H\)-free if it contains no induced subgraph isomorphic to \(H\). We prove new complexity results for the two classical cycle transversal problems Feedback Vertex Set and Odd Cycle Transversal by showing that they can be solved in polynomial time on \((sP_1+ P_3)\)-free graphs for every integer \(s\ge 1\). We show the same result for the variants Connected Feedback Vertex Set and Connected Odd Cycle Transversal. We also prove that the latter two problems are polynomial-time solvable on cographs; this was already known for Feedback Vertex Set and Odd Cycle Transversal. We complement these results by proving that Odd Cycle Transversal and Connected Odd Cycle Transversal are NP-complete on \((P_2+ P_5,P_6)\)-free graphs.Paired-domination problem on distance-hereditary graphs.https://zbmath.org/1459.052422021-05-28T16:06:00+00:00"Lin, Ching-Chi"https://zbmath.org/authors/?q=ai:lin.ching-chi"Ku, Keng-Chu"https://zbmath.org/authors/?q=ai:ku.keng-chu"Hsu, Chan-Hung"https://zbmath.org/authors/?q=ai:hsu.chan-hungSummary: A paired-dominating set of a graph \(G\) is a dominating set \(S\) of \(G\) such that the subgraph of \(G\) induced by \(S\) has a perfect matching. \textit{T. W. Haynes} and \textit{P. J. Slater} [Networks 32, No. 3, 199--206 (1998; Zbl 0997.05074)] introduced the concept of paired-domination and showed that the problem of determining minimum paired-dominating sets is NP-complete on general graphs. Ever since then many algorithmic results are studied on some important classes of graphs. In this paper, we extend the results by providing an \(O(n^2)\)-time algorithm on distance-hereditary graphs.Environmental impact analysis of hub-and-spoke network operation.https://zbmath.org/1459.900372021-05-28T16:06:00+00:00"Tian, Yong"https://zbmath.org/authors/?q=ai:tian.yong"Sun, Mengyuan"https://zbmath.org/authors/?q=ai:sun.mengyuan"Wan, Lili"https://zbmath.org/authors/?q=ai:wan.lili"Hang, Xu"https://zbmath.org/authors/?q=ai:hang.xuSummary: The hub-and-spoke network has demonstrated its economies of scale and scope in the rapid development of the civil aviation industry. In order to fit the development trend of green civil aviation, a series of environmental problems such as fuel consumption and pollutant emissions caused by air transportation cannot be ignored. Firstly, this paper selects six cities of Shenyang, Beijing, Qingdao, Zhengzhou, Guangzhou, and Nanjing as the research objects, collects the passenger flow and the distance information of the corresponding segment, determines the location of the hub airport, analyzes the operating environment of the aircraft in the hub-and-spoke network, establishes an aircraft emission assessment model, and calculates the mass of aircraft emissions and fuel consumption. Secondly, based on the calculation results, the comparison of aircraft emissions and fuel consumption between the hub-and-spoke network and the point-to-point network shows that the total carbon monoxide (CO) emissions are reduced by 35.84\%, the total hydrocarbon compounds (HC) emissions are increased by 68.82\%, and the total nitrogen oxides \((\mathrm{NO_x})\) emissions are increased by 24.87\%. The total mass of pollutants (including CO, HC, and \(\mathrm{NO_x})\) decreased by 29.37\%, and the total fuel consumption decreased by 68.17\%. In general, the use of a hub-and-spoke network reduces the pollutant emissions and fuel consumption of aircraft as a whole while ensuring the lowest passenger transportation cost. Finally, based on the current international situation and the enhancement of people's awareness of environmental protection, a summary analysis of the hub-and-spoke network and the point-to-point network is obtained, and some enlightenment and research significance are obtained.On the vanishing of discrete singular cubical homology for graphs.https://zbmath.org/1459.050542021-05-28T16:06:00+00:00"Barcelo, Hélène"https://zbmath.org/authors/?q=ai:barcelo.helene"Greene, Curtis"https://zbmath.org/authors/?q=ai:greene.curtis"Jarrah, Abdul Salam"https://zbmath.org/authors/?q=ai:jarrah.abdul-salam"Welker, Volkmar"https://zbmath.org/authors/?q=ai:welker.volkmarRamsey numbers involving large books.https://zbmath.org/1459.053302021-05-28T16:06:00+00:00"Lin, Qizhong"https://zbmath.org/authors/?q=ai:lin.qizhong"Liu, Xiudi"https://zbmath.org/authors/?q=ai:liu.xiudiClustering of sparse data via network communities -- a prototype study of a large online market.https://zbmath.org/1459.911162021-05-28T16:06:00+00:00"Reichardt, Jörg"https://zbmath.org/authors/?q=ai:reichardt.jorg"Bornholdt, Stefan"https://zbmath.org/authors/?q=ai:bornholdt.stefanSome results on the graph associated to a lattice with given a filter.https://zbmath.org/1459.050862021-05-28T16:06:00+00:00"Malekpour, S."https://zbmath.org/authors/?q=ai:malekpour.shahide"Bazigaran, B."https://zbmath.org/authors/?q=ai:bazigaran.behnamSummary: In this paper, we study some graph-theoretical properties of \(\Gamma_S(L)\), a graph which the vertex set is all elements of a finite lattice \(L\) and two distinct vertices \(a\) and \(b\) are adjacent if and only if \(a\vee b \in S\), where \(S\) is a \(\wedge\)-closed subset of \(L\). As a consequence of our work, some results in \textit{M. Afkhami} and \textit{K. Khashyarmanesh} [Bull. Malays. Math. Sci. Soc. (2) 37, No. 1, 261--269 (2014; Zbl 1286.05031)] are extended to the case that \(S\) is a filter.The distance Laplacian spectral radius of clique trees.https://zbmath.org/1459.051972021-05-28T16:06:00+00:00"Zhang, Xiaoling"https://zbmath.org/authors/?q=ai:zhang.xiaoling"Zhou, Jiajia"https://zbmath.org/authors/?q=ai:zhou.jiajiaSummary: The distance Laplacian matrix of a connected graph \(G\) is defined as \(\mathcal{L}\left( G\right)=\mathrm{Tr}\left( G\right)-D\left( G\right)\), where \(D\left( G\right)\) is the distance matrix of \(G\) and \(\mathrm{Tr}\left( G\right)\) is the diagonal matrix of vertex transmissions of \(G\). The largest eigenvalue of \(\mathcal{L}\left( G\right)\) is called the distance Laplacian spectral radius of \(G\). In this paper, we determine the graphs with maximum and minimum distance Laplacian spectral radius among all clique trees with \(n\) vertices and \(k\) cliques. Moreover, we obtain \(n\) vertices and \(k\) cliques.Activity ageing in growing networks.https://zbmath.org/1459.053122021-05-28T16:06:00+00:00"Lambiotte, R."https://zbmath.org/authors/?q=ai:lambiotte.renaudAn elementary proof of a matrix tree theorem for directed graphs.https://zbmath.org/1459.051022021-05-28T16:06:00+00:00"De Leenheer, Patrick"https://zbmath.org/authors/?q=ai:de-leenheer.patrickIn this paper, an elementary proof of a generalization of Kirchhoff's matrix tree theorem to directed and weighted graphs is given. This proof is based on the Laplacian matrix associated to the graph.
Reviewer: Shariefuddin Pirzada (Srinagar)More on the sixth coefficient of the matching polynomial in regular graphs.https://zbmath.org/1459.051282021-05-28T16:06:00+00:00"Alikhani, S."https://zbmath.org/authors/?q=ai:alikhani.saeid"Soltani, N."https://zbmath.org/authors/?q=ai:soltani.nedaSummary: A matching set \(M\) in a graph \(G\) is a collection of edges of \(G\) such that no two edges from \(M\) share a vertex. The matching polynomial of \(G\) of order \(n\) is defined by \(\mu (G,x) = \sum^{\lfloor\frac{n}{2}\rfloor}_{r=0} (-1)^r \rho (G,r) x^{n-2r}\), where \(\rho(G,r)\) is the number of matching sets of \(G\) with cardinality \(r\) and \(p(G,0)\) is considered to be one. A graph that is characterized by its matching polynomial is said to be matching unique. In this paper, we consider some parameters related to the matching of regular graphs. We find the sixth coefficient of the matching polynomial of regular graphs. As a consequence, every cubic graph of order 10 is matching unique.Algebraic connectivity and disjoint vertex subsets of graphs.https://zbmath.org/1459.051892021-05-28T16:06:00+00:00"Sun, Yan"https://zbmath.org/authors/?q=ai:sun.yan"Li, Faxu"https://zbmath.org/authors/?q=ai:li.faxuSummary: It is well known that the algebraic connectivity of a graph is the second small eigenvalue of its Laplacian matrix. In this paper, we mainly research the relationships between the algebraic connectivity and the disjoint vertex subsets of graphs, which are presented through some upper bounds on algebraic connectivity.A guide to temporal networks. 2nd edition.https://zbmath.org/1459.600022021-05-28T16:06:00+00:00"Masuda, Naoki"https://zbmath.org/authors/?q=ai:masuda.naoki"Lambiotte, Renaud"https://zbmath.org/authors/?q=ai:lambiotte.renaudPublisher's description: Network science offers a powerful language to represent and study complex systems composed of interacting elements -- from the Internet to social and biological systems. The book presents recent theoretical and modelling progress in the emerging field of temporally varying networks and provides connections between the different areas of knowledge required to address this multi-disciplinary subject. After an introduction to key concepts on networks and stochastic dynamics, the authors guide the reader through a coherent selection of mathematical and computational tools for network dynamics. Perfect for students and professionals, this book is a gateway to an active field of research developing between the disciplines of applied mathematics, physics and computer science, with applications in others including social sciences, neuroscience and biology.
This second edition extensively expands upon the coverage of the first edition as the authors expertly present recent theoretical and modelling progress in the emerging field of temporal networks, providing the keys to (and connections between) the different areas of knowledge required to address this multi-disciplinary problem.
See the review of the first edition in [Zbl 1376.60001].Irregularity measures for metal-organic networks.https://zbmath.org/1459.050502021-05-28T16:06:00+00:00"Guo, Xuan"https://zbmath.org/authors/?q=ai:guo.xuan"Chu, Yu-Ming"https://zbmath.org/authors/?q=ai:chu.yuming"Hashmi, Muhammad Khalid"https://zbmath.org/authors/?q=ai:hashmi.muhammad-khalid"Virk, Abaid Ur Rehman"https://zbmath.org/authors/?q=ai:virk.abaid-ur-rehman"Li, Jingjng"https://zbmath.org/authors/?q=ai:li.jingjngSummary: Topological index plays an important role in predicting physicochemical properties of a molecular structure. With the help of the topological index, we can associate a single number with a molecular graph. Drugs and other chemical compounds are frequently demonstrated as different polygonal shapes, trees, graphs, etc. In this paper, we will compute irregularity indices for metal-organic networks.On the rank of a random binary matrix.https://zbmath.org/1459.150372021-05-28T16:06:00+00:00"Cooper, Colin"https://zbmath.org/authors/?q=ai:cooper.colin"Frieze, Alan"https://zbmath.org/authors/?q=ai:frieze.alan-m"Pegden, Wesley"https://zbmath.org/authors/?q=ai:pegden.wesleySummary: We study the rank of a random \(n \times m\) matrix \(\mathbf{A}_{n,m;k}\) with entries from \(GF(2)\), and exactly \(k\) unit entries in each column, the other entries being zero. The columns are chosen independently and uniformly at random from the set of all \(\binom{n}{k}\) such columns.
We obtain an asymptotically correct estimate for the rank as a function of the number of columns \(m\) in terms of \(c,n,k\), and where \(m=cn/k\). The matrix \(\mathbf{A}_{n,m;k}\) forms the vertex-edge incidence matrix of a \(k\)-uniform random hypergraph \(H\). The rank of \(\mathbf{A}_{n,m;k}\) can be expressed as follows. Let \(|C_2|\) be the number of vertices of the 2-core of \(H\), and \(|E(C_2)|\) the number of edges. Let \(m^*\) be the value of \(m\) for which \(|C_2|= |E(C_2)|\). Then w.h.p. for \(m<m^*\) the rank of \(\mathbf{A}_{n,m;k}\) is asymptotic to \(m\), and for \(m \ge m^*\) the rank is asymptotic to \(m-|E(C_2)|+|C_2|\).
In addition, assign i.i.d. \(U[0,1]\) weights \(X_i, i \in{1,2,\ldots m}\) to the columns, and define the weight of a set of columns \(S\) as \(X(S)=\sum_{j \in S} X_j\). Define a basis as a set of \(n-\mathbb{1} (k\text{ even})\) linearly independent columns. We obtain an asymptotically correct estimate for the minimum weight basis. This generalises the well-known result of \textit{A. M. Frieze} [Discrete Appl. Math. 10, 47--56 (1985; Zbl 0578.05015)] that, for \(k=2\), the expected length of a minimum weight spanning tree tends to \(\zeta(3)\sim 1.202\).Spreading dynamics in complex networks.https://zbmath.org/1459.911412021-05-28T16:06:00+00:00"Pei, Sen"https://zbmath.org/authors/?q=ai:pei.sen"Makse, Hernán A."https://zbmath.org/authors/?q=ai:makse.hernan-aGraph cycles and Olympiad problems.https://zbmath.org/1459.051382021-05-28T16:06:00+00:00"Suksompong, Warut"https://zbmath.org/authors/?q=ai:suksompong.warutSummary: We show how certain basic results about cycles in directed and undirected graphs can be used to solve some clever problems that appeared in major mathematics competitions.Recognizing Cayley digraphs.https://zbmath.org/1459.051212021-05-28T16:06:00+00:00"McDowell, Eric L."https://zbmath.org/authors/?q=ai:mcdowell.eric-lSummary: A criterion is developed that decides in a finite number of steps whether a finite digraph is a Cayley digraph of some group. In the process, a class of algebraic structure more general than groups are considered that arise naturally from digraphs that are ``almost'' Cayley.Synchronizability of multilayer star and star-ring networks.https://zbmath.org/1459.341252021-05-28T16:06:00+00:00"Deng, Yang"https://zbmath.org/authors/?q=ai:deng.yang"Jia, Zhen"https://zbmath.org/authors/?q=ai:jia.zhen"Yang, Feimei"https://zbmath.org/authors/?q=ai:yang.feimeiSummary: Synchronization of multilayer complex networks is one of the important frontier issues in network science. In this paper, we strictly derived the analytic expressions of the eigenvalue spectrum of multilayer star and star-ring networks and analyzed the synchronizability of these two networks by using the master stability function (MSF) theory. In particular, we investigated the synchronizability of the networks under different interlayer coupling strength, and the relationship between the synchronizability and structural parameters of the networks (i.e., the number of nodes, intralayer and interlayer coupling strengths, and the number of layers) is discussed. Finally, numerical simulations demonstrated the validity of the theoretical results.On the maximum weight independent set problem in graphs without induced cycles of length at least five.https://zbmath.org/1459.052342021-05-28T16:06:00+00:00"Chudnovsky, Maria"https://zbmath.org/authors/?q=ai:chudnovsky.maria"Pilipczuk, Marcin"https://zbmath.org/authors/?q=ai:pilipczuk.marcin"Pilipczuk, Michał"https://zbmath.org/authors/?q=ai:pilipczuk.michal"Thomassé, Stéphan"https://zbmath.org/authors/?q=ai:thomasse.stephanPartitioning edge-colored hypergraphs into few monochromatic tight cycles.https://zbmath.org/1459.052532021-05-28T16:06:00+00:00"Bustamante, Sebastián"https://zbmath.org/authors/?q=ai:bustamante.sebastian"Corsten, Jan"https://zbmath.org/authors/?q=ai:corsten.jan"Frankl, Nóra"https://zbmath.org/authors/?q=ai:frankl.nora"Pokrovskiy, Alexey"https://zbmath.org/authors/?q=ai:pokrovskiy.alexey"Skokan, Jozef"https://zbmath.org/authors/?q=ai:skokan.jozefA sufficient condition for planar graphs of maximum degree 6 to be totally 7-colorable.https://zbmath.org/1459.050952021-05-28T16:06:00+00:00"Zhu, Enqiang"https://zbmath.org/authors/?q=ai:zhu.enqiang"Rao, Yongsheng"https://zbmath.org/authors/?q=ai:rao.yongshengSummary: A total \(k\)-coloring of a graph is an assignment of \(k\) colors to its vertices and edges such that no two adjacent or incident elements receive the same color. The total coloring conjecture (TCC) states that every simple graph \(G\) has a total \(\left( \Delta \left( G\right) + 2\right)\)-coloring, where \(\Delta\left( G\right)\) is the maximum degree of \(G\). This conjecture has been confirmed for planar graphs with maximum degree at least 7 or at most 5, i.e., the only open case of TCC is that of maximum degree 6. It is known that every planar graph \(G\) of \(\Delta\left( G\right)\geq9\) or \(\Delta\left( G\right)\in\left\{ 7,8\right\}\) with some restrictions has a total \(\left( \Delta \left( G\right) + 1\right)\)-coloring. In particular, in (Shen and Wang, 2009), the authors proved that every planar graph with maximum degree 6 and without 4-cycles has a total 7-coloring. In this paper, we improve this result by showing that every diamond-free and house-free planar graph of maximum degree 6 is totally 7-colorable if every 6-vertex is not incident with two adjacent four cycles or three cycles of size \(p\), \(q\), \(\ell\) for some \(\left\{ p, q, \ell\right\}\in\left\{ \left\{ 3,4,4\right\}, \left\{ 3,3,4\right\}\right\}\).On Grundy and b-chromatic number of some families of graphs: a comparative study.https://zbmath.org/1459.050872021-05-28T16:06:00+00:00"Masih, Zoya"https://zbmath.org/authors/?q=ai:masih.zoya"Zaker, Manouchehr"https://zbmath.org/authors/?q=ai:zaker.manouchehrSummary: The Grundy and the b-chromatic number of graphs are two important chromatic parameters. The Grundy number of a graph \(G\), denoted by \(\Gamma (G)\) is the worst case behavior of greedy (First-Fit) coloring procedure for \(G\) and the b-chromatic number \(\text{b}(G)\) is the maximum number of colors used in any color-dominating coloring of \(G\). Because the nature of these colorings are different they have been studied widely but separately in the literature. This paper presents a comparative study of these coloring parameters. There exists a sequence \(\{G_n\}_{n\geq 1}\) with limited b-chromatic number but \(\Gamma (G_n)\rightarrow\infty\). We obtain families of graphs \(\mathcal{F}\) such that for some adequate function \(f(.)\), \(\Gamma (G)\le f(\text{b}(G))\), for each graph \(G\) from the family. This verifies a previous conjecture for these families.Wiener indices of maximal \(k\)-degenerate graphs.https://zbmath.org/1459.050482021-05-28T16:06:00+00:00"Bickle, Allan"https://zbmath.org/authors/?q=ai:bickle.allan"Che, Zhongyuan"https://zbmath.org/authors/?q=ai:che.zhongyuanSummary: A graph is maximal \(k\)-degenerate if each induced subgraph has a vertex of degree at most \(k\) and adding any new edge to the graph violates this condition. In this paper, we provide sharp lower and upper bounds on Wiener indices of maximal \(k\)-degenerate graphs of order \(n\geq k\geq 1\). A graph is chordal if every induced cycle in the graph is a triangle and chordal maximal \(k\)-degenerate graphs of order \(n\geq k\) are \(k\)-trees. For \(k\)-trees of order \(n\geq 2k+2\), we characterize all extremal graphs for the upper bound.Fractional matchings, component-factors and edge-chromatic critical graphs.https://zbmath.org/1459.052632021-05-28T16:06:00+00:00"Klopp, Antje"https://zbmath.org/authors/?q=ai:klopp.antje"Steffen, Eckhard"https://zbmath.org/authors/?q=ai:steffen.eckhardSummary: The first part of the paper studies star-cycle factors of graphs. It characterizes star-cycle factors of a graph \(G\) and proves upper bounds for the minimum number of \(K_{1,2}\)-components in a \(\{K_{1,1}, K_{1,2}, C_n:n\geq 3\}\)-factor of a graph \(G\). Furthermore, it shows where these components are located with respect to the Gallai-Edmonds decomposition of \(G\) and it characterizes the edges which are not contained in any \(\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}\)-factor of \(G\). The second part of the paper proves that every edge-chromatic critical graph \(G\) has a \(\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}\)-factor, and the number of \(K_{1,2}\)-components is bounded in terms of its fractional matching number. Furthermore, it shows that for every edge \(e\) of \(G\), there is a \(\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}\)-factor \(F\) with \(e \in E(F)\). Consequences of these results for Vizing's critical graph conjectures are discussed.On generalisations of the AVD conjecture to digraphs.https://zbmath.org/1459.050992021-05-28T16:06:00+00:00"Bensmail, Julien"https://zbmath.org/authors/?q=ai:bensmail.julien"McInerney, Fionn"https://zbmath.org/authors/?q=ai:mc-inerney.fionnSummary: Given an undirected graph, in the AVD (edge-colouring) Conjecture, the goal is to find a proper edge-colouring with the least number of colours such that every two adjacent vertices are incident to different sets of colours. More precisely, the conjecture says that, a few exceptions apart, every graph \(G\) should admit such an edge-colouring with at most \(\Delta (G)+2\) colours. Several aspects of interest behind this problem have been investigated over the recent years, including verifications of the conjecture for particular graph classes, general approximations of the conjecture, and multiple generalisations. In this paper, following a recent work of \textit{E. Sopena} and \textit{M. Woźniak} [``A note on the neighbour-distinguishing index of digraphs'', Preprint, \url{arXiv:1909.10240}], generalisations of the AVD Conjecture to digraphs are investigated. More precisely, four of the several possible ways of generalising the conjecture are focused upon. We completely settle one of our four variants, while, for the three remaining ones, we provide partial results.A size condition for diameter two orientable graphs.https://zbmath.org/1459.050612021-05-28T16:06:00+00:00"Cochran, Garner"https://zbmath.org/authors/?q=ai:cochran.garner"Czabarka, Éva"https://zbmath.org/authors/?q=ai:czabarka.eva"Dankelmann, Peter"https://zbmath.org/authors/?q=ai:dankelmann.peter"Székely, László"https://zbmath.org/authors/?q=ai:szekely.laszlo-aSummary: It was conjectured by \textit{K. M. Koh} and \textit{E. G. Tay} [ibid. 18, No. 4, 745--756 (2002; Zbl 1009.05063)] that for \(n\geq 5\) every simple graph of order \(n\) and size at least \(\binom{n}{2}-n+5\) has an orientation of diameter two. We prove this conjecture and hence determine for every \(n\geq 5\) the minimum value of \(m\) such that every graph of order \(n\) and size \(m\) has an orientation of diameter two.Measuring similarity between connected graphs: the role of induced subgraphs and complementarity eigenvalues.https://zbmath.org/1459.051422021-05-28T16:06:00+00:00"Seeger, Alberto"https://zbmath.org/authors/?q=ai:seeger.alberto"Sossa, David"https://zbmath.org/authors/?q=ai:sossa.davidSummary: This work elaborates on the old problem of measuring the degree of similarity, say \(\mathfrak{f}(G,H)\), between a pair of connected graphs \(G\) and \(H\), not necessarily of the same order. The choice of a similarity index \(\mathfrak{f}\) depends essentially on the graph properties that are considered as important in a given context. As relevant information on a graph, one may consider for instance its degree sequence, its characteristic polynomial, and so on. We explore some new similarity indices based on nonstandard spectral information contained in the graphs under comparison. By nonstandard spectral information in a graph, we mean the set of complementarity eigenvalues of the adjacency matrix. From such a spectral perspective, two distinct graphs \(G\) and \(H\) are viewed as highly similar if they share a large number of complementarity eigenvalues. This basic idea will be cast in a rigorous mathematical formalism.Embeddings of a graph into a surface with different weak chromatic numbers.https://zbmath.org/1459.052122021-05-28T16:06:00+00:00"Enami, Kengo"https://zbmath.org/authors/?q=ai:enami.kengo"Noguchi, Kenta"https://zbmath.org/authors/?q=ai:noguchi.kentaSummary: A weak coloring of a graph \(G\) embedded on a surface is a vertex coloring of \(G\) such that no face is monochromatic. The weak chromatic number of \(G\) is the minimum number \(k\) such that \(G\) has a weak \(k\)-coloring. \textit{A. Kündgen} and \textit{R. Ramamurthi} [J. Comb. Theory, Ser. B 85, No. 2, 307--337 (2002; Zbl 1029.05057)] conjectured that for each positive integer \(k\), there is a graph that has two different embeddings on the same surface whose weak chromatic numbers differ by at least \(k\). In this paper, we answer this conjecture affirmatively in two ways.Anti-Ramsey problems in complete bipartite graphs for \(t\) edge-disjoint rainbow spanning trees.https://zbmath.org/1459.052002021-05-28T16:06:00+00:00"Jia, Yuxing"https://zbmath.org/authors/?q=ai:jia.yuxing"Lu, Mei"https://zbmath.org/authors/?q=ai:lu.mei"Zhang, Yi"https://zbmath.org/authors/?q=ai:zhang.yi.6Summary: Let \(r(K_{p,q},t)\) be the maximum number of colors in an edge-coloring of the complete bipartite graph \(K_{p,q}\) not having \(t\) edge-disjoint rainbow spanning trees. We prove that \(r(K_{p,p},1)=p^2-2p+2\) for \(p\ge 4\) and \(r(K_{p,q},1)=pq-2q+1\) for \(p>q\ge 4\). Let \(t\ge 2\). We also show that \(r(K_{p,p},t)=p^2-2p+t+1\) for \(p \ge 2t+\sqrt{3t-3}+4\) and \(r(K_{p,q},t)=pq-2q+t\) for \(p>q\geq 2t+\sqrt{3t-2}+4\).Sharp upper bounds on the \(k\)-independence number in graphs with given minimum and maximum degree.https://zbmath.org/1459.052442021-05-28T16:06:00+00:00"O, Suil"https://zbmath.org/authors/?q=ai:o.suil"Shi, Yongtang"https://zbmath.org/authors/?q=ai:shi.yongtang"Taoqiu, Zhenyu"https://zbmath.org/authors/?q=ai:taoqiu.zhenyuSummary: The \(k\)-independence number of a graph \(G\) is the maximum size of a set of vertices at pairwise distance greater than \(k\). In this paper, for each positive integer \(k\), we prove sharp upper bounds for the \(k\)-independence number in an \(n\)-vertex connected graph with given minimum and maximum degree.On stability of the independence number of a certain distance graph.https://zbmath.org/1459.053082021-05-28T16:06:00+00:00"Ogarok, P. A."https://zbmath.org/authors/?q=ai:ogarok.p-a"Raigorodskii, A. M."https://zbmath.org/authors/?q=ai:raigorodskii.andrei-mSummary: We study the asymptotic behavior of the independence number of a random subgraph of a certain \((r, s)\)-distance graph. We provide upper and lower bounds for the critical edge survival probability under which a phase transition occurs, i.e., large new independent sets appear in the subgraph, which did not exist in the original graph.Proof of Stahl's conjecture in some new cases.https://zbmath.org/1459.050902021-05-28T16:06:00+00:00"Osztényi, József"https://zbmath.org/authors/?q=ai:osztenyi.jozsefGiven positive integers \(s\) and \(t\), an \(s\)-tuple coloring of a graph \(G\) with \(t\) colors is an assignment of \(s\) distinct colors to each vertex of \(G\) such that no two adjacent vertices share a color. The smallest integer \(t\) for which \(G\) has an \(s\)-tuple coloring with \(t\) colors is called the \(s\)th-multichromatic number of \(G\) and denoted by \(\chi_{s}(G)\). For positive integers \(m\), \(n\) with \(m \geq 2n\), the Kneser graph \(KG_{m,n}\) has as vertices all \(n\)-subsets of \(\{1, \ldots, m\}\), two vertices being connected by an edge if the corresponding subsets are disjoint. \textit{S. Stahl} [J. Comb. Theory, Ser. B 20, 185--203 (1976; Zbl 0293.05115)] formulated the conjecture that for all \(2n \le m\) and \(q\), \(0 \le r < n\), \(\chi_{nq-r}(KG_{m,n})=mq-2r\). \textit{L. Lovász} [J. Comb. Theory, Ser. A 25, 319--324 (1978; Zbl 0418.05028)] has proved the conjecture for \(nq-r=1\). Stahl [loc. cit.] showed it for \(r=0\) and also that \(\chi_{s+1} \geq \chi_{s}+2\) for any positive integer \(s\), which proved the conjecture for \(1 < s \le n\). \textit{S. Stahl} [loc. cit.; Discrete Math. 185, No. 1--3, 287--291 (1998; Zbl 0956.05045)] also settled the cases \(n=2,3\) and \(m=2n+1\), and \textit{J. Kincses} et al. [Eur. J. Comb. 34, No. 2, 502--511 (2013; Zbl 1254.05116)] the case \(m=10\) and \(n=4\). Finally, the conjecture is known to hold for \(s=qn\). In this paper, a new lower bound on the multichromatic number of Kneser graphs \(KG_{m,n}\) is presented and shown to be sharp for graph parameters \(2n < m < 3n\) and cases \(qn - \frac{n}{m-2n} < s < qn\) hereby confirming Stahl's conjecture [loc. cit.] for these new cases.
Reviewer: Reinhardt Euler (Brest)Proportional choosability of complete bipartite graphs.https://zbmath.org/1459.050882021-05-28T16:06:00+00:00"Mudrock, Jeffrey A."https://zbmath.org/authors/?q=ai:mudrock.jeffrey-a"Hewitt, Jade"https://zbmath.org/authors/?q=ai:hewitt.jade"Shin, Paul"https://zbmath.org/authors/?q=ai:shin.paul"Smith, Collin"https://zbmath.org/authors/?q=ai:smith.collinSummary: Proportional choosability is a list analogue of equitable coloring that was introduced in [\textit{H. Kaul} et al., Discrete Math. 342, No. 8, 2371--2383 (2019; Zbl 1418.05064)]. The smallest \(k\) for which a graph \(G\) is proportionally \(k\)-choosable is the proportional choice number of \(G\), and it is denoted \(\chi_{\mathrm{pc}}(G)\). In the first ever paper on proportional choosability, it was shown that when \(2\leq n\leq m\), \(\max\{n+1,1+\lceil m/2\rceil\}\leq\chi_{\mathrm{pc}}(K_{n,m})\leq n+m-1\). In this note we improve on this result by showing that \(\max\{n+1,\lceil n/2\rceil+\lceil m/2\rceil\}\leq\chi_{\mathrm{pc}}(K_{n,m})\leq n+m-1-\lfloor m/3\rfloor\). In the process, we prove some new lower bounds on the proportional choice number of complete multipartite graphs. We also present several interesting open questions.On \(\ell\)-distance-balanced product graphs.https://zbmath.org/1459.050632021-05-28T16:06:00+00:00"Jerebic, Janja"https://zbmath.org/authors/?q=ai:jerebic.janja"Klavžar, Sandi"https://zbmath.org/authors/?q=ai:klavzar.sandi"Rus, Gregor"https://zbmath.org/authors/?q=ai:rus.gregorSummary: A graph \(G\) is \(\ell\)-distance-balanced if for each pair of vertices \(x\) and \(y\) at a distance \(\ell\) in \(G\), the number of vertices closer to \(x\) than to \(y\) is equal to the number of vertices closer to \(y\) than to \(x\). A complete characterization of \(\ell\)-distance-balanced corona products is given and characterization of lexicographic products for \(\ell\geq 3\), thus complementing known results for \(\ell\in\{1,2\}\) and correcting an earlier related assertion. A sufficient condition on \(H\) which guarantees that \(K_n\square H\) is \(\ell\)-distance-balanced is given, and it is proved that if \(K_n\square H\) is \(\ell\)-distance-balanced, then \(H\) is an \(\ell\)-distance-balanced graph. A known characterization of 1-distance-balanced graphs is extended to \(\ell\)-distance-balanced graphs, again correcting an earlier claimed assertion.Strict neighbor-distinguishing index of subcubic graphs.https://zbmath.org/1459.050782021-05-28T16:06:00+00:00"Gu, Jing"https://zbmath.org/authors/?q=ai:gu.jing"Wang, Weifan"https://zbmath.org/authors/?q=ai:wang.wei-fan"Wang, Yiqiao"https://zbmath.org/authors/?q=ai:wang.yiqiao"Wang, Ying"https://zbmath.org/authors/?q=ai:wang.ying.2|wang.ying.3|wang.ying.8|wang.ying.5|wang.ying.6|wang.ying.4|wang.ying|wang.ying.1Summary: A proper edge coloring of a graph \(G\) is strict neighbor-distinguishing if for any two adjacent vertices \(u\) and \(v\), the set of colors used on the edges incident to \(u\) and the set of colors used on the edges incident to \(v\) are not included with each other. The strict neighbor-distinguishing index of \(G\) is the minimum number \(\chi^\prime_{\text{snd}}(G)\) of colors in a strict neighbor-distinguishing edge coloring of \(G\). In this paper, we prove that every connected subcubic graph \(G\) with \(\delta (G)\geq 2\) has \(\chi^\prime_{\text{snd}}(G)\leq 7\), and moreover \(\chi^\prime_{\text{snd}}(G)=7\) if and only if \(G\) is a graph obtained from the graph \(K_{2,3}\) by inserting a 2-vertex into one edge.Ramsey and Gallai-Ramsey numbers for two classes of unicyclic graphs.https://zbmath.org/1459.052052021-05-28T16:06:00+00:00"Wang, Zhao"https://zbmath.org/authors/?q=ai:wang.zhao"Mao, Yaping"https://zbmath.org/authors/?q=ai:mao.yaping"Magnant, Colton"https://zbmath.org/authors/?q=ai:magnant.colton"Zou, Jinyu"https://zbmath.org/authors/?q=ai:zou.jinyuSummary: Given a graph \(G\) and a positive integer \(k\), define the Gallai-Ramsey number to be the minimum number of vertices \(n\) such that any \(k\)-edge coloring of \(K_n\) contains either a rainbow (all different colored) triangle or a monochromatic copy of \(G\). In this paper, we consider two classes of unicyclic graphs, the star with an extra edge and the path with a triangle at one end. We provide the 2-color Ramsey numbers for these two classes of graphs and use these to obtain general upper and lower bounds on the Gallai-Ramsey numbers.Upper bounds on the \(k\)-tuple (Roman) domination number of a graph.https://zbmath.org/1459.052392021-05-28T16:06:00+00:00"Henning, Michael A."https://zbmath.org/authors/?q=ai:henning.michael-anthony"Rad, Nader Jafari"https://zbmath.org/authors/?q=ai:jafari-rad.naderSummary: \textit{D. Rautenbach} and \textit{L. Volkmann} [Appl. Math. Lett. 20, No. 1, 98--102 (2007; Zbl 1137.05054)], gave an upper bound for the \(k\)-tuple domination number of a graph. \textit{N. J. Rad} [J. Comb. Math. Comb. Comput. 111, 177--184 (2019; Zbl 1452.05129)] presented an improvement of the above bound using the Caro-Wei Theorem. In this paper, using the well-known Brooks' Theorem for vertex coloring and vertex covers, we improve the above bounds on the \(k\)-tuple domination number under some certain conditions. In the special case \(k=1\), we improve the upper bounds for the domination number [\textit{V. I. Arnautov}, Prikl. Mat. Programm. 11, 3--8 (1974; Zbl 0297.05131); \textit{C. Payan}, Cah. Cent. Étud. Rech. Opér. 17, 307--317 (1975; Zbl 0341.05126)] and the Roman domination number [\textit{E. J. Cockayne} et al., Discrete Math. 278, No. 1--3, 11--22 (2004; Zbl 1036.05034)]. We also improve bounds given by \textit{A. Hansberg} and \textit{L. Volkmann} [Discrete Appl. Math. 157, No. 7, 1634--1639 (2009; Zbl 1179.05081)] for Roman \(k\)-domination number, and \textit{N. J. Rad} and \textit{H. Rahbani} [Discuss. Math., Graph Theory 39, No. 1, 41--53 (2019; Zbl 1401.05224)] for double Roman domination number.Questions on color-critical subgraphs.https://zbmath.org/1459.050892021-05-28T16:06:00+00:00"Newman, Nicholas"https://zbmath.org/authors/?q=ai:newman.nicholas"Noble, Matthew"https://zbmath.org/authors/?q=ai:noble.matthewSummary: In our work, we define a \(k\)-tuple of positive integers \((x_1,\dots,x_k)\) to be a \(\chi\)-sequence if there exists a \(k\)-chromatic graph \(G\) such that for each \(i\in\{1,\dots,k\}\), the order of a minimum \(i\)-chromatic subgraph of \(G\) is equal to \(x_i\). Denote by \(\mathcal{X}_k\) the set of all \(\chi\)-sequences of length \(k\). A very difficult question is to determine, for a given \((x_1,\dots,x_k)\in\mathcal{X}_k\), the set of all integers \(y\) such that \((x_1,\dots,x_k,y)\in\mathcal{X}_{k+1}\). We propose a few variants of this question and elaborate upon a number of partial results along the way.An injective version of the 1-2-3 conjecture.https://zbmath.org/1459.052832021-05-28T16:06:00+00:00"Bensmail, Julien"https://zbmath.org/authors/?q=ai:bensmail.julien"Li, Bi"https://zbmath.org/authors/?q=ai:li.bi"Li, Binlong"https://zbmath.org/authors/?q=ai:li.binlongSummary: In this work, we introduce and study a new graph labelling problem standing as a combination of the 1-2-3 Conjecture and injective colouring of graphs, which also finds connections with the notion of graph irregularity. In this problem, the goal, given a graph \(G\), is to label the edges of \(G\) so that every two vertices having a common neighbour get incident to different sums of labels. We are interested in the minimum \(k\) such that \(G\) admits such a \(k\)-labelling. We suspect that almost all graphs \(G\) can be labelled this way using labels \(1,\dots,\Delta(G)\). Towards this speculation, we provide bounds on the maximum label value needed in general. In particular, we prove that using labels \(1,\dots,\Delta(G)\) is indeed sufficient when \(G\) is a tree, a particular cactus, or when its injective chromatic number \(\chi_{\text{i}}(G)\) is equal to \(\Delta(G)\).Graphs with diameter 2 and large total domination number.https://zbmath.org/1459.052362021-05-28T16:06:00+00:00"Dubickas, Artūras"https://zbmath.org/authors/?q=ai:dubickas.arturasSummary: In this paper we show that for each sufficiently large \(n\) there exist graphs \(G\) of order \(n\) and diameter 2 whose total domination number \(\gamma_t(G)\) is greater than \(\sqrt{(3n\log n)/8}-\sqrt{n}\). On the other hand, it is shown that the total domination number of a graph of order \(n\geqslant 3\) and diameter 2 is always less than \(\sqrt{(n\log n)/2}+\sqrt{n/2}\).Decompositions of 6-regular bipartite graphs into paths of length six.https://zbmath.org/1459.052542021-05-28T16:06:00+00:00"Chu, Yanan"https://zbmath.org/authors/?q=ai:chu.yanan"Fan, Genghua"https://zbmath.org/authors/?q=ai:fan.genghua"Zhou, Chuixiang"https://zbmath.org/authors/?q=ai:zhou.chuixiangSummary: Let \(T\) be a tree with \(m\) edges. It was conjectured that every \(m\)-regular bipartite graph can be decomposed into edge-disjoint copies of \(T\). In this paper, we prove that every 6-regular bipartite graph can be decomposed into edge-disjoint paths with 6 edges. As a consequence, every 6-regular bipartite graph on \(n\) vertices can be decomposed into \(\frac{n}{2}\) paths, which is related to the well-known Gallai's Conjecture: every connected graph on \(n\) vertices can be decomposed into at most \(\frac{n+1}{2}\) paths.Counting locally flat-foldable origami configurations via 3-coloring graphs.https://zbmath.org/1459.050752021-05-28T16:06:00+00:00"Chiu, Alvin"https://zbmath.org/authors/?q=ai:chiu.alvin"Hoganson, William"https://zbmath.org/authors/?q=ai:hoganson.william"Hull, Thomas C."https://zbmath.org/authors/?q=ai:hull.thomas-c"Wu, Sylvia"https://zbmath.org/authors/?q=ai:wu.sylviaSummary: Origami, where two-dimensional sheets are folded into complex structures, is rich with combinatorial and geometric structure, most of which remains to be fully understood. In this paper we consider flat origami, where the sheet of material is folded into a two-dimensional object, and consider the mountain (convex) and valley (concave) creases that result, called a MV assignment of the crease pattern. An open problem is to count the number locally valid MV assignments \(\mu\) of a given flat-foldable crease pattern \(C\), where locally valid means that each vertex will fold flat under \(\mu\) with no self-intersections of the folded material. In this paper we solve this problem for a large family of crease patterns by creating a planar graph \(C^\ast\) whose 3-colorings are in one-to-one correspondence with the locally valid MV assignments of \(C\). This reduces the problem of enumerating locally valid MV assignments to the enumeration of 3-colorings of graphs.Stability theorems for graph vulnerability parameters.https://zbmath.org/1459.053222021-05-28T16:06:00+00:00"Yatauro, Michael"https://zbmath.org/authors/?q=ai:yatauro.michaelSummary: Given a graph property \(P\), \textit{J. A. Bondy} and \textit{V. Chvatal} [Discrete Math. 15, 111--135 (1976; Zbl 0331.05138)] defined \(P\) to be \(k\)-stable if for any nonadjacent \(u,v\in V(G)\), whenever \(G+uv\) has the property \(P\) and \(d(u)+d(v)\ge k\), then \(G\) itself has the property \(P\). The smallest such \(k\) is called the stability of \(P\). We consider the graph parameters integrity, toughness, tenacity, and binding number. For each of these parameters, we provide the stability for the property that \(G\) has a value for that parameter which is at least \(c\), for some fixed \(c\). We also demonstrate how stability can lead to degree sequence theorems and compare these results to known degree sequence theorems that are best possible in a certain sense.Hamiltonicity in prime sum graphs.https://zbmath.org/1459.051492021-05-28T16:06:00+00:00"Chen, Hong-Bin"https://zbmath.org/authors/?q=ai:chen.hongbin"Fu, Hung-Lin"https://zbmath.org/authors/?q=ai:fu.hunglin"Guo, Jun-Yi"https://zbmath.org/authors/?q=ai:guo.junyiSummary: For any positive integer \(n\), we define the prime sum graph \(G_n=(V,E)\) of order \(n\) with the vertex set \(V=\{1,2,\dots,n\}\) and \(E=\{ij: i+j\text{ is prime}\}\). Filz in 1982 posed a conjecture that \(G_{2n}\) is Hamiltonian for any \(n\geq 2\), i.e., the set of integers \(\{1,2,\dots,2n\}\) can be represented as a cyclic rearrangement so that the sum of any two adjacent integers is a prime number. With a fundamental result in graph theory and a recent breakthrough on the twin prime conjecture, we prove that Filz's conjecture is true for infinitely many cases.Optimal online edge coloring of planar graphs with advice.https://zbmath.org/1459.682332021-05-28T16:06:00+00:00"Mikkelsen, Jesper W."https://zbmath.org/authors/?q=ai:mikkelsen.jesper-wSummary: Using the framework of advice complexity, we study the amount of knowledge about the future that an online algorithm needs to color the edges of a graph optimally, i.e., using as few colors as possible. For graphs of maximum degree \(\Delta \), it follows from Vizing's Theorem that \(O(m\log \Delta)\) bits of advice suffice to achieve optimality, where \(m\) is the number of edges. We show that for graphs of bounded degeneracy (a class of graphs including e.g. trees and planar graphs), only \(O(m)\) bits of advice are needed to compute an optimal solution online, independently of how large \(\Delta \) is. On the other hand, we show that \(\Omega (m)\) bits of advice are necessary just to achieve a competitive ratio better than that of the best deterministic online algorithm without advice. Furthermore, we consider algorithms which use a fixed number of advice bits per edge (our algorithm for graphs of bounded degeneracy belongs to this class of algorithms). We show that for bipartite graphs,
any such algorithm must use at least \(\Omega (m\log \Delta)\) bits of advice to achieve optimality.
For the entire collection see [Zbl 1316.68024].Rainbow antistrong connection in tournaments.https://zbmath.org/1459.050792021-05-28T16:06:00+00:00"Hu, Yumei"https://zbmath.org/authors/?q=ai:hu.yumei"Wei, Yarong"https://zbmath.org/authors/?q=ai:wei.yarongSummary: An arc-coloured digraph is rainbow antistrong connected if any two distinct vertices \(u, v\) are connected by both a forward antidirected \((u, v)\)-trail and a forward antidirected \((v, u)\)-trail which do not use two arcs with the same colour. The rainbow antistrong connection number of a digraph \(D\) is the minimum number of colours needed to make the digraph rainbow antistrong connected, denoted by \(\overset{\rightarrow }{\operatorname{rac}}(D)\). An arc-coloured digraph is strong rainbow antistrong connected if any two distinct vertices \(u, v\) are connected by both a forward antidirected \((u, v)\)-geodesic trail and a forward antidirected \((v, u)\)-geodesic trail which do not use two arcs with the same colour. The
strong rainbow antistrong connection number of a digraph \(D\), denoted by \(\overset{\rightarrow }{\operatorname{srac}}(D)\), is the minimum number of colours needed to make the digraph strong rainbow antistrong connected. In this paper, we prove that for any antistrong tournament \(T_n\) with \(n\) vertices \(\overset{\rightarrow }{\operatorname{rac}}(T_n)\ge 3\) and \(\overset{\rightarrow }{\operatorname{srac}}(T_n)\ge 3\), and we construct tournaments \(T_n\) with \(\overset{\rightarrow }{\operatorname{rac}}(T_n)=\overset{\rightarrow }{\operatorname{srac}}(T_n)=3\) for every \(n\ge 18\). Then, we prove that for any antistrong tournament \(T_n\) whose diameter is at least \(4, \overset{\rightarrow }{\operatorname{rac}}(T_n)\le 7\), and we construct tournaments \(T_n\) whose diameter is 3 with \(\overset{\rightarrow }{\operatorname{rac}}(T_n)=7\) for every \(n\ge 5\).On well-dominated graphs.https://zbmath.org/1459.052282021-05-28T16:06:00+00:00"Anderson, Sarah E."https://zbmath.org/authors/?q=ai:anderson.sarah-e"Kuenzel, Kirsti"https://zbmath.org/authors/?q=ai:kuenzel.kirsti"Rall, Douglas F."https://zbmath.org/authors/?q=ai:rall.douglas-fSummary: A graph is well-dominated if all of its minimal dominating sets have the same cardinality. It is proved that there are exactly eleven connected, well-dominated, triangle-free graphs whose domination number is at most 3. We prove that at least one of the factors is well-dominated if the Cartesian product of two graphs is well-dominated. In addition, we show that the Cartesian product of two connected, triangle-free graphs is well-dominated if and only if both graphs are complete graphs of order 2. Under the assumption that at least one of the connected graphs \(G\) or \(H\) has no isolatable vertices, we prove that the direct product of \(G\) and \(H\) is well-dominated if and only if either \(G=H=K_3\) or \(G=K_2\) and \(H\) is either the 4-cycle or the corona of a connected graph. Furthermore, we show that the disjunctive product of two connected graphs is well-dominated if and only if one of the factors is a complete graph and the other factor has domination number at most 2.End-vertices of graph search algorithms.https://zbmath.org/1459.681602021-05-28T16:06:00+00:00"Kratsch, Dieter"https://zbmath.org/authors/?q=ai:kratsch.dieter"Liedloff, Mathieu"https://zbmath.org/authors/?q=ai:liedloff.mathieu"Meister, Daniel"https://zbmath.org/authors/?q=ai:meister.danielSummary: Is it possible to force a graph search algorithm to visit a selected vertex as last? \textit{D. G. Corneil} et al. [Discrete Appl. Math. 158, No. 5, 434--443 (2010; Zbl 1225.05229)] showed that this end-vertex decision problem is NP-complete for Lexicographic Breadth-First Search (LexBFS). \textit{P. Charbit} et al. [Discrete Math. Theor. Comput. Sci. 16, No. 2, 57--72 (2014; Zbl 1301.05329)] extended the intractability result, and showed that the end-vertex problem is hard also for BFS, DFS, and LexDFS. We ask for positive results, and study algorithmic and combinatorial questions. We show that the end-vertex problem for BFS and DFS can be solved in \(\mathcal{O}^*(2^n)\) time, hereby improving upon the straightforward and currently best known running-time bound of \(\mathcal{O}^*(n!)\). We also determine conditions that preserve end-vertices in subgraphs when extending to larger graphs. Such results are of interest in algorithm design, when applying techniques such as dynamic programming and divide-and-conquer.
For the entire collection see [Zbl 1316.68024].Algorithms solving the matching cut problem.https://zbmath.org/1459.681592021-05-28T16:06:00+00:00"Kratsch, Dieter"https://zbmath.org/authors/?q=ai:kratsch.dieter"Le, Van Bang"https://zbmath.org/authors/?q=ai:le-van-bang.Summary: In a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be NP-complete. This paper provides a first branching algorithm solving Matching Cut in time \(O^*(2^{n/2})=O^*(1.4143^n)\) for an \(n\)-vertex input graph, and shows that Matching Cut parameterized by vertex cover number \(\tau (G)\) can be solved by a single-exponential algorithm in time \(2^{\tau (G)} O(n^2)\). Moreover, the paper also gives a polynomially solvable case for Matching Cut which covers previous known results on graphs of maximum degree three, line graphs, and claw-free graphs.
For the entire collection see [Zbl 1316.68024].Unit ball graphs on geodesic spaces.https://zbmath.org/1459.052162021-05-28T16:06:00+00:00"Kuroda, Masamichi"https://zbmath.org/authors/?q=ai:kuroda.masamichi"Tsujie, Shuhei"https://zbmath.org/authors/?q=ai:tsujie.shuheiSummary: Consider finitely many points in a geodesic space. If the distance of two points is less than a fixed threshold, then we regard these two points as ``near''. Connecting near points with edges, we obtain a simple graph on the points, which is called a unit ball graph. If the space is the real line, then it is known as a unit interval graph. Unit ball graphs on a geodesic space describe geometric characteristics of the space in terms of graphs. In this article, we show that every unit ball graph on a geodesic space is (strongly) chordal if and only if the space is an \(\mathbb{R}\)-tree and that every unit ball graph on a geodesic space is (claw, net)-free if and only if the space is a connected manifold of dimension at most 1. As a corollary, we prove that the collection of unit ball graphs essentially characterizes the real line and the unit circle.Spiders and their kin: an investigation of Stanley's chromatic symmetric function for spiders and related graphs.https://zbmath.org/1459.053372021-05-28T16:06:00+00:00"Foley, Angèle M."https://zbmath.org/authors/?q=ai:foley.angele-m"Kazdan, Joshua"https://zbmath.org/authors/?q=ai:kazdan.joshua"Kröll, Larissa"https://zbmath.org/authors/?q=ai:kroll.larissa"Martínez Alberga, Sofía"https://zbmath.org/authors/?q=ai:martinez-alberga.sofia"Melnyk, Oleksii"https://zbmath.org/authors/?q=ai:melnyk.oleksii"Tenenbaum, Alexander"https://zbmath.org/authors/?q=ai:tenenbaum.alexanderSummary: We study the chromatic symmetric functions of graph classes related to spiders, namely generalized spider graphs (line graphs of spiders), and what we call horseshoe crab graphs. We show that no two generalized spiders have the same chromatic symmetric function, thereby extending the work of \textit{J. L. Martin} et al. [J. Comb. Theory, Ser. A 115, No. 2, 237--253 (2008; Zbl 1133.05020)]. Additionally, we establish that a subclass of generalized spiders, which we call generalized nets, has no \(e\)-positive members, providing a more general counterexample to the necessity of the claw-free condition. We use yet another class of generalized spiders to construct a counterexample to a problem involving the \(e\)-positivity of claw-free, \(P_4\)-sparse graphs, showing that \textit{S. Tsujie}'s result on the \(e\)-positivity of claw-free [Graphs Comb. 34, No. 5, 1037--1048 (2018; Zbl 1402.05081)], \(P_4\)-free graphs cannot be extended to graphs in this set. Finally, we investigate the \(e\)-positivity of another type of graphs, the horseshoe crab graphs (a class of unit interval graphs), and prove the positivity of all but one of the coefficients. This has close connections to the work of \textit{D. D. Gebhard} and \textit{B. E. Sagan} [J. Algebr. Comb. 13, No. 3, 227--255 (2001; Zbl 0979.05105)] and \textit{S. Cho} and \textit{J. Huh} [SIAM J. Discrete Math. 33, No. 4, 2286--2315 (2019; Zbl 1428.05310)].Strongly spanning trailable graphs with small circumference and Hamilton-connected claw-free graphs.https://zbmath.org/1459.051512021-05-28T16:06:00+00:00"Liu, Xia"https://zbmath.org/authors/?q=ai:liu.xia"Xiong, Liming"https://zbmath.org/authors/?q=ai:xiong.liming"Lai, Hong-Jian"https://zbmath.org/authors/?q=ai:lai.hong-jianSummary: A graph \(G\) is strongly spanning trailable if for any \(e_1=u_1v_1, e_2=u_2v_2\in E(G)\) (possibly \(e_1=e_2), G(e_1, e_2)\), which is obtained from \(G\) by replacing \(e_1\) by a path \(u_1v_{e_1}v_1\) and by replacing \(e_2\) by a path \(u_2v_{e_2}v_2\), has a spanning \((v_{e_1}, v_{e_2})\)-trail. A graph \(G\) is Hamilton-connected if there is a spanning path between any two vertices of \(V(G)\). In this paper, we first show that every 2-connected 3-edge-connected graph with circumference at most 8 is strongly spanning trailable with an exception of order 8. As applications, we prove that every 3-connected \(\{K_{1, 3},N_{1,2,4}\}\)-free graph is Hamilton-connected and every 3-connected \(\{K_{1,3},P_{10}\}\)-free graph is Hamilton-connected with a well-defined exception. The last two results extend the results in [\textit{Z. Hu} and \textit{S. Zhang}, ibid. 32, No. 2, 685--705 (2016; Zbl 1338.05134)] and [\textit{Q. Bian} et al., ibid. 30, No. 5, 1099--1122 (2014; Zbl 1298.05189)].Supereulerian graphs with constraints on the matching number and minimum degree.https://zbmath.org/1459.051472021-05-28T16:06:00+00:00"Algefari, Mansour J."https://zbmath.org/authors/?q=ai:algefari.mansour-j"Lai, Hong-Jian"https://zbmath.org/authors/?q=ai:lai.hong-jianSummary: A graph is supereulerian if it has a spanning eulerian subgraph. We show that a connected simple graph \(G\) with \(n=|V(G)|\geq 2\) and \(\delta (G)\geq\alpha^\prime(G)\) is supereulerian if and only if \(G\neq K_{1,n-1}\) if \(n\) is even or \(G\neq K_{2,n-2}\) if \(n\) is odd. Consequently, every connected simple graph \(G\) with \(\delta (G)\geq\alpha'(G)\) has a hamiltonian line graph.The three-point function of planar quadrangulations.https://zbmath.org/1459.821022021-05-28T16:06:00+00:00"Bouttier, J."https://zbmath.org/authors/?q=ai:bouttier.jeremie"Guitter, E."https://zbmath.org/authors/?q=ai:guitter.emmanuelEfficiently testing \(T\)-interval connectivity in dynamic graphs.https://zbmath.org/1459.681532021-05-28T16:06:00+00:00"Casteigts, Arnaud"https://zbmath.org/authors/?q=ai:casteigts.arnaud"Klasing, Ralf"https://zbmath.org/authors/?q=ai:klasing.ralf"Neggaz, Yessin M."https://zbmath.org/authors/?q=ai:neggaz.yessin-m"Peters, Joseph G."https://zbmath.org/authors/?q=ai:peters.joseph-gSummary: Many types of dynamic networks are made up of durable entities whose links evolve over time. When considered from a \textit{global} and \textit{discrete} standpoint, these networks are often modelled as evolving graphs, i.e. a sequence of static graphs \(\mathcal{{G}}=\{G_1,G_2,\dots,G_{\delta}\}\) such that \(G_i=(V,E_i)\) represents the network topology at time step \(i\). Such a sequence is said to be \(T\)-interval connected if for any \(t\in [1, \delta -T+1]\) all graphs in \(\{G_t,G_{t+1},\dots,G_{t+T-1}\}\) share a common connected spanning subgraph. In this paper, we consider the problem of deciding whether a given sequence \(\mathcal{{G}}\) is \(T\)-interval connected for a given \(T\). We also consider the related problem of finding the largest \(T\) for which a given \(\mathcal{{G}}\) is \(T\)-interval connected. We assume that the changes between two consecutive graphs are arbitrary, and that two operations, \textit{binary intersection} and \textit{connectivity testing}, are available to solve the
problems. We show that \(\varOmega (\delta)\) such operations are required to solve both problems, and we present optimal \(O(\delta)\) online algorithms for both problems.
For the entire collection see [Zbl 1316.68024].Imperfect bifurcations in opinion dynamics under external fields.https://zbmath.org/1459.911342021-05-28T16:06:00+00:00"Freitas, Francisco"https://zbmath.org/authors/?q=ai:freitas.francisco"Vieira, Allan R."https://zbmath.org/authors/?q=ai:vieira.allan-r"Anteneodo, Celia"https://zbmath.org/authors/?q=ai:anteneodo.celiaA refined complexity analysis of finding the most vital edges for undirected shortest paths.https://zbmath.org/1459.681522021-05-28T16:06:00+00:00"Bazgan, Cristina"https://zbmath.org/authors/?q=ai:bazgan.cristina"Nichterlein, André"https://zbmath.org/authors/?q=ai:nichterlein.andre"Niedermeier, Rolf"https://zbmath.org/authors/?q=ai:niedermeier.rolfSummary: We study the NP-hard Shortest Path Most Vital Edges problem arising in the context of analyzing network robustness. For an undirected graph with positive integer edge lengths and two designated vertices \(s\) and \(t\), the goal is to delete as few edges as possible in order to increase the length of the (new) shortest \(st\)-path as much as possible. This scenario has been mostly studied from the viewpoint of approximation algorithms and heuristics, while we particularly introduce a parameterized and multivariate point of view. We derive refined tractability as well as hardness results, and identify numerous directions for future research. Among other things, we show that increasing the shortest path length by at least one is much easier than to increase it by at least two.
For the entire collection see [Zbl 1316.68024].Locating-total domination number of cacti graphs.https://zbmath.org/1459.052492021-05-28T16:06:00+00:00"Wei, Jianxin"https://zbmath.org/authors/?q=ai:wei.jianxin"Ahmad, Uzma"https://zbmath.org/authors/?q=ai:ahmad.uzma"Hameed, Saira"https://zbmath.org/authors/?q=ai:hameed.saira"Hanif, Javaria"https://zbmath.org/authors/?q=ai:hanif.javariaSummary: For a connected graph \(J\), a subset \(W\subseteq V\left( J\right)\) is termed as a locating-total dominating set if for \(a\in V\left( J\right),N\left( a\right)\cap W\neq\phi \), and for \(a,b\in V\left( J\right)-W,N\left( a\right)\cap W\neq N\left( b\right)\cap W \). The number of elements in a smallest such subset is termed as the locating-total domination number of \(J\). In this paper, the locating-total domination number of unicyclic graphs and bicyclic graphs are studied and their bounds are presented. Then, by using these bounds, an upper bound for cacti graphs in terms of their order and number of cycles is estimated. Moreover, the exact values of this domination variant for some families of cacti graphs including tadpole graphs and rooted products are also determined.Posets with series parallel orders and strict-double-bound graphs.https://zbmath.org/1459.052192021-05-28T16:06:00+00:00"Tashiro, Shin-ichiro"https://zbmath.org/authors/?q=ai:tashiro.shin-ichiro"Ogawa, Kenjiro"https://zbmath.org/authors/?q=ai:ogawa.kenjiro"Tsuchiya, Morimasa"https://zbmath.org/authors/?q=ai:tsuchiya.morimasaSummary: For a poset \(P= (X,\le_P)\), the strict-double-bound graph of \(P\) is the graph sDB\((P)\) on \(V(\text{sDB}(P))= X\) for which vertices \(u\) and \(v\) of sDB\((P)\) are adjacent if and only if \(u\ne v\) and there exist elements \(x\in X\) distinct from \(u\) and \(v\) such that \(x \le_P u\le_P y\) and \(x\le_P v\le_P y\). A poset \(P\) is a series parallel order if and only if it contains no induced subposet isomorphic to \(N\)-poset.
In this paper, we deal with strict-double-bound graphs of series parallel orders. In particular, we show that if \(P_3\) is contained as an induced subgraph in a strict-double-bound graph of a series parallel order, it is contained in either of \(C_4\), 3-pan, \(K_{1,3}\) or \(K_4-e\). As a consequence of this result, we can claim that a strict-double graph of a series parallel order is \(P_4\)-free. Furthermore, we study sufficient conditions for a strict-double-bound graph of a series parallel order to be an interval graph, difference graph or Meyniel graph.On the Steiner quadruple system with ten points.https://zbmath.org/1459.050232021-05-28T16:06:00+00:00"Brier, Robert"https://zbmath.org/authors/?q=ai:brier.robert"Bryant, Darryn"https://zbmath.org/authors/?q=ai:bryant.darryn-eSummary: The Steiner quadruple system on ten points, SQS(10), is constructed with points corresponding to the ten triangle factors of the complete graph on six vertices. This construction shows that the Tutte-Coxeter graph is obtained from the SQS(10) by taking the blocks as vertices, and joining two blocks by an edge when they are disjoint.Honeycomb toroidal graphs.https://zbmath.org/1459.051162021-05-28T16:06:00+00:00"Alspach, Brian"https://zbmath.org/authors/?q=ai:alspach.brian-rSummary: Honeycomb toroidal graphs are bipartite trivalent Cayley graphs on generalized dihedral groups. We examine the two historical threads leading to these graphs, some of the properties that have been established, and some open problems.Double jump peg solitaire on graphs.https://zbmath.org/1459.052072021-05-28T16:06:00+00:00"Beeler, Robert A."https://zbmath.org/authors/?q=ai:beeler.robert-a"Gray, Aaron D."https://zbmath.org/authors/?q=ai:gray.aaron-dSummary: Peg solitaire is a game in which pegs are placed in every hole but one and the player jumps over pegs along rows or columns to remove them. Usually, the goal is to have a single peg remaining. In a paper by \textit{R. A. Beeler} and \textit{D. P. Hoilman} [Discrete Math. 311, No. 20, 2198--2202 (2011; Zbl 1230.05211)], this game is generalized to graphs. In this paper, we consider a variation in which each peg must be jumped twice in order to be removed. For this variation, we consider the solvability of several graph families. For our major results, we characterize solvable joins of graphs and show that the Cartesian product of solvable graphs is likewise solvable.Two results on \(k\)-\((2,1)\)-total choosability of planar graphs.https://zbmath.org/1459.052912021-05-28T16:06:00+00:00"Song, Yan"https://zbmath.org/authors/?q=ai:song.yan"Sun, Lei"https://zbmath.org/authors/?q=ai:sun.leiTwo Moore's theorems for graphs.https://zbmath.org/1459.051232021-05-28T16:06:00+00:00"Mednykh, Alexander"https://zbmath.org/authors/?q=ai:mednykh.alexander-d"Mednykh, Ilya"https://zbmath.org/authors/?q=ai:mednykh.ilya-aSummary: Let \(X\) be a finite connected graph, possibly with loops and multiple edges. An automorphism group of \(X\) acts purely harmonically if it acts freely on the set of directed edges of \(X\) and has no invertible edges. Define a genus \(g\) of the graph \(X\) to be the rank of the first homology group. A finite group acting purely harmonically on a graph of genus \(g\) is a natural discrete analogue of a finite group of automorphisms acting on a Riemann surface of genus \(g\). In the present paper, we investigate cyclic group \(\mathbb{Z}_n\) acting purely harmonically on a graph \(X\) of genus \(g\) with fixed points. Given subgroup \(\mathbb{Z}_d< \mathbb{Z}_n\), we find the signature of orbifold \(X=\mathbb{Z}_d\) through the signature of orbifold \(X= \mathbb{Z}_n\). As a result, we obtain formulas for the number of fixed points for generators of group \(\mathbb{Z}_d\) and for genus of orbifold \(X= \mathbb{Z}_d\). For Riemann surfaces, similar results were obtained earlier by \textit{M. J. Moore} [Can. J. Math. 22, 922--932 (1970; Zbl 0218.30021)].Strongly regular graphs with nonprincipal eigenvalue 4 and its extensions.https://zbmath.org/1459.050652021-05-28T16:06:00+00:00"Makhnev, A. A."https://zbmath.org/authors/?q=ai:makhnev.aleksandr-a|makhnev.a-a-junSummary: J. Koolen suggested the problem of investigation distance-regular graphs with strongly regular local subgraphs having the second eigenvalue \(\leq t\) for some natural number \(t\). Earlier the Koolen problem for \(t=3\) was solved. In this paper it is developed the programm of investigation distance-regular graphs with strongly regular local subgraphs having the second eigenvalue \(r\), \(3<r\leq4\).Integer flows and modulo orientations of signed graphs.https://zbmath.org/1459.051072021-05-28T16:06:00+00:00"Han, Miaomiao"https://zbmath.org/authors/?q=ai:han.miaomiao"Li, Jiaao"https://zbmath.org/authors/?q=ai:li.jiaao"Luo, Rong"https://zbmath.org/authors/?q=ai:luo.rong"Shi, Yongtang"https://zbmath.org/authors/?q=ai:shi.yongtang"Zhang, Cun-Quan"https://zbmath.org/authors/?q=ai:zhang.cunquanStrong chordality of graphs with possible loops.https://zbmath.org/1459.052382021-05-28T16:06:00+00:00"Hell, Pavol"https://zbmath.org/authors/?q=ai:hell.pavol"Hernández-Cruz, César"https://zbmath.org/authors/?q=ai:hernandez-cruz.cesar"Huang, Jing"https://zbmath.org/authors/?q=ai:huang.jing"Lin, Jephian C.-H."https://zbmath.org/authors/?q=ai:lin.jephian-chin-hungThe size Ramsey number of graphs with bounded treewidth.https://zbmath.org/1459.052012021-05-28T16:06:00+00:00"Kamcev, Nina"https://zbmath.org/authors/?q=ai:kamcev.nina"Liebenau, Anita"https://zbmath.org/authors/?q=ai:liebenau.anita"Wood, David R."https://zbmath.org/authors/?q=ai:wood.david-ronald"Yepremyan, Liana"https://zbmath.org/authors/?q=ai:yepremyan.lianaLarge induced matchings in random graphs.https://zbmath.org/1459.052972021-05-28T16:06:00+00:00"Cooley, Oliver"https://zbmath.org/authors/?q=ai:cooley.oliver"Draganić, Nemanja"https://zbmath.org/authors/?q=ai:draganic.nemanja"Kang, Mihyun"https://zbmath.org/authors/?q=ai:kang.mihyun"Sudakov, Benny"https://zbmath.org/authors/?q=ai:sudakov.bennyTransitive tournament tilings in oriented graphs with large minimum total degree.https://zbmath.org/1459.051012021-05-28T16:06:00+00:00"DeBiasio, Louis"https://zbmath.org/authors/?q=ai:debiasio.louis"Lo, Allan"https://zbmath.org/authors/?q=ai:lo.allan-siu-lun"Molla, Theodore"https://zbmath.org/authors/?q=ai:molla.theodore"Treglown, Andrew"https://zbmath.org/authors/?q=ai:treglown.andrewA new approach to finding the extra connectivity of graphs.https://zbmath.org/1459.051442021-05-28T16:06:00+00:00"Zhu, Qiang"https://zbmath.org/authors/?q=ai:zhu.qiang"Ma, Fang"https://zbmath.org/authors/?q=ai:ma.fang"Guo, Guodong"https://zbmath.org/authors/?q=ai:guo.guodong"Wang, Dajin"https://zbmath.org/authors/?q=ai:wang.dajinSummary: The connectivity of a graph is the minimum number of nodes, whose removal will cause the graph disconnected. It is one of the basic and important properties of the graph, and can be used as a metric of the graph's robustness. Since the connectivity of a graph is upper-bounded by its minimal node-degree, which can be small, the notion of conditional connectivity has been proposed in the past to more realistically reflect a graph's robustness. The extra connectivity is such a conditional connectivity. In this paper, we introduce a new approach to finding the extra connectivity of the hypercube, a well-known regular graph model for networks of parallel computers. We will show that the results on the isoperimetric inequalities for hypercubes can be used to obtain their extra connectivity. That is, using isoperimetric inequalities, for an \(n\)-dimensional hypercube \(Q_n\) with \(n\geq 5\), the \(h\)-extra connectivity is equal to its minimum \((h+1)\)-vertex boundary number for \(1\leq h\leq n-4\) and \(n+1\leq h\leq 2n-4\). For \(n-3\leq h\leq n\), the \(h\)-extra connectivity of hypercube equals its minimum \((n-2)\)-vertex boundary number. The paper's result for the first time establishes a relationship between the isoperimetric inequalities and the extra connectivity of graphs.Every planar graph is 1-defective \((9,2)\)-paintable.https://zbmath.org/1459.052102021-05-28T16:06:00+00:00"Han, Ming"https://zbmath.org/authors/?q=ai:han.ming"Kierstead, H. A."https://zbmath.org/authors/?q=ai:kierstead.henry-a"Zhu, Xuding"https://zbmath.org/authors/?q=ai:zhu.xudingSummary: Assume \(G\) is a graph, and \(k\), \(d\), \(m\) are natural numbers. The \(d\)-defective \((k,m)\)-painting game on \(G\) is played by two players: Lister and Painter. Initially, each vertex has \(k\) tokens and is uncolored. In each round, Lister chooses a set \(M\) of vertices and removes one token from each chosen vertex. Painter colors a subset \(X\) of \(M\) which induces a subgraph \(G[X]\) of maximum degree at most \(d\). A vertex \(v\) is fully colored if \(v\) has received \(m\) colors. Lister wins if at the end of some round, there is a vertex with no more tokens left and is not fully colored. Otherwise, at some round, all vertices are fully colored and Painter wins. We say \(G\) is \(d\)-defective \((k,m)\)-paintable if Painter has a winning strategy in this game. We prove that every planar graph is 1-defective \((9,2)\)-paintable. In addition, our winning strategy also implies that every planar graph is 1-defective \((\lceil\frac{9}{2}m\rceil,m)\)-paintable for any positive \(m\).On star 5-colorings of sparse graphs.https://zbmath.org/1459.050762021-05-28T16:06:00+00:00"Choi, Ilkyoo"https://zbmath.org/authors/?q=ai:choi.ilkyoo"Park, Boram"https://zbmath.org/authors/?q=ai:park.boramSummary: A star \(k\)-coloring of a graph \(G\) is a proper (vertex) \(k\)-coloring of \(G\) such that the vertices on a path of length three receive at least three colors. Given a graph \(G\), its star chromatic number, denoted \(\chi_s(G)\), is the minimum integer \(k\) for which \(G\) admits a star \(k\)-coloring. Studying star coloring of sparse graphs is an active area of research, especially in terms of the maximum average degree of a graph; the maximum average degree, denoted \(\operatorname{mad}(G)\), of a graph \(G\) is \(\max\{\frac{2|E(H)|}{|V(H)|}:H\subset G\}\). It is known that for a graph \(G\), if \(\operatorname{mad}(G) < \frac{8}{3}\), then \(\chi_s(G)\leq 6\) [\textit{A. Kündgen} and \textit{C. Timmons}, J. Graph Theory 63, No. 4, 324--337 (2010; Zbl 1209.05090)], and if \(\operatorname{mad}(G) < \frac{18}{7}\) and its girth is at least 6, then \(\chi_s(G)\leq 5\) [\textit{Y. Bu} et al., J. Graph Theory 62, No. 3, 201--219 (2009; Zbl 1179.05042)]. We improve both results by showing that for a graph \(G\), if \(\operatorname{mad}(G)\leq\frac{8}{3}\), then \(\chi_s (G)\leq 5\). As an immediate corollary, we obtain that a planar graph with girth at least 8 has a star 5-coloring, improving the best known girth condition for a planar graph to have a star 5-coloring [Kündgen and Timmons, loc. cit.; \textit{C. Timmons}, Electron. J. Comb. 15, No. 1, Research Paper R124, 17 p. (2008; Zbl 1165.05326)].Distances in graphs of girth 6 and generalised cages.https://zbmath.org/1459.050572021-05-28T16:06:00+00:00"Alochukwu, Alex"https://zbmath.org/authors/?q=ai:alochukwu.alex"Dankelmann, Peter"https://zbmath.org/authors/?q=ai:dankelmann.peterSummary: In this paper we present bounds on the radius and diameter of graphs of girth at least 6 and for \((C_4,C_5)\)-free graphs, i.e., graphs not containing cycles of length 4 or 5. We show that the diameter of a graph \(G\) of girth at least 6 is at most \(\frac{3n}{\delta^2-\delta +1}-1\), and the radius is at most \(\frac{3n}{2(\delta^2-\delta +1)}+10\), where \(n\) is the order and \(\delta\) the minimum degree of \(G\). If \(\delta -1\) is a prime power, then both bounds are sharp apart from an additive constant.
For graphs of large maximum degree \(\varDelta\), we show that these bounds can be improved to \(\frac{3n-\varDelta\delta}{\delta^2-\delta +1}-\frac{3(\delta -1)\sqrt{\varDelta(\delta -2)}}{\delta^2-\delta +1}+10\) for the diameter, and \(\frac{3n -3\varDelta\delta}{2(\delta^2 -\delta +1)}-\frac{3(\delta -1)\sqrt{\varDelta(\delta -2)}}{2(\delta^2-\delta +1)}+22\) for the radius. We further show that only slightly weaker bounds hold for \((C_4,C_5)\)-free graphs.
As a by-product we obtain a result on a generalisation of cages. For given \(\delta,\varDelta\in\mathbb{N}\) with \(\varDelta\geq\delta\) let \(n(\delta,\varDelta,g)\) be the minimum order of a graph of girth \(g\), minimum degree \(\delta\) and maximum degree \(\varDelta\). Then \(n(\delta,\varDelta,6)\geq\varDelta\delta+(\delta-1)\sqrt{\varDelta(\delta-2)}+\frac{3}{2}\). If \(\delta -1\) is a prime power, then there exist infinitely many values of \(\varDelta\) such that, for \(\delta\) constant and \(\varDelta\) large, \(n(\delta,\varDelta,6)=\delta\varDelta +O(\sqrt{\varDelta} )\).On three outer-independent domination related parameters in graphs.https://zbmath.org/1459.052432021-05-28T16:06:00+00:00"Mojdeh, Doost Ali"https://zbmath.org/authors/?q=ai:mojdeh.doost-ali"Peterin, Iztok"https://zbmath.org/authors/?q=ai:peterin.iztok"Samadi, Babak"https://zbmath.org/authors/?q=ai:samadi.babak"Yero, Ismael G."https://zbmath.org/authors/?q=ai:yero.ismael-gSummary: Let \(G\) be a graph and let \(S\subseteq V(G)\). The set \(S\) is a double outer-independent dominating set of \(G\) if \(|N[v]\cap S|\geq 2\) for all \(v\in V(G)\), and \(V(G)\setminus S\) is independent. Similarly, \(S\) is a 2-outer-independent dominating set, if every vertex from \(V(G)\setminus S\) has at least two neighbors in \(S\) and \(V(G)\setminus S\) is independent. Finally, \(S\) is a total outer-independent dominating set if every vertex from \(V(G)\) has a neighbor in \(S\) and the complement of \(S\) is an independent set. The double, total or 2-outer-independent domination number of \(G\) is the smallest possible cardinality of any double, total or 2-outer-independent dominating set of \(G\), respectively. In this paper, the 2-outer-independent, the total outer-independent and the double outer-independent domination numbers of graphs are investigated. We prove some Nordhaus-Gaddum type inequalities, derive their computational complexities and present several bounds for them.Various characterizations of throttling numbers.https://zbmath.org/1459.052332021-05-28T16:06:00+00:00"Carlson, Joshua"https://zbmath.org/authors/?q=ai:carlson.joshua"Kritschgau, Jürgen"https://zbmath.org/authors/?q=ai:kritschgau.jurgenSummary: Zero forcing can be described as a graph process that uses a color change rule in which vertices change white vertices to blue. The throttling number of a graph minimizes the sum of the number of vertices initially colored blue and the number of time steps required to color the entire graph. Positive semidefinite (PSD) zero forcing is a commonly studied variant of standard zero forcing that alters the color change rule. This paper introduces a method for extending a graph using a PSD zero forcing process. Using this extension method, graphs with PSD throttling number at most \(t\) are characterized as specific minors of the Cartesian product of complete graphs and trees. A similar characterization is obtained for the minor monotone floor of PSD zero forcing. Finally, the set of connected graphs on \(n\) vertices with throttling number at least \(n-k\) is characterized by forbidding a finite family of induced subgraphs. These forbidden subgraphs are constructed for standard throttling.Circuit \(k\)-covers of signed graphs.https://zbmath.org/1459.051112021-05-28T16:06:00+00:00"Chen, Jing"https://zbmath.org/authors/?q=ai:chen.jing.1|chen.jing.5|chen.jing.3|chen.jing.2|chen.jing.4"Fan, Genghua"https://zbmath.org/authors/?q=ai:fan.genghuaSummary: Let \(G\) be a signed graph and \(\mathcal{F}\) a set of signed circuits in \(G\). For an edge \(e\in E(G)\), \(\mathcal{F}(e)\) denotes the number of signed circuits in \(\mathcal{F}\) that contain \(e\). \(\mathcal{F}\) is called a circuit-cover of \(G\) if \(\mathcal{F}(e) > 0\) for each \(e\in E(G)\), and a circuit \(k\)-cover of \(G\) if \(\mathcal{F}(e)=k\) for each \(e\in E(G)\). \(G\) is coverable if it has a circuit-cover. The existence of a circuit-cover in \(G\) is equivalent to the existence of a nowhere-zero flow in \(G\). For a coverable signed graph \(G\), it is proved in this paper that if each maximal 2-edge-connected subgraph of \(G\) is eulerian, then \(G\) has a circuit 6-cover, consisting of four circuit-covers of \(G\), and as an immediate consequence, \(G\) has a circuit-cover of length at most \(\frac{3}{2}|E(G)|\), generalizing a known result on signed eulerian graphs. New results on circuit \(k\)-covers are obtained and applied to estimating bounds on the lengths of shortest circuit-covers of signed graphs.Unrooted non-binary tree-based phylogenetic networks.https://zbmath.org/1459.050422021-05-28T16:06:00+00:00"Fischer, Mareike"https://zbmath.org/authors/?q=ai:fischer.mareike"Herbst, Lina"https://zbmath.org/authors/?q=ai:herbst.lina"Galla, Michelle"https://zbmath.org/authors/?q=ai:galla.michelle"Long, Yangjing"https://zbmath.org/authors/?q=ai:long.yangjing"Wicke, Kristina"https://zbmath.org/authors/?q=ai:wicke.kristinaSummary: Phylogenetic networks are a generalization of phylogenetic trees allowing for the representation of non-treelike evolutionary events such as hybridization. Typically, such networks have been analyzed based on their `level', i.e. based on the complexity of their 2-edge-connected components. However, recently the question of how `treelike' a phylogenetic network is has become the center of attention in various studies. This led to the introduction of tree-based networks, i.e. networks that can be constructed from a phylogenetic tree, called the base tree, by adding additional edges. While the concept of tree-basedness was originally introduced for rooted phylogenetic networks, it has recently also been considered for unrooted networks. In the present study, we compare and contrast findings obtained for unrooted binary tree-based networks to unrooted non-binary networks. In particular, while it is known that up to level 4 all unrooted binary networks are tree-based, we show that in the case of non-binary networks, this result only holds up to level 3.Shannon capacity and the categorical product.https://zbmath.org/1459.050922021-05-28T16:06:00+00:00"Simonyi, Gábor"https://zbmath.org/authors/?q=ai:simonyi.gaborSummary: Shannon OR-capacity \(C_{\mathrm{OR}}(G)\) of a graph \(G\), that is the traditionally more often used Shannon AND-capacity of the complementary graph, is a homomorphism monotone graph parameter therefore \(C_{\mathrm{OR}}(F\times G)\leqslant\min\{C_{\mathrm{OR}}(F),C_{\mathrm{OR}}(G)\}\) holds for every pair of graphs, where \(F\times G\) is the categorical product of graphs \(F\) and \(G\). Here we initiate the study of the question when could we expect equality in this inequality. Using a strong recent result of \textit{J. Zuiddam} [Combinatorica 39, No. 5, 1173--1184 (2019; Zbl 1449.05208)], we show that if this ``Hedetniemi-type'' equality is not satisfied for some pair of graphs then the analogous equality is also not satisfied for this graph pair by some other graph invariant that has a much ``nicer'' behavior concerning some different graph operations. In particular, unlike Shannon OR-capacity or the chromatic number, this other invariant is both multiplicative under the OR-product and additive under the join operation, while it is also nondecreasing along graph homomorphisms. We also present a natural lower bound on \(C_{\mathrm{OR}}(F\times G)\) and elaborate on the question of how to find graph pairs for which it is known to be strictly less than the upper bound \(\min\{C_{\mathrm{OR}}(F),C_{\mathrm{OR}}(G)\}\). We present such graph pairs using the properties of Paley graphs.Determining the circular flow number of a cubic graph.https://zbmath.org/1459.051092021-05-28T16:06:00+00:00"Lukoťka, Robert"https://zbmath.org/authors/?q=ai:lukotka.robertSummary: A circular nowhere-zero \(r\)-flow on a bridgeless graph \(G\) is an orientation of the edges and an assignment of real values from \([1, r-1]\) to the edges in such a way that the sum of incoming values equals the sum of outgoing values for every vertex. The circular flow number, \(\phi_c(G)\), of \(G\) is the infimum over all values \(r\) such that \(G\) admits a nowhere-zero \(r\)-flow. A flow has its underlying orientation. If we subtract the number of incoming and the number of outgoing edges for each vertex, we get a mapping \(V(G) \to \mathbb{Z} \), which is its underlying balanced valuation. In this paper we describe efficient and practical polynomial algorithms to turn balanced valuations and orientations into circular nowhere zero \(r\)-flows they underlie with minimal \(r\). Using this algorithm one can determine the circular flow number of a graph by enumerating balanced valuations. For cubic graphs we present an algorithm that determines \(\phi_c(G)\) in case that \(\phi_c(G) \leqslant 5\) in time \(O(2^{0.6\cdot|V(G)|})\). If \(\phi_c(G) > 5\), then the algorithm determines that \(\phi_c(G) > 5\) and thus the graph is a counterexample to Tutte's 5-flow conjecture. The key part is a procedure that generates all (not necessarily proper) 2-vertex-colourings without a monochromatic path on three vertices in \(O(2^{0.6\cdot|V(G)|})\) time. We also prove that there is at most \(2^{0.6\cdot|V(G)|}\) of them.Uniquely \(D\)-colourable digraphs with large girth. II: Simplification via generalization.https://zbmath.org/1459.050812021-05-28T16:06:00+00:00"Kayll, P. Mark"https://zbmath.org/authors/?q=ai:kayll.peter-mark"Parsa, Esmaeil"https://zbmath.org/authors/?q=ai:parsa.esmaeilSummary: We prove that for every digraph \(D\) and every choice of positive integers \(k, \ell\) there exists a digraph \(D^\ast\) with girth at least \(\ell\) together with a surjective acyclic homomorphism \(\psi\colon D^\ast\to D\) such that: (i) for every digraph \(C\) of order at most \(k\), there exists an acyclic homomorphism \(D^\ast\to C\) if and only if there exists an acyclic homomorphism \(D\to C\); and (ii) for every \(D\)-pointed digraph \(C\) of order at most \(k\) and every acyclic homomorphism \(\varphi\colon D^\ast\to C\) there exists a unique acyclic homomorphism \(f\colon D\to C\) such that \(\varphi=f\circ\psi \). This implies the main results in [\textit{A. Harutyunyan} et al., Can. J. Math. 64, No. 6, 1310--1328 (2012; Zbl 1254.05057)] analogously with how the work [\textit{J. Nešetřil} and \textit{X. Zhu}, J. Comb. Theory, Ser. B 90, No. 1, 161--172 (2004; Zbl 1033.05044)] generalizes and extends [\textit{X. Zhu}, J. Graph Theory 23, No. 1, 33--41 (1996; Zbl 0864.05037)].Largest component and node fault tolerance for grids.https://zbmath.org/1459.052142021-05-28T16:06:00+00:00"Przybyło, Jakub"https://zbmath.org/authors/?q=ai:przybylo.jakub"Żak, Andrzej"https://zbmath.org/authors/?q=ai:zak.andrzejSummary: A graph \(G\) is called \(t\)-node fault tolerant with respect to \(H\) if \(G\) still contains a subgraph isomorphic to \(H\) after removing any \(t\) of its vertices. The least value of \(|E(G)|-|E(H)|\) among all such graphs \(G\) is denoted by \(\Delta(t,H)\). We study fault tolerance with respect to some natural architectures of a computer network, i.e. the \(d\)-dimensional toroidal grids and the hypercubes. We provide the first non-trivial lower bounds for \(\Delta(1,H)\) in these cases. For this aim we establish a general connection between the notion of fault tolerance and the size of a largest component of a graph. In particular, we give for all values of \(k\) (and \(n)\) a lower bound on the order of the largest component of any graph obtained from \(C_n\Box C_n\) via removal of \(k\) of its vertices, which is in general optimal.Polynomially bounding the number of minimal separators in graphs: reductions, sufficient conditions, and a dichotomy theorem.https://zbmath.org/1459.052772021-05-28T16:06:00+00:00"Milanič, Martin"https://zbmath.org/authors/?q=ai:milanic.martin"Pivač, Nevena"https://zbmath.org/authors/?q=ai:pivac.nevenaSummary: A graph class is said to be tame if graphs in the class have a polynomially bounded number of minimal separators. Tame graph classes have good algorithmic properties, which follow, for example, from an algorithmic metatheorem of \textit{F. V. Fomin} et al. [SIAM J. Comput. 44, No. 1, 54--87 (2015; Zbl 1357.05144)]. We show that a hereditary graph class \(\mathcal{G}\) is tame if and only if the subclass consisting of graphs in \(\mathcal{G}\) without clique cutsets is tame. This result and Ramsey's theorem lead to several types of sufficient conditions for a graph class to be tame. In particular, we show that any hereditary class of graphs of bounded clique cover number that excludes some complete prism is tame, where a complete prism is the Cartesian product of a complete graph with a \(K_2\). We apply these results, combined with constructions of graphs with exponentially many minimal separators, to develop a dichotomy theorem separating tame from non-tame graph classes within the family of graph classes defined by sets of forbidden induced subgraphs with at most four vertices.The graph of 4-ary simplex codes of dimension 2.https://zbmath.org/1459.050642021-05-28T16:06:00+00:00"Kwiatkowski, Mariusz"https://zbmath.org/authors/?q=ai:kwiatkowski.mariusz"Pankov, Mark"https://zbmath.org/authors/?q=ai:pankov.markSummary: We give a complete description of the distance relation on the graph of 4-ary simplex codes of dimension 2. This is a connected graph of diameter 3. For every vertex we determine the sets of all vertices at distance \(i\in\{1,2,3\}\) and describe their symmetries.Distributed coordination on state-dependent fuzzy graphs.https://zbmath.org/1459.930182021-05-28T16:06:00+00:00"Oyedeji, Mojeed O."https://zbmath.org/authors/?q=ai:oyedeji.mojeed-o"Mahmoud, Magdi S."https://zbmath.org/authors/?q=ai:mahmoud.magdi-sadik-mostafa"Xia, Yuanqing"https://zbmath.org/authors/?q=ai:xia.yuanqingSummary: Multiagent systems are increasingly becoming popular among researchers spanning multiple fields of study. However, existing studies only models communication interaction between agents as either fixed or switching topologies described by crisp graphs supported by algebraic graph theories. In this paper, we propose an alternative approach to describing agent interactions using fuzzy graphs. Our approach is aimed at opening up new research avenues and defining new problems in coordination control especially in terms of dynamics between agents' states, graph topologies and coordination objectives. This paper studies distributed coordination on fuzzy graphs where the edge-weights modeling network topologies are dependent on the states of the agents in the network. In hindsight, the network weights are adjustable based on the situational state of the agents. First, we introduce the concept of fuzzy graphs and give some distinguishing features from the crisp or fixed graphs. Next, we provide some membership functions to define the state-dependent weights and finally we use some simulations to demonstrate the convergence of the proposed consensus algorithms especially for cases where the agents are subject to system failures.On proper (strong) rainbow connection of graphs.https://zbmath.org/1459.050802021-05-28T16:06:00+00:00"Jiang, Hui"https://zbmath.org/authors/?q=ai:jiang.hui"Li, Wenjing"https://zbmath.org/authors/?q=ai:li.wenjing.1"Li, Xueliang"https://zbmath.org/authors/?q=ai:li.xueliang"Magnant, Colton"https://zbmath.org/authors/?q=ai:magnant.coltonIn this paper, the authors consider finite, undirected graphs without loops or parallel edges. Let \(G=(V, E)\) be a graph. A mapping \(c:E\rightarrow \{1,\dots,t\}\) is a proper \(t\)-edge-coloring, if adjacent edges of \(G\) receive different colors. A path \(P\) of \(G\) is an R-path with respect to \(c\), if any two edges of \(P\) have different colors.
For a graph \(G\) let \(\operatorname{prc}(G)\) be the smallest \(t\), such that \(G\) admits a \(t\)-edge-coloring with the property that for any two different vertices \(u\) and \(v\) of \(G\) there is an R-path connecting \(u\) and \(v\). Moreover, let \(\operatorname{psrc}(G)\) be the smallest \(t\), such that \(G\) admits a \(t\)-edge-coloring with the property that for any two different vertices \(u\) and \(v\) of \(G\) there is an R-path connecting \(u\) and \(v\) that is a shortest \(u\)-\(v\)-path.
In this paper, the authors characterize the graphs \(G\) with \(\operatorname{prc}(G)=|E|\) or \(\operatorname{prc}(G)=|E|-1\). Then they show that for any \(n\geq 1\) and \(p_1,\dots,p_n\geq 2\), the Cartesian product of \(K_{p_1},\dots,K_{p_n}\), denoted by \(G\), satisfies \(\operatorname{prc}(G)=\operatorname{psrc}(G)=\chi^\prime(G)\). Here \(\chi^\prime(G)\) is the chromatic index of \(G\). In the end of the paper, the authors present some sufficient conditions for a graph \(H\) to satisfy \(\operatorname{prc}(H)=\operatorname{rc}(H)\). Here, \(\operatorname{rc}(H)\) denotes the smallest \(t\) for which there is a mapping \(c:E(H)\rightarrow \{1,\dots,t\}\) (not necessarily a proper edge-coloring) such that for any two distinct vertices of \(H\) there is an R-path connecting these two vertices.
Reviewer: Vahan Mkrtchyan (L'Aquila)On the star chromatic index of generalized Petersen graphs.https://zbmath.org/1459.050962021-05-28T16:06:00+00:00"Zhu, Enqiang"https://zbmath.org/authors/?q=ai:zhu.enqiang"Shao, Zehui"https://zbmath.org/authors/?q=ai:shao.zehuiLet \(G\) be a graph. A proper \(\ell\)-edge coloring is a map \(c:E(G)\rightarrow\{1,\dots,\ell\}\) such that every two different incident edges \(uv\) and \(vw\) receive different colors, that is \(c(uv)\neq c(vw)\). A proper \(\ell\)-edge coloring is called a star \(\ell\)-edge coloring if edges of any \(P_r\), \(r\geq 5\), or any \(C_q\), \(q\geq 4\), receive more than two colors. The star chromatic index \(\chi^\prime_s(G)\) of \(G\) is the minimum \(\ell\) such that there exists a proper star \(\ell\)-edge coloring of \(G\).
Let \(k\) and \(n\), \(n\geq 2k+1\), be two positive integers. The generalized Peterson graph \(P(n,k)\) has vertex set
\[V(P(n,k))=\{v_1,\dots,v_n\}\cup\{u_1,\dots,u_n\}\]
and edge set \(E(P(n,k))\) equals to
\[\{v_iv_{i+1}:i\in\{1,\dots,n\}\}\cup\{v_iu_i:i\in\{1,\dots,n\}\}\cup\{u_iu_{i+k}:i\in\{1,\dots,n\}\},\]
where all the operations are taken modulo \(n\). Generalized Petersen graphs are one of the most important cubic graph families.
This work deals with star chromatic index of generalized Petersen graphs. They characterize those Petersen graphs for which \(\chi^\prime_s(P(n,k))=4\). This happens when \(n\) is a multiple of four and \(k\) is odd. Among others they show that they have \(\chi^\prime_s(P(n,k))=5\) for ``almost all'' of them. Among the exceptions, it is not clear weather they have a star \(5\)-edge coloring or not.
Reviewer: Iztok Peterin (Maribor)The minimum size of a graph with given tree connectivity.https://zbmath.org/1459.050452021-05-28T16:06:00+00:00"Sun, Yuefang"https://zbmath.org/authors/?q=ai:sun.yuefang|sun.yuefang.1"Sheng, Bin"https://zbmath.org/authors/?q=ai:sheng.bin"Jin, Zemin"https://zbmath.org/authors/?q=ai:jin.zeminThe paper under review investigates generalizations of connectivity, where a path between two vertices is changed to a tree containing some \(k\) vertices. Given a graph \(G\) a set of vertices \(S\subseteq V(G)\), let \( \kappa_G(S)\) denote the maximum number of trees, which are (1) subgraphs of \(G\), (2) pairwise edge-disjoint, (3) the common vertices of any two of the trees is exactly \(S\). (Such trees are called internally disjoint.) Similarly, let \( \lambda_G(S)\) denote the maximum number of trees, which are (1) subgraphs of \(G\), (2) pairwise edge-disjoint. (Such trees are called edge-disjoint.) Let \(\kappa_k(G)\) and \(\lambda_k(G)\) defined as \(\kappa_k(G) =\min \{ \kappa_G(S) : S\subseteq V(G), |S|=k\}\) and \(\lambda_k(G) =\min \{ \lambda_G(S) : S\subseteq V(G), |S|=k\}\).
Let \(f(n, k, t)\) (\(g(n, k, t)\)) be the minimum size of a connected graph \(G\) with order \(n\) and \(\kappa_k(G) = t\) (\(\lambda_k(G) = t\)), where \(3 \leq k \leq n\) and \(1 \leq t \leq n-\lceil k/2\rceil\). When \(t\) is arbitrary, the paper obtains upper bounds of both \(f(n, k, t)\) and \(g(n, k, t)\) for \(k =3, 4, 5\), and shows that these bounds can be attained. When \(k\) is arbitrary, the paper obtains an upper bound of \(g(n, k, t)\) for \(t =1, 2, 3, 4\) and an upper bound of \(f(n, k, t)\) for \(t =1, 2, 3\). These bounds can also be attained.
Reviewer: László A. Székely (Columbia)Integrability of graph combinatorics via random walks and heaps of dimers.https://zbmath.org/1459.053002021-05-28T16:06:00+00:00"Di Francesco, P."https://zbmath.org/authors/?q=ai:di-francesco.philippe"Guitter, E."https://zbmath.org/authors/?q=ai:guitter.emmanuelGeneralized group-subgroup pair graphs.https://zbmath.org/1459.051182021-05-28T16:06:00+00:00"Kimoto, Kazufumi"https://zbmath.org/authors/?q=ai:kimoto.kazufumiSummary: A regular finite graph is called a Ramanujan graph if its zeta function satisfies an analog of the Riemann Hypothesis. Such a graph has a small second eigenvalue so that it is used to construct cryptographic hash functions. Typically, explicit family of Ramanujan graphs are constructed by using Cayley graphs. In the paper, we introduce a generalization of Cayley graphs called generalized group-subgroup pair graphs, which are a generalization of group-subgroup pair graphs defined by \textit{C. Reyes-Bustos} [Linear Algebra Appl. 488, 320--349 (2016; Zbl 1326.05063)]. We study basic properties, especially spectra of them.
For the entire collection see [Zbl 1457.94002].Changing and unchanging of the domination number of a graph: path addition numbers.https://zbmath.org/1459.052472021-05-28T16:06:00+00:00"Samodivkin, Vladimir"https://zbmath.org/authors/?q=ai:samodivkin.vladimir-d|samodivkin.vladimirLet \(G\) be a graph and \(u\) and \(v\) two distinct vertices of \(G\). By \(G_{u,v,k-2}\) one denotes a graph obtained from \(G\) by adding a path \(ux_1x_2\dots x_{k-2}v\) of length \(k\) between \(u\) and \(v\). Set \(D\subseteq V(G)\) is a dominating set of \(G\) if every vertex from \(V(G)-D\) has a neighbor in \(D\). The domination number \(\gamma(G)\) is the minimum cardinality of a dominating set of \(G\).
The present work is a study of the relation between \(\gamma(G)\) and \(\gamma(G_{u,v,k})\) for different \(k\). The minimum and maximum number \(k\) for which \(\gamma(G)<\gamma(G_{u,v,k})\) for arbitrary (adjacent or nonadjacent) \(u\) and \(v\) is also considered.
Reviewer: Iztok Peterin (Maribor)Selfish versus unselfish optimization of network creation.https://zbmath.org/1459.910252021-05-28T16:06:00+00:00"Schneider, Johannes J."https://zbmath.org/authors/?q=ai:schneider.johannes-j"Kirkpatrick, Scott"https://zbmath.org/authors/?q=ai:kirkpatrick.scottCops and robbers on toroidal chess graphs.https://zbmath.org/1459.052112021-05-28T16:06:00+00:00"Nicholson, Neil R."https://zbmath.org/authors/?q=ai:nicholson.neil-r"Hahn, Allyson"https://zbmath.org/authors/?q=ai:hahn.allysonSummary: We investigate multiple variants of the game Cops and Robbers. Playing it on an \(n\times n\) toroidal chess graph, the game is varied by defining moves for cops and robbers differently, always mimicking moves of certain chess pieces. In these cases, the cop number is completely determined.Loops of any size and Hamilton cycles in random scale-free networks.https://zbmath.org/1459.821002021-05-28T16:06:00+00:00"Bianconi, Ginestra"https://zbmath.org/authors/?q=ai:bianconi.ginestra"Marsili, Matteo"https://zbmath.org/authors/?q=ai:marsili.matteoDouble phase transition of the Ising model in core-periphery networks.https://zbmath.org/1459.820822021-05-28T16:06:00+00:00"Chen, Hanshuang"https://zbmath.org/authors/?q=ai:chen.hanshuang"Zhang, Haifeng"https://zbmath.org/authors/?q=ai:zhang.haifeng"Shen, Chuansheng"https://zbmath.org/authors/?q=ai:shen.chuanshengThe effects of spatial constraints on the evolution of weighted complex networks.https://zbmath.org/1459.052942021-05-28T16:06:00+00:00"Barrat, Alain"https://zbmath.org/authors/?q=ai:barrat.alain"Barthélemy, Marc"https://zbmath.org/authors/?q=ai:barthelemy.marc"Vespignani, Alessandro"https://zbmath.org/authors/?q=ai:vespignani.alessandroA sufficient condition for the existence of restricted fractional \((g, f)\)-factors in graphs.https://zbmath.org/1459.052752021-05-28T16:06:00+00:00"Zhou, S."https://zbmath.org/authors/?q=ai:zhou.shanchang|zhou.shumei|zhou.shuheng|zhou.siyun|zhou.shouwei|zhou.shengyao|zhou.shiming|zhou.shaojie|zhou.songping|zhou.shigang|zhou.shunxing|zhou.shijian|zhou.shihang|zhou.suhong|zhou.shaorui|zhou.shengan|zhou.shengyu|zhou.shouzi|zhou.shu|zhou.shizhong|zhou.shenggao|zhou.shanggang|zhou.shiqiong|zhou.shude|zhou.shunxian|zhou.shengsheng|zhou.shuange|zhou.shulu|zhou.shuna|zhou.sean|zhou.shiqi|zhou.shuisheng|zhou.siyuan|zhou.shuyun|zhou.shutang|zhou.sanming|zhou.shenglin|zhou.shusen|zhou.shiaokang|zhou.shuangzhen|zhou.suihua|zhou.shixing|zhou.shijun|zhou.shaopu|zhou.si|zhou.shixuan|zhou.shaoling|zhou.sheng|zhou.shi|zhou.shiguo|zhou.shuangfeng|zhou.shenghan|zhou.shuangyou|zhou.shifan|zhou.shuquan|zhou.shirong|zhou.shaolei|zhou.shuhan|zhou.shihua|zhou.shaonan|zhou.shuyan|zhou.shuqin|zhou.shengtian|zhou.shutao|zhou.suyun|zhou.shifei|zhou.shisheng|zhou.shibo|zhou.shaoyan|zhou.senqiang|zhou.siwang|zhou.siping|zhou.sun|zhou.shangchao|zhou.shan|zhou.shengfeng|zhou.shuangxi|zhou.sumei|zhou.sujing|zhou.suhua|zhou.shaolin|zhou.shuwei|zhou.shengmin|zhou.suiping|zhou.shansheng|zhou.shengtong|zhou.shujing|zhou.shunhua|zhou.song|zhou.shuhang|zhou.shibing|zhou.shangming|zhou.shengfan|zhou.shujia|zhou.shihao|zhou.shiyuan|zhou.sanping|zhou.shenglong|zhou.shengming|zhou.suxin|zhou.sangsang|zhou.shenfan|zhou.samson|zhou.shangbo|zhou.shumin|zhou.shuzi|zhou.shengbing|zhou.shouming|zhou.shidong|zhou.shunmei|zhou.shuqi|zhou.shaofu|zhou.shunping|zhou.shoujun|zhou.sufang|zhou.shenghua|zhou.shanyou|zhou.shuiping|zhou.su|zhou.shidi|zhou.shujun|zhou.shurong|zhou.shuke|zhou.shengfang|zhou.shijie|zhou.shaowei|zhou.shuya|zhou.shimin|zhou.shuwang|zhou.sha|zhou.shaoyang|zhou.sishou|zhou.suwu|zhou.shiyu|zhou.shaosheng|zhou.sen|zhou.shouhang|zhou.sizhong|zhou.shulin|zhou.shucheng|zhou.siqi|zhou.shuxia|zhou.shouze|zhou.shaowu|zhou.shasha|zhou.shouqin|zhou.suwen|zhou.shaoquan|zhou.siyi|zhou.shengchao|zhou.suqian|zhou.shi-ping|zhou.shengwei|zhou.shengbin|zhou.shuangshuang|zhou.siyu|zhou.shenjie|zhou.shuming|zhou.shuqing|zhou.saijun|zhou.siqing|zhou.shuang|zhou.shuo|zhou.shuaihu|zhou.shensheng|zhou.shiwu|zhou.shihong|zhou.shaohua|zhou.shaoqian|zhou.shiwei|zhou.shun|zhou.sulian|zhou.shuzhi|zhou.shilin|zhou.shangli|zhou.shuyu|zhou.shixiang|zhou.shenghai|zhou.shuwen|zhou.shengxi|zhou.shujuan|zhou.shengli|zhou.sanyu|zhou.shengyi|zhou.shaobo|zhou.shanshan|zhou.shuigeng|zhou.shanyu|zhou.shengwu|zhou.shanxue|zhou.shuanggui|zhou.siyong|zhou.shengjuan|zhou.shudao|zhou.shuai|zhou.shenlin|zhou.suquan|zhou.shipeng|zhou.shigui"Sun, Z."https://zbmath.org/authors/?q=ai:sun.zhenhua|sun.ziyue|sun.zhongmiao|sun.zhengyi|sun.zhaobo|sun.zhijie|sun.zichen|sun.zhaohui|sun.zhiyong|sun.zhanquan|sun.ziqiang|sun.zhenguo|sun.zhenquan|sun.zhenyu|sun.zhanxue|sun.ziyan|sun.zihua|sun.zhihao|sun.zhenqiu|sun.zhubin|sun.zhihu|sun.zhiren|sun.zhiqiang|sun.zhongxi|sun.zhengya|sun.zhihua|sun.zhidong|sun.zhonghua|sun.zhuo|sun.zhen|sun.ziwen|sun.zongyang|sun.zhendong|sun.zhiling|sun.zhongxiu|sun.zhenying|sun.zhenqi|sun.zhenan|sun.zonghai|sun.zhoujun|sun.zhiwei|sun.zhenxu|sun.zhuoxin|sun.zhili|sun.zilai|sun.zaiguan|sun.zhimeng|sun.zimo|sun.zhongfeng|sun.zhicheng|sun.zhengce|sun.zhengwei|sun.zhaoxu|sun.zhaobin|sun.zhuling|sun.zepeng|sun.zhongkui|sun.zhenyang|sun.zongjuan|sun.zhao|sun.zheng|sun.zhengkang|sun.zhengshun|sun.zhiying|sun.zhiyuan|sun.zengguo|sun.zonghan|sun.zongliang|sun.zhihong|sun.zhitian|sun.zhenlong|sun.zhilin|sun.zhanwei|sun.zhaohonh|sun.zhishuai|sun.zongguang|sun.zhufeng|sun.zhongkang|sun.zhizhong|sun.zhengzhi|sun.zaidong|sun.zhenxi|sun.zhaohong|sun.zhiyang|sun.zhongyang|sun.zuochen|sun.zhanyu|sun.zhe|sun.zhimin|sun.zongqi|sun.zhanshan|sun.zhengxing|sun.zhengying|sun.zhijian|sun.zongming|sun.zhengjia|sun.zhongyao|sun.zhiping|sun.zhensheng|sun.zhongfei|sun.zhengdong|sun.zhaochun|sun.zhixue|sun.zhaonan|sun.zhiming|sun.ziqi|sun.zhonggui|sun.zeyu|sun.zhonghong|sun.zichun|sun.zhangfeng|sun.zhongqin|sun.zhaorui|sun.zhenlu|sun.zhongbo|sun.zhongyuan|sun.zhijun|sun.zuoyu|sun.zhenwen|sun.zezhen|sun.zhelei|sun.ziyao|sun.zhenzhong|sun.zhaochen|sun.zhongtao|sun.zuo|sun.zhipeng|sun.zhenyao|sun.zhuoer|sun.zhongju|sun.zhichang|sun.zhongqi|sun.zixing|sun.zhixin|sun.zhaohao|sun.zhenzu|sun.zhonggao|sun.zengzhuo|sun.zhongde|sun.zhenyuan|sun.zhong|sun.zairan|sun.zhongmin|sun.zhaocai|sun.zhi|sun.zhanli|sun.zhouhong|sun.zongyao|sun.zhongguo|sun.zhi-wei.1|sun.zhifeng|sun.zuyao|sun.zhangpeng|sun.zhengqi|sun.zhibin|sun.zhongbin|sun.zuolei|sun.zhihui|sun.zongyuan|sun.zhaomei|sun.zhuoming|sun.zhaocheng|sun.zhang|sun.zhongwei|sun.zhiguo|sun.zhi-wei|sun.zhengjie|sun.zhaowei|sun.zongxuan|sun.ziyuan|sun.zhiye|sun.zhigang|sun.zhiyu|sun.zengqi|sun.zongli"Pan, Q."https://zbmath.org/authors/?q=ai:pan.qiangyou|pan.quanru|pan.qinxue|pan.qiongqiong|pan.qingfang|pan.qianyao|pan.qunhua|pan.qiuhui|pan.qingchao|pan.quanxiang|pan.qiaoqiao|pan.quanke|pan.qiyuan|pan.qingfei|pan.qunxing|pan.qiang|pan.qielu|pan.qingqing|pan.qiuyue|pan.qing|pan.qihong|pan.qiao|pan.qingxian|pan.qingnian|pan.qianqian|pan.qun|pan.quyu|pan.qi|pan.qiujin|pan.qiwen|pan.qijun|pan.qian|pan.qie|pan.qishu|pan.qiong|pan.quan|pan.qishengSummary: In an NFV network, the availability of resource scheduling can be transformed to the existence of the fractional factor in the corresponding NFV network graph. Researching on the existence of special fractional factors in network structure can help to construct the NFV network with efficient application of resources. Let \(h: E(G) \rightarrow [0, 1]\) be a function. We write \(d_G^h(x)=\sum \limits_{e\ni x}h(e)\). We call a graph \(F_h\) with vertex set \(V(G)\) and edge set \(E_h\) a fractional \((g, f)\)-factor of \(G\) with indicator function \(h\) if \(g(x)\le{d}_G^h(x)\le f(x)\) holds for any \(x \in V(G)\), where \(E_h = \{e : e \in E(G), h(e) > 0\}\). We say that \(G\) has property \(E(m, n)\) with respect to a fractional \((g, f)\)-factor if for any two sets of independent edges \(M\) and \(N\) with \(\vert M \vert = m\), \(\vert N \vert = n\), and \(M\cap N=\emptyset\), \(G\) admits a fractional \((g, f)\)-factor \(F_h\) with \(h(e) = 1\) for any \(e \in M\) and \(h(e) = 0\) for any \(e \in N\). The concept of property \(E(m, n)\) with respect to a fractional \((g, f)\)-factor corresponds to the structure of an NFV network where certain channels are occupied or damaged in some period of time. In this paper, we consider the resource scheduling problem in NFV networks using graph theory, and show a neighborhood union condition for a graph to have property \(E(1, n)\) with respect to a fractional \((g, f)\)-factor. Furthermore, it is shown that the lower bound on the neighborhood union condition in the main result is the best possible in some sense.Hoffman's ratio bound.https://zbmath.org/1459.051722021-05-28T16:06:00+00:00"Haemers, Willem H."https://zbmath.org/authors/?q=ai:haemers.willem-hSummary: Hoffman's ratio bound is an upper bound for the independence number of a regular graph in terms of the eigenvalues of the adjacency matrix. The bound has proved to be very useful and has been applied many times. Hoffman did not publish his result, and for a great number of users the emergence of Hoffman's bound is a black hole. With his note I hope to clarify the history of this bound and some of its generalizations.Proof of a conjecture on extremal spectral radii of blow-up graphs.https://zbmath.org/1459.051802021-05-28T16:06:00+00:00"Lou, Zhenzhen"https://zbmath.org/authors/?q=ai:lou.zhenzhen"Zhai, Mingqing"https://zbmath.org/authors/?q=ai:zhai.mingqingSummary: A blow-up of a graph \(G\), is obtained from \(G\) by replacing each vertex \(v_i\) of \(G\) with an independent set \(V_i\) of \(n_i\) vertices and joining each vertex in \(V_i\) with each vertex in \(V_j\) provided \(v_iv_j\in E(G)\). Let \(\mathcal{M}_n(G)\) be the set of all the blow-ups of \(G\) such that each \(n_i\geq 1\) and \(\sum_{i=1}^{|G|}n_i=n\). \textit{J. Monsalve} and \textit{J. Rada} [ibid. 609, 1--11 (2021; Zbl 1458.05152)] proposed three conjectures on the extremal spectral radii of \(\mathcal{M}_n(P_k)\) and \(\mathcal{M}_n(C_k)\). In this paper, we confirm the first two conjectures and construct an infinite class of graphs to illustrate the third conjecture is not true.Lower bounds for the spectral norm of digraphs.https://zbmath.org/1459.051702021-05-28T16:06:00+00:00"García, Jazmín"https://zbmath.org/authors/?q=ai:garcia.jazmin"Monsalve, Juan"https://zbmath.org/authors/?q=ai:monsalve.juan-daniel"Rada, Juan"https://zbmath.org/authors/?q=ai:rada.juanSummary: Let \(D\) be a digraph with \(n\) vertices and \(\sigma_1(D)\geq\sigma_2(D)\geq\cdots\geq\sigma_n(D)\) the singular values of the adjacency matrix \(\mathcal{A}\) of \(D\). The spectral norm of \(D\) is \(\sigma_1(D)\) and the trace norm of \(D\) is \(\|D\|_\ast=\sum_{i=1}^n\sigma_i(D)\). In this paper we find lower bounds for the spectral norm of a digraph in terms of the structure of the digraph. Moreover, we introduce the concept of almost regular digraphs (extension of the well known almost regular graphs), and show that the lower bounds are attained precisely in almost regular digraphs. When we apply this theory to graphs, we recover well known lower bounds for the spectral radius of graphs. Also, we give a new upper bound for the trace norm of a digraph. Moreover, we determine the digraphs for which this bound is sharp: sink-source complete bipartite digraphs or symmetric balanced incomplete block designs (BIBD).Maximizing the signless Laplacian spectral radius of \(k\)-connected graphs with given diameter.https://zbmath.org/1459.051752021-05-28T16:06:00+00:00"Huang, Peng"https://zbmath.org/authors/?q=ai:huang.peng"Li, Jianxi"https://zbmath.org/authors/?q=ai:li.jianxi"Shiu, Wai Chee"https://zbmath.org/authors/?q=ai:shiu.waichee|shiu.wai-cheeSummary: Let \(\mathcal{G}_{n,k}^D\) be the set of all \(k\)-connected graphs of order \(n\) with diameter \(D\). In this paper, we determine the graphs with maximal signless Laplacian spectral radius among all graphs in \(\mathcal{G}_{n,k}^D\).On the cover time of \(\lambda\)-biased walk on supercritical Galton-Watson trees.https://zbmath.org/1459.050402021-05-28T16:06:00+00:00"Bai, Tianyi"https://zbmath.org/authors/?q=ai:bai.tianyiSummary: In this paper, we study the time required for a \(\lambda\)-biased \((\lambda>1)\) walk to visit all the vertices of a supercritical Galton-Watson tree up to generation \(n\). Inspired by the extremal landscape approach in [\textit{A. Cortines}, \textit{O. Louidor} and \textit{S. Saglietti}, ``A scaling limit for the cover time of the binary tree'', Preprint, \url{arXiv:1812.10101}] for the simple random walk on binary trees, we establish the scaling limit of the cover time in the biased setting.,The perimeter of large planar Voronoi cells: a double-stranded random walk.https://zbmath.org/1459.600232021-05-28T16:06:00+00:00"Hilhorst, H. J."https://zbmath.org/authors/?q=ai:hilhorst.hendrik-janNetwork community detection using modularity density measures.https://zbmath.org/1459.053112021-05-28T16:06:00+00:00"Chen, Tianlong"https://zbmath.org/authors/?q=ai:chen.tianlong"Singh, Pramesh"https://zbmath.org/authors/?q=ai:singh.pramesh"Bassler, Kevin E."https://zbmath.org/authors/?q=ai:bassler.kevin-eFrom standard alpha-stable Lévy motions to horizontal visibility networks: dependence of multifractal and Laplacian spectrum.https://zbmath.org/1459.601032021-05-28T16:06:00+00:00"Zou, Hai-Long"https://zbmath.org/authors/?q=ai:zou.hai-long"Yu, Zu-Guo"https://zbmath.org/authors/?q=ai:yu.zuguo"Anh, Vo"https://zbmath.org/authors/?q=ai:anh.vo-ngoc|anh.vo-v"Ma, Yuan-Lin"https://zbmath.org/authors/?q=ai:ma.yuanlinThe random fractional matching problem.https://zbmath.org/1459.901382021-05-28T16:06:00+00:00"Lucibello, Carlo"https://zbmath.org/authors/?q=ai:lucibello.carlo"Malatesta, Enrico M."https://zbmath.org/authors/?q=ai:malatesta.enrico-m"Parisi, Giorgio"https://zbmath.org/authors/?q=ai:parisi.giorgio"Sicuro, Gabriele"https://zbmath.org/authors/?q=ai:sicuro.gabrieleRapid mixing for lattice colourings with fewer colours.https://zbmath.org/1459.820232021-05-28T16:06:00+00:00"Achlioptas, Dimitris"https://zbmath.org/authors/?q=ai:achlioptas.dimitris"Molloy, Mike"https://zbmath.org/authors/?q=ai:molloy.michael-k"Moore, Cristopher"https://zbmath.org/authors/?q=ai:moore.cristopher"Van Bussel, Frank"https://zbmath.org/authors/?q=ai:van-bussel.frankThe theoretical capacity of the parity source coder.https://zbmath.org/1459.680582021-05-28T16:06:00+00:00"Ciliberti, Stefano"https://zbmath.org/authors/?q=ai:ciliberti.stefano"Mézard, Marc"https://zbmath.org/authors/?q=ai:mezard.marcStudy of prime graph of a ring.https://zbmath.org/1459.051242021-05-28T16:06:00+00:00"Pawar, Kishor"https://zbmath.org/authors/?q=ai:pawar.kishor-fakira"Joshi, Sandeep"https://zbmath.org/authors/?q=ai:joshi.sandeep-sSummary: The notion of a prime graph of a ring \(R, (PG(R))\) was first introduced by \textit{S. Bhavanari} et al. [J. Comb. Inf. Syst. Sci. 35, No. 1--2, Part 1, 27--42 (2010; Zbl 1271.16039)]. In this paper, we introduce the notion of ``Complement of a Prime Graph of a Ring \(R\)', denote it by \((PG(R))^c\) and find the degree of vertices in \(PG(R)\) and \((PG(R))^c\) for the ring \(\mathbb Z_n\) and the number of triangles in \(PG(R)\) and \((PG(R))^c\). It is proved that for any \(n \geq 6\) which not a prime then \(gr(PG(\mathbb Z_n ))=3\). If \(n\) is any prime number or \(n=4\) then \(gr(PG(\mathbb Z_n))= \infty \).Quantum walks.https://zbmath.org/1459.810692021-05-28T16:06:00+00:00"Konno, Norio"https://zbmath.org/authors/?q=ai:konno.norioFrom the text: This manuscript aims to introduce a basic quantum walk model and its characteristics with reference to recent literature. In the first two sections, we will describe a typical 2-state quantum walk model in \(\mathbb{Z}\), where \(\mathbb{Z}\) is the set of integers. Then, in Sections 3 and 4, we will discuss 3-state and multiple states and space-inhomogeneous models, respectively. In Section 5, we will describe a model on a half-line related to the CMV matrix. In Section 6, we will briefly discuss quantum
walks on a graph. We will introduce future topics in Section 7 and propose a future project in the final section.\par\vspace{1mm}
Quantum walks can be analyzed using combinatorial method, Fourier analysis, the stationary phase method, generating function method, CGMV method, and graph zeta method, and by clarifying their mutual relations, there is a further possibility for detailed research related to the behavior of multiple states or on a general graph. This should help in constructing theory in a quantum system
that corresponds to the traditional Markov process theory. We demonstrate the future flow of quantum walk research centered around the discrete-time case that the author currently has in mind below. This may be helpful in finding a research\(L(p,q)\)-label coloring problem with application to channel allocation.https://zbmath.org/1459.052892021-05-28T16:06:00+00:00"Liu, Zhenbin"https://zbmath.org/authors/?q=ai:liu.zhenbin"Wu, Yuqiang"https://zbmath.org/authors/?q=ai:wu.yuqiangSummary: In this paper, the \(L(p,q)\)-coloring problem of the graph is studied with application to channel allocation of the wireless network. First, by introducing two new logical operators, some necessary and sufficient conditions for solving the \(L(p,q)\)-coloring problem are given. Moreover, it is noted that all solutions of the obtained logical equations are corresponding to each coloring scheme. Second, by using the semitensor product, the necessary and sufficient conditions are converted to an algebraic form. Based on this, all coloring schemes can be obtained through searching all column indices of the zero columns. Finally, the obtained result is applied to analyze channel allocation of the wireless network. Furthermore, an illustration example is given to show the effectiveness of the obtained results in this paper.A mathematical approach on representation of competitions: competition cluster hypergraphs.https://zbmath.org/1459.911482021-05-28T16:06:00+00:00"Samanta, Sovan"https://zbmath.org/authors/?q=ai:samanta.sovan"Muhiuddin, G."https://zbmath.org/authors/?q=ai:muhiuddin.gulam|muhiuddin.ghulam"Alanazi, Abdulaziz M."https://zbmath.org/authors/?q=ai:alanazi.abdulaziz-m"Das, Kousik"https://zbmath.org/authors/?q=ai:das.kousikSummary: Social networks are represented using graph theory. In this case, individuals in a social network are assumed as nodes. Sometimes institutions or groups are also assumed as nodes. Institutions and such groups are assumed as cluster nodes that contain individuals or simple nodes. Hypergraphs have hyperedges that include more than one node. In this study, cluster hypergraphs are introduced to generalize the concept of hypergraphs, where cluster nodes are allowed. Sometimes competitions in the real world are done as groups. Cluster hypergraphs are used to represent such kinds of competitions. Competition cluster hypergraphs of semidirected graphs (a special type of mixed graphs called semidirected graphs, where the directed and undirected edges both are allowed) are introduced, and related properties are discussed. To define competition cluster hypergraphs, a few properties of semidirected graphs are established. Some associated terms on semidirected graphs are studied. At last, a numerical application is illustrated.A short note on inverse sum indeg index of graphs.https://zbmath.org/1459.050472021-05-28T16:06:00+00:00"Balachandran, Selvaraj"https://zbmath.org/authors/?q=ai:balachandran.selvaraj"Elumalai, Suresh"https://zbmath.org/authors/?q=ai:elumalai.suresh"Mansour, Toufik"https://zbmath.org/authors/?q=ai:mansour.toufikEigenvalue estimates of the \(p\)-Laplacian on finite graphs.https://zbmath.org/1459.051932021-05-28T16:06:00+00:00"Wang, Yu-Zhao"https://zbmath.org/authors/?q=ai:wang.yuzhao"Huang, Huimin"https://zbmath.org/authors/?q=ai:huang.huiminSummary: In this paper, we study the eigenvalue of \(p\)-Laplacian on finite graphs. Under generalized curvature dimensional condition, we obtain a lower bound of the first nonzero eigenvalue of \(p\)-Laplacian. Moreover, a upper bound of the largest \(p\)-Laplacian eigenvalue is derived.The comparative analysis of metric and edge metric dimension of some subdivisions of the wheel graph.https://zbmath.org/1459.050662021-05-28T16:06:00+00:00"Raza, Zahid"https://zbmath.org/authors/?q=ai:raza.zahid"Bataineh, M. S."https://zbmath.org/authors/?q=ai:bataineh.mohammad-saleh|bataineh.mohammad-s-a|bataineh.mohammed-s-aGromov-Hausdorff distances to simplexes and some applications to discrete optimisation.https://zbmath.org/1459.053582021-05-28T16:06:00+00:00"Ivanov, Aleksandr Olegovich"https://zbmath.org/authors/?q=ai:ivanov.aleksandr-olegovich"Tuzhilin, Alekseĭ Avgustinovich"https://zbmath.org/authors/?q=ai:tuzhilin.alexey-aSummary: Relations between Gromov-Hausdorff distance and Discrete Optimisation problems are discussed. We use the Gromov-Hausdorff distances to single-distance metric space for solving the following problems: calculation of lengths of minimum spanning tree edges of a finite metric space; generalised Borsuk problem; chromatic number and clique cover number of a simple graph calculation problems.Self-adjointness of perturbed bi-Laplacians on infinite graphs.https://zbmath.org/1459.052212021-05-28T16:06:00+00:00"Milatovic, Ognjen"https://zbmath.org/authors/?q=ai:milatovic.ognjenSummary: We give a sufficient condition for the essential self-adjointness of a perturbation of the square of the magnetic Laplacian on an infinite weighted graph. The main result is applicable to graphs whose degree function is not necessarily bounded. The result allows perturbations that are not necessarily bounded from below by a constant.The second-minimum Wiener index of cacti with given cycles.https://zbmath.org/1459.051312021-05-28T16:06:00+00:00"Deng, Hanyuan"https://zbmath.org/authors/?q=ai:deng.hanyuan"Keerthi Vasan, G. C."https://zbmath.org/authors/?q=ai:keerthi-vasan.g-c"Balachandran, S."https://zbmath.org/authors/?q=ai:balachandran.selvarajOn images and pre-images in a graph of the composition of independent uniform random mappings.https://zbmath.org/1459.053052021-05-28T16:06:00+00:00"Mironkin, V. O."https://zbmath.org/authors/?q=ai:mironkin.v-oSummary: We study the probability characteristics of the random mapping graph \(f_{\left[k\right]} \) -- the composition of \(k\) independent equiprobable random mappings \(f_1, \ldots, f_k \), where \(f_i\colon \left\{1,\ldots,n\right\}\to \left\{1,\ldots,n\right\}\), \(n,k\in\mathbb{N}\), \(i=1,\ldots,n\). The following results are obtained. For any fixed \(x,y\in S=\{1,\ldots,n\}\), \(x\ne y\),
\[ \mathcal{P}\{f_{\left[k\right]}(x)=f_{\left[k\right]}(y)\}=\sum\limits_{\begin{smallmatrix}s_1,\ldots,s_{k-1}\in\mathbb{N}\colon\\2\geqslant s_1\geqslant\ldots\geqslant s_{k-1} \end{smallmatrix}}\dfrac{q(2,s_1)}{n^{s_{k-1}-1}}\prod\limits_{i=1}^{k-2}q(s_i,s_{i+1}),\]
where \(q(a,b)=\text{C}_n^{n-b} \left(\dfrac{b}{n}\right)^a \sum\limits_{l=0}^b\text{C}_b^l(-1)^l\left(1-\dfrac{l}{b}\right)^a\). For any fixed \(x\in S\),
\begin{gather*} \mathcal{P}\{ x\in f_{\left[k\right]}(S)\}=\frac1{n}\sum\limits_{l=1}^n{\left(\dfrac{(n)_l}{n^l} \right)^k}+\\ +\sum\limits_{l=1}^{n-2}\sum\limits_{t=1}^{n-l-1}\sum\limits_{m=1}^{n-t-l}(-1)^{m-1}\text{C}_{n-1}^m\sum\limits_{\begin{smallmatrix}s_1,\ldots,s_{k-1}\in\mathbb{N}\colon\\m\geqslant s_1\geqslant\ldots\geqslant s_{k-1} \end{smallmatrix}}\dfrac{q(m,s_1)}{n^{s_{k-1}}}\prod\limits_{i=1}^{k-2}q(s_i,s_{i+1})V^{\left\{k,m\right\}}_{s_1,\ldots,s_{k-1}}, \end{gather*}
where
\begin{gather*} V^{\left\{k,m\right\}}_{s_1,\ldots,s_{k-1}}=\mathcal{P}\{x\in H_{f_{\left[k\right]}}^{\left(t,l\right)}\bigm| D^{\left\{k\right\}}_{s_1,\ldots,s_{k-1},1}\left(y_1,\ldots,y_m\right),f_{\left[k\right]}\left(y_1\right)=x \}=\\ =\frac{1}{n}\prod\limits_{i=m+1}^{t+l+m-1}{\left( 1-\dfrac{i}{n} \right)}\prod\limits_{i=1}^{k-1}\prod\limits_{j=s_i+1}^{t+l+s_i-2}{\left( 1-\dfrac{j}{n} \right)}\sum\limits_{v=0}^{k-1}\prod\limits_{u=1}^v{\left( 1-\dfrac{t+l+s_u-1}{n} \right)}, \end{gather*}
\(H_f^{\left(t,l\right)}\) is \(t\)-th layer of cycles of length \(l\) in graph \(G_f, D^{\left\{k\right\}}_{s_1,\ldots,s_k}(y_1,\ldots,y_m)=\bigcap\limits_{i=1}^k \{|\{f_{\left[i\right]}(y_1),\ldots,f_{\left[i\right]}(y_m)\}|=s_i\} \), and \((n)_z=n(n-1)\dots(n-z+1)\). For any fixed \(x\in S\setminus S^\prime\) and for any \(r\in \{1,\ldots,n-1\}\), \(S^\prime\subseteq S\), \(|S^\prime|=r\), \(z\in \{1,\ldots,n\} \),
\begin{gather*} \mathcal{P}\{\tau_{f_{\left[k\right]}}(x)=z,\mathcal{R}_{f_{\left[k\right]}}(x)\cap S^\prime=\emptyset \}=\\ =\left(1-\left(1-\frac{z}{n}\right)\left( 1-\frac{z-1}{n} \right)^{k-1}\right)\left(\frac{\left(n\right)_{z-1}}{n^{z-1}} \right)^{k-1}\frac{\left(n\right)_{r+z}}{n^{z-1}\left(n\right)_{r+1}}, \end{gather*}
where \(\mathcal{R}_{f_{\left[k\right]}}(x)\) is the aperiodicity segment of vertex \(x\) in the graph of mapping \(f_{\left[k\right]}\), \(\tau_{f_{\left[k\right]}}(x)=\min\{ t\in \mathbb{N}\colon{f_{\left[k\right]}}^t(x)\in \{ x,{f_{\left[k\right]}}(x),\dots,{f_{\left[k\right]}}^{t-1}(x) \}\} \). For any fixed \(x,y\in S\), \(x\ne y\), and for any \(r\in\{1,\ldots,n\} \),
\[ \mathcal{P}\{y \in (f_{\left[k\right]})^{-r}(x)\}=\frac1n\left(1-\frac1{n-1}\sum\limits_{z\in Q_r\setminus\{1\}}\left(\dfrac{(n)_z}{n^z}\right)^k\right),\]
where \(Q_r=\{m\in \mathbb{N}\colon m|r\} \).Constructing all nonisomorphic supergraphs with isomorphism rejection.https://zbmath.org/1459.052132021-05-28T16:06:00+00:00"Kamil, I. A."https://zbmath.org/authors/?q=ai:kamil.ikhab-aabdzhuldzhabbar-kamil"Sudani, H. H. K."https://zbmath.org/authors/?q=ai:sudani.hayder-hussein-karim"Lobov, A. A."https://zbmath.org/authors/?q=ai:lobov.aleksandr-andreevich"Abrosimov, M. B."https://zbmath.org/authors/?q=ai:abrosimov.mikhail-borisovichSummary: An important trend in graph theory is the construction of graphs with given properties without directly checking for isomorphism. Programs that perform such constructions are called generators. Generators of undirected graphs with a given number of vertices, trees, regular graphs, bipartite graphs, tournaments, etc. are well known. A graph \(G = (V, \alpha)\) is a subgraph of a graph \(H = (W, \beta)\) if all vertices and edges of \(G\) belong to \(H\), that is, \(V \subseteq W\) and \(\alpha \subseteq \beta \). If \(G\) is a subgraph of \(H\), then \(H\) is a supergraph of \(G\). In researches on graph theory, often the properties of a graph are studied through some of its parts. The inverse problem also appears: to construct graphs with given part as subgraph. Such a problem occurs, for example, in the study of fault-tolerant design of discrete systems. The algorithm for constructing for a given graph all its supergraphs with a given number of edges without checking for isomorphism is described. A special matrix code (route or M-code) and an algorithm for generating supergraphs by the method of canonical representatives based on it are proposed. The concept of the method of canonical representatives is that one representative in each class of isomorphic graphs is selected and is called canonical. A canonical representative function (canonical form) is a function \(c\) with the properties that \(c(G) = c (H)\) if and only if \(G\) and \(H\) are isomorphic. The graph \(c(G)\) is called the canonical representative of the graph \(G\). We introduce a routing code (M-code) for the graph \(H\) relative to the graph \(G\) and denote \(C_G (H)\). Given the adjacency matrices of the graphs \(G\) and \(H\), construct the code \(C_G (H)\) in two steps: first, write out the elements of the adjacency matrix of the graph \(H\) at the corresponding positions of which there is \(1\) in the adjacency matrix of the graph \(G\), then those with a value of 0. For undirected graphs, only elements located above the main diagonal are written out. Optimization of the proposed method and issues related to its parallel implementation are discussed.On the minimal diameter of closed hyperbolic surfaces.https://zbmath.org/1459.052952021-05-28T16:06:00+00:00"Budzinski, Thomas"https://zbmath.org/authors/?q=ai:budzinski.thomas"Curien, Nicolas"https://zbmath.org/authors/?q=ai:curien.nicolas"Petri, Bram"https://zbmath.org/authors/?q=ai:petri.bramSummary: We prove that the minimal diameter of a closed orientable hyperbolic surface of genus \(g\) is asymptotic to \(\log g\) as \(g\to\infty\). The proof relies on a random construction, which we analyze using lattice-point counting theory and the exploration of random trivalent graphs.Spanning paths and cycles in triangle-free graphs.https://zbmath.org/1459.050432021-05-28T16:06:00+00:00"Mafuta, P."https://zbmath.org/authors/?q=ai:mafuta.phillip"Mushanyu, J."https://zbmath.org/authors/?q=ai:mushanyu.josiahSummary: Let \(G\) be a simple, connected, triangle-free graph with minimum degree \(\delta\) and leaf number \(L(G)\). We prove that if \(L(G) \leq 2\delta - 1\), then \(G\) is either Hamiltonian or \(G \in \mathcal{F}_2\), where \(\mathcal{F}_2\) is the class of non-Hamiltonian graphs with leaf number \(2 \delta - 1\). Further, if \(L(G) \leq 2\delta \), we show that \(G\) is traceable or \(G \in \mathcal{F}_3\). The results, apart from strengthening theorems in [\textit{P. Mafuta} et al., Acta Math. Hung. 152, No. 1, 217--226 (2017; Zbl 1389.05094); \textit{P. Mafuta} et al., Discrete Appl. Math. 255, 278--282 (2019; Zbl 1405.05088)] for this class of graphs, provide a sufficient condition for a triangle-free graph to be Hamiltonian or traceable based on leaf number and minimum degree.Injective coloring of generalized Petersen graphs.https://zbmath.org/1459.050842021-05-28T16:06:00+00:00"Li, Zepeng"https://zbmath.org/authors/?q=ai:li.zepeng"Shao, Zehui"https://zbmath.org/authors/?q=ai:shao.zehui"Zhu, Enqiang"https://zbmath.org/authors/?q=ai:zhu.enqiangSummary: An injective coloring of a graph is a vertex coloring where two vertices have distinct c olors if a path of length two exists between them. The injective chromatic number \(\chi_i(G)\) of a graph \(G\) is the smallest number \(k\) such that \(G\) admits an injective coloring with \(k\) colors. \textit{G. Hahn} et al. [Discrete Math. 256, No. 1--2, 179--192 (2002; Zbl 1007.05046)] proved that \(\Delta \leq \chi_i (G) \leq \Delta^2 - \Delta + 1\) for any graph \(G\), where \(\Delta\) is the maximum degree of \(G\). For a constant \(c \geq 0\), determining the injective chromatic number of which graphs is at most \(\Delta + c\) is an interesting problem. In this paper, we investigate the injective colorings of generalized Petersen graphs \(P(n, k)\). We prove that \(\chi_i (P(n, k)) \leq 5\) for any generalized Petersen graph \(P(n, k)\) and \(\chi_i (P(n, k)) = 3\) if \(n \equiv 0 \pmod 3\) and \(k \not\equiv 0\pmod 3\). Furthermore, we determine the precise injective chromatic numbers of \(P(n, 1)\) and \(P(n, 2)\).The discrete strategy improvement algorithm for parity games and complexity measures for directed graphs.https://zbmath.org/1459.680812021-05-28T16:06:00+00:00"Canavoi, Felix"https://zbmath.org/authors/?q=ai:canavoi.felix"Grädel, Erich"https://zbmath.org/authors/?q=ai:gradel.erich"Rabinovich, Roman"https://zbmath.org/authors/?q=ai:rabinovich.romanSummary: For some time the discrete strategy improvement algorithm due to Jurdzinski and Voge had been considered as a candidate for solving parity games in polynomial time. However, it has recently been proved by Oliver Friedmann that the strategy improvement algorithm requires super-polynomially many iteration steps, for all popular local improvements rules, including switch-all (also with Fearnley's snare memorisation), switch-best, random-facet, random-edge, switch-half, least-recently-considered, and Zadeh's Pivoting rule.
We analyse the examples provided by Friedmann in terms of complexity measures for directed graphs such as treewidth, DAG-width, Kelly-width, entanglement, directed pathwidth, and cliquewidth. It is known that for every class of parity games on which one of these parameters is bounded, the winning regions can be efficiently computed. It turns out that with respect to almost all of these measures, the complexity of Friedmann's counterexamples is bounded, and indeed in most cases by very small numbers. This analysis strengthens in some sense Friedmann's results and shows that the discrete strategy improvement algorithm is even more limited than one might have thought. Not only does it require super-polynomial running time in the general case, where the problem of polynomial-time solvability is open, it even has super-polynomial lower time bounds on natural classes of parity games on which efficient algorithms are known.
For the entire collection see [Zbl 1392.68016].Semigraphs.https://zbmath.org/1459.052172021-05-28T16:06:00+00:00"Sampathkumar, E."https://zbmath.org/authors/?q=ai:sampathkumar.eFrom the introduction: The notion of semigraph is a generalization of that of a graph. While generalizing a structure, one naturally looks for one in which every concept/idea
in the structure has a natural generalization. Semigraph is such a natural generalization of graph, and it resembles graph when drawn in a plane.
Semigraphs are defined, illustrated by a number of examples. We have a
variety of definitions of each concept like adjacency, degrees etc. In fact, the
beauty of semigraphs lies in the variety of definitions/concepts, all of which
coincide for graphs.
For the entire collection see [Zbl 1437.05006].Bounded degree and planar spectra.https://zbmath.org/1459.030362021-05-28T16:06:00+00:00"Dawar, Anuj"https://zbmath.org/authors/?q=ai:dawar.anuj"Kopczyński, Eryk"https://zbmath.org/authors/?q=ai:kopczynski.erykSummary: The finite spectrum of a first-order sentence is the set of positive integers that are the sizes of its models. The class of finite spectra is known to be the same as the complexity class NE. We consider the spectra obtained by limiting models to be either planar (in the graph-theoretic sense) or by bounding the degree of elements. We show that the class of such spectra is still surprisingly rich by establishing that significant fragments of NE are included among them. At the same time, we establish non-trivial upper bounds showing that not all sets in NE are obtained as planar or bounded-degree spectra.Fast unfolding of communities in large networks.https://zbmath.org/1459.911302021-05-28T16:06:00+00:00"Blondel, Vincent D."https://zbmath.org/authors/?q=ai:blondel.vincent-d"Guillaume, Jean-Loup"https://zbmath.org/authors/?q=ai:guillaume.jean-loup"Lambiotte, Renaud"https://zbmath.org/authors/?q=ai:lambiotte.renaud"Lefebvre, Etienne"https://zbmath.org/authors/?q=ai:lefebvre.etienneThe percolation transition in correlated hypergraphs.https://zbmath.org/1459.821142021-05-28T16:06:00+00:00"Bradde, Serena"https://zbmath.org/authors/?q=ai:bradde.serena"Bianconi, Ginestra"https://zbmath.org/authors/?q=ai:bianconi.ginestraMost graphs are knotted.https://zbmath.org/1459.053022021-05-28T16:06:00+00:00"Ichihara, Kazuhiro"https://zbmath.org/authors/?q=ai:ichihara.kazuhiro"Mattman, Thomas W."https://zbmath.org/authors/?q=ai:mattman.thomas-wUnknotting numbers and crossing numbers of spatial embeddings of a planar graph.https://zbmath.org/1459.570032021-05-28T16:06:00+00:00"Akimoto, Yuta"https://zbmath.org/authors/?q=ai:akimoto.yuta"Taniyama, Kouki"https://zbmath.org/authors/?q=ai:taniyama.koukiRandom tree growth by vertex splitting.https://zbmath.org/1459.052992021-05-28T16:06:00+00:00"David, F."https://zbmath.org/authors/?q=ai:david.filippo|david.florin|david.francois"Dukes, W. M. B."https://zbmath.org/authors/?q=ai:dukes.w-m-b"Jonsson, T."https://zbmath.org/authors/?q=ai:jonsson.thordur"Stefánsson, S. Ö."https://zbmath.org/authors/?q=ai:stefansson.sigurdur-ornPersistent homology of complex networks.https://zbmath.org/1459.550042021-05-28T16:06:00+00:00"Horak, Danijela"https://zbmath.org/authors/?q=ai:horak.danijela"Maletić, Slobodan"https://zbmath.org/authors/?q=ai:maletic.slobodan"Rajković, Milan"https://zbmath.org/authors/?q=ai:rajkovic.milanOn subgroup perfect codes in Cayley graphs.https://zbmath.org/1459.051262021-05-28T16:06:00+00:00"Zhang, Junyang"https://zbmath.org/authors/?q=ai:zhang.junyang"Zhou, Sanming"https://zbmath.org/authors/?q=ai:zhou.sanmingSummary: A perfect code in a graph \(\Gamma=(V,E)\) is a subset \(C\) of \(V\) such that no two vertices in \(C\) are adjacent and every vertex in \(V\setminus C\) is adjacent to exactly one vertex in \(C\). A subgroup \(H\) of a group \(G\) is called a subgroup perfect code of \(G\) if there exists a Cayley graph of \(G\) which admits \(H\) as a perfect code. Equivalently, \(H\) is a subgroup perfect code of \(G\) if there exists an inverse-closed subset \(A\) of \(G\) containing the identity element such that \((A,H)\) is a tiling of \(G\) in the sense that every element of \(G\) can be uniquely expressed as the product of an element of \(A\) and an element of \(H\). In this paper we obtain multiple results on subgroup perfect codes of finite groups, including a few necessary and sufficient conditions for a subgroup of a finite group to be a subgroup perfect code, a few results involving 2-subgroups in the study of subgroup perfect codes, and several results on subgroup perfect codes of metabelian groups, generalized dihedral groups, nilpotent groups and 2-groups.Cut-edges and regular factors in regular graphs of odd degree.https://zbmath.org/1459.052652021-05-28T16:06:00+00:00"Kostochka, Alexandr V."https://zbmath.org/authors/?q=ai:kostochka.alexandr-v"Raspaud, André"https://zbmath.org/authors/?q=ai:raspaud.andre"Toft, Bjarne"https://zbmath.org/authors/?q=ai:toft.bjarne"West, Douglas B."https://zbmath.org/authors/?q=ai:west.douglas-b"Zirlin, Dara"https://zbmath.org/authors/?q=ai:zirlin.daraSummary: We study \(2k\)-factors in \((2r+1)\)-regular graphs. \textit{D. Hanson} et al. [Ars Comb. 50, 23--32 (1998; Zbl 0963.05049)] proved that every \((2r+1)\)-regular graph with at most \(2r\) cut-edges has a 2-factor. We generalize their result by proving for \(k\le (2r+1)/3\) that every \((2r+1)\)-regular graph with at most \(2r-3(k-1)\) cut-edges has a \(2k\)-factor. Both the restriction on \(k\) and the restriction on the number of cut-edges are sharp. We characterize the graphs that have exactly \(2r-3(k-1)+1\) cut-edges but no \(2k\)-factor. For \(k>(2r+1)/3\), there are graphs without cut-edges that have no \(2k\)-factor, as studied by \textit{B. Bollobás} et al. [J. Graph Theory 9, No. 1, 97--103 (1985; Zbl 0612.05049)].Mean-field theory of graph neural networks in graph partitioning.https://zbmath.org/1459.053192021-05-28T16:06:00+00:00"Kawamoto, Tatsuro"https://zbmath.org/authors/?q=ai:kawamoto.tatsuro"Tsubaki, Masashi"https://zbmath.org/authors/?q=ai:tsubaki.masashi"Obuchi, Tomoyuki"https://zbmath.org/authors/?q=ai:obuchi.tomoyukiArithmetical structures on paths with a doubled edge.https://zbmath.org/1459.052862021-05-28T16:06:00+00:00"Glass, Darren"https://zbmath.org/authors/?q=ai:glass.darren-b"Wagner, Joshua"https://zbmath.org/authors/?q=ai:wagner.joshuaSummary: An arithmetical structure on a graph is given by a labeling of the vertices that satisfies certain divisibility properties. In this note, we look at several families of graphs and attempt to give counts on the number of arithmetical structures for graphs in these families.Fast hyperbolic mapping based on the hierarchical community structure in complex networks.https://zbmath.org/1459.650592021-05-28T16:06:00+00:00"Wang, Zuxi"https://zbmath.org/authors/?q=ai:wang.zuxi"Sun, Lingjie"https://zbmath.org/authors/?q=ai:sun.lingjie"Cai, Menglin"https://zbmath.org/authors/?q=ai:cai.menglin"Xie, Pengcheng"https://zbmath.org/authors/?q=ai:xie.pengchengPushable chromatic number of graphs with degree constraints.https://zbmath.org/1459.050722021-05-28T16:06:00+00:00"Bensmail, Julien"https://zbmath.org/authors/?q=ai:bensmail.julien"Das, Sandip"https://zbmath.org/authors/?q=ai:das.sandip"Nandi, Soumen"https://zbmath.org/authors/?q=ai:nandi.soumen"Paul, Soumyajit"https://zbmath.org/authors/?q=ai:paul.soumyajit"Pierron, Théo"https://zbmath.org/authors/?q=ai:pierron.theo"Sen, Sagnik"https://zbmath.org/authors/?q=ai:sen.sagnik"Sopena, Éric"https://zbmath.org/authors/?q=ai:sopena.ericAn oriented graph \(\overset{\rightarrow}{G}\) is a loopless digraph withour opposite edges. The oriented chromatic number of \(\overset{\rightarrow}{G}\), denoted \(\chi_o(\overset{\rightarrow}{G})\), is the minimum order of an orientged graph \(\overset{\rightarrow}{H}\) such that there exists a homomorphism \(\overset{\rightarrow}{G}\rightarrow\overset{\rightarrow}{H}\). Two graphs \(\overset{\rightarrow}{G}\) and \(\overset{\rightarrow}{G^\prime}\) are in push relation if one can be transformed into the other by pushing some vertices, where pushing a vertex means changing the orientation of all the arcs incident to it. The pushable chromatic number of \(\overset{\rightarrow}{G}\), denoted \(\chi_p(\overset{\rightarrow}{G})\), is the minimum order of an orientged graph \(\overset{\rightarrow}{H}\) such that there exists a homomorphism \(\overset{\rightarrow}{G^\prime}\rightarrow\overset{\rightarrow}{H}\) for some \(\overset{\rightarrow}{G^\prime}\) being in push relation with \(\overset{\rightarrow}{G}\). It has been known that \(\chi_p(\overset{\rightarrow}{G})\leq\chi_o(\overset{\rightarrow}{G})\leq2\chi_p(\overset{\rightarrow}{G})\) for every oriented graph \(\overset{\rightarrow}{G}\). The authors consider the degree-constrained families of graphs and prove in particular that for every connected graph \(\overset{\rightarrow}{G}\) with \(\Delta\geq 29\), \(2^{\Delta/2-1}\leq \chi_p(\overset{\rightarrow}{G})\leq (\Delta-3)(\Delta-1)2^{\Delta-1}+2\) and \(2^{\Delta/2}\leq \chi_o(\overset{\rightarrow}{G})\leq (\Delta-3)(\Delta-1)2^{\Delta}+2\). For graphs with maximum degree \(\Delta=3\), one has \(6\leq \chi_p(\overset{\rightarrow}{G})\leq 7\), while in the case of graphs with maximum average degree \(\operatorname{mad}(G)<3\), \(5\leq \chi_p(\overset{\rightarrow}{G})\leq 7\) holds. Finally, planar graphs with girth at leasy \(6\) satisfy \(4\leq \chi_p(\overset{\rightarrow}{G})\leq 7\).
Reviewer: Marcin Anholcer (Poznań)Probabilistic zero forcing on random graphs.https://zbmath.org/1459.053012021-05-28T16:06:00+00:00"English, Sean"https://zbmath.org/authors/?q=ai:english.sean"MacRury, Calum"https://zbmath.org/authors/?q=ai:macrury.calum"Prałat, Paweł"https://zbmath.org/authors/?q=ai:pralat.pawelThis paper studies probabilistic zero forcing on ER random graphs. Given a graph \(G\) and a set of blue vertices \(Z\) in \(G\), the process of zero forcing involves a coloring process in which a blue vertex \(u\) formes a white vertex \(v\) to become blue if \(v\) is the only neighbor of \(u\) that is white. In probabilistic zero forcing of forcing to become blue is given by \(|N(u)\cap Z|/d_u\), where \(d_u\) is the degree of \(u\) and \(N(u)\) is the neighborhood. Let \(\operatorname{pt}(G,Z)\) be the propagation time of a probabilistic zero forcing with initial blue set \(Z\). If \(pn\gg \ln n\) then for any vertex \(v\) in the random graph \(G(n,p)\), it is shown that almost surely \(\operatorname{pt}(G(n,p),v)\le(1+o(1))(\log_2\log_2 n+\log_3(1/p))\) and \(\operatorname{pt}(G(n,p),v)\ge(1+o(1))\max(\log_2\log_2 n, \log_4(1/p))\).
Reviewer: Yilun Shang (Newcastle)Deciding the on-line chromatic number of a graph with pre-coloring is PSPACE-complete.https://zbmath.org/1459.680782021-05-28T16:06:00+00:00"Kudahl, Christian"https://zbmath.org/authors/?q=ai:kudahl.christianSummary: In an on-line coloring, the vertices of a graph are revealed one by one. An algorithm assigns a color to each vertex after it is revealed. When a vertex is revealed, it is also revealed which of the previous vertices it is adjacent to. The on-line chromatic number of a graph, \(G\), is the smallest number of colors an algorithm will need when on-line-coloring \(G\). The algorithm may know \(G\), but not the order in which the vertices are revealed. The problem of determining if the on-line chromatic number of a graph is less than or equal to \(k\), given a pre-coloring, is shown to be PSPACE-complete.
For the entire collection see [Zbl 1316.68024].An adaptive upper bound on the Ramsey numbers \(R(3,\dots,3)\).https://zbmath.org/1459.051992021-05-28T16:06:00+00:00"Eliahou, Shalom"https://zbmath.org/authors/?q=ai:eliahou.shalomSummary: Since [\textit{X. Xu} et al., ``Upper bounds for Ramsey numbers \(R_n(3)\) and Schur numbers'', Math. Econ. 19, No. 1, 81--84 (2002)], the best known upper bound on the Ramsey numbers \(R_n(3) = R(3,\dots, 3)\) is \(R_n(3) \le n!(e - 1/6) + 1\) for all \(n \ge 4\). It is based on the current estimate \(R_4(3) \le 62\). We show here how any closing-in on \(R_4(3)\) yields an improved upper bound on \(R_n(3)\) for all \(n \ge 4\). For instance, with our present adaptive bound, the conjectured value \(R_4(3) = 51\) implies \(R_n(3) \le n!(e - 5/8) + 1\) for all \(n \ge 4\).Encoding labelled \(p\)-Riordan graphs by words and pattern-avoiding permutations.https://zbmath.org/1459.050022021-05-28T16:06:00+00:00"Iamthong, Kittitat"https://zbmath.org/authors/?q=ai:iamthong.kittitat"Jung, Ji-Hwan"https://zbmath.org/authors/?q=ai:jung.ji-hwan"Kitaev, Sergey"https://zbmath.org/authors/?q=ai:kitaev.sergeySummary: The notion of a \(p\)-Riordan graph generalizes that of a Riordan graph, which, in turn, generalizes the notions of a Pascal graph and a Toeplitz graph. In this paper we introduce the notion of a \(p\)-Riordan word, and show how to encode \(p\)-Riordan graphs by \(p\)-Riordan words. For special important cases of Riordan graphs (the case \(p=2)\) and oriented Riordan graphs (the case \(p=3)\) we provide alternative encodings in terms of pattern-avoiding permutations and certain balanced words, respectively. As a bi-product of our studies, we provide an alternative proof of a known enumerative result on closed walks in the cube.Rainbow numbers for \(x_1+x_2=kx_3\) in \(\mathbb{Z}_n\).https://zbmath.org/1459.053272021-05-28T16:06:00+00:00"Bevilacqua, Erin"https://zbmath.org/authors/?q=ai:bevilacqua.erin"King, Samuel"https://zbmath.org/authors/?q=ai:king.samuel"Kritschgau, Jürgen"https://zbmath.org/authors/?q=ai:kritschgau.jurgen"Tait, Michael"https://zbmath.org/authors/?q=ai:tait.michael"Tebon, Suzannah"https://zbmath.org/authors/?q=ai:tebon.suzannah"Young, Michael"https://zbmath.org/authors/?q=ai:young.michael-e|young.michael-jSummary: In this work, we investigate the fewest number of colors needed to guarantee a rainbow solution to the equation \(x_1+x_2=kx_3\) in \(\mathbb{Z}_n\). This value is called the rainbow number and is denoted by \(\text{rb}(\mathbb{Z}_n, k)\) for positive integer values of \(n\) and \(k\). We find that \(\text{rb}(\mathbb{Z}_p,1) = 4\) for all primes greater than 3 and that \(\text{rb}(\mathbb{Z}_n,1)\) can be determined from the prime factorization of \(n\). Furthermore, when \(k\) is prime, \(rb(\mathbb{Z}_n, k)\) can be determined from the prime factorization of \(n\).Algorithm of multidimensional data transmission using extremal uniform hypergraphs.https://zbmath.org/1459.940032021-05-28T16:06:00+00:00"Egorova, E. K."https://zbmath.org/authors/?q=ai:egorova.e-k"Mokryakov, A. V."https://zbmath.org/authors/?q=ai:mokryakov.a-v"Suvorova, A. A."https://zbmath.org/authors/?q=ai:suvorova.a-a"Tsurkov, V. I."https://zbmath.org/authors/?q=ai:tsurkov.vladimir-iSummary: Recently, unmanned aerial vehicles are being used more often for reconnaissance in combat conditions. Communication with unmanned aerial vehicles must be carried out in a confidential mode to protect against the interception of control, while being fast and resistant to attacks from outside. A new algorithm for transferring confidential data, based on the properties of extremal uniform hypergraphs, is proposed.A characterization of the sum and integral sum labellings of some classes of graphs.https://zbmath.org/1459.052852021-05-28T16:06:00+00:00"Federico Elizeche, Edgar"https://zbmath.org/authors/?q=ai:elizeche.edgar-federico"Tripathi, Amitabha"https://zbmath.org/authors/?q=ai:tripathi.amitabhaSummary: A finite simple graph \(G\) is called a sum graph (respectively, integral sum graph) if there is a bijection \(f\) from the vertices of \(G\) to a set of positive integers (respectively, integers) \(S\) such that \(uv\) is an edge of \(G\) if and only if \(f(u)+f(v) \in S\). For a connected graph \(G\), the sum number (respectively, integral sum number) of \(G\), \(\sigma (G)\) (respectively, \(\zeta(G)\)), is the minimum number of isolated vertices that must be added to \(G\) so that the resulting graph is a sum graph (respectively, integral sum graph). The spum (respectively, the integral spum) of a graph \(G\) is the minimum difference between the largest and smallest integer in any set \(S\) that corresponds to a sum graph (respectively, integral sum graph) containing \(G\). The integral radius of a graph \(G\) is the minimum \(r=r(G)\) for which there exists an integral labelling lying in the interval \([-r, r]\). We characterize sum and integral sum labellings of complete graphs, symmetric complete bipartite graphs and star graphs, and deduce the spum, integral spum, and integral radius for these classes of graphs.Problems and invariants connected with bicliques and multicliques of graphs.https://zbmath.org/1459.052412021-05-28T16:06:00+00:00"Lepin, V. V."https://zbmath.org/authors/?q=ai:lepin.v-v"Duginov, O. I."https://zbmath.org/authors/?q=ai:duginov.o-iSummary: It is given a survey of selected results on graph problems and invariants related to bicliques (complete bipartite subgraphs) and multicliques (complete multipartite subgraphs) of graphs. Besides, new heuristics for solving the minimum biclique cover problem are given.On biclique covering number of the Cartesian product of graphs.https://zbmath.org/1459.052682021-05-28T16:06:00+00:00"Lepin, V. V."https://zbmath.org/authors/?q=ai:lepin.v-v"Duginov, O. I."https://zbmath.org/authors/?q=ai:duginov.o-iSummary: The paper is dealt with the biclique cover number (i.e. minimal number of complete bipartite subgraphs of a graph needed to cover the edge set of the graph) of the Cartesian product of two graphs. It is obtained upper bounds on the biclique cover number for the Cartesian product of graphs. It is given the formula for exact value of the biclique cover number for the Cartesian product of \(P_n\) and \(K_2\), \(C_n\) and \(K_2\), \(P_n\) and \(P_n\).Linear recurrences for cylindrical networks.https://zbmath.org/1459.051042021-05-28T16:06:00+00:00"Galashin, Pavel"https://zbmath.org/authors/?q=ai:galashin.pavel"Pylyavskyy, Pavlo"https://zbmath.org/authors/?q=ai:pylyavskyy.pavloSummary: We prove a general theorem that gives a linear recurrence for tuples of paths in every cylindrical network. This can be seen as a cylindrical analog of the Lindström-Gessel-Viennot theorem. We illustrate the result by applying it to Schur functions, plane partitions, and domino tilings.\(H\)-coverings of path-amalgamated ladders and fans.https://zbmath.org/1459.052742021-05-28T16:06:00+00:00"Xiong, Yijun"https://zbmath.org/authors/?q=ai:xiong.yijun"Wang, Huajun"https://zbmath.org/authors/?q=ai:wang.huajun"Umar, Muhammad Awais"https://zbmath.org/authors/?q=ai:umar.muhammad-awais"Chu, Yu-Ming"https://zbmath.org/authors/?q=ai:chu.yuming"Ali, Basharat Rehman"https://zbmath.org/authors/?q=ai:ali.basharat-rehman"Naseem, Maria"https://zbmath.org/authors/?q=ai:naseem.mariaSummary: Let \(\mathbb{G}\) be a connected, simple graph with finite vertices \(v\) and edges \(e\). A family \(\left\{ \mathbb{G}_1, \mathbb{G}_2, \ldots, \mathbb{G}_p\right\}\subset\mathbb{G}\) of subgraphs such that for all \(e\in E\), \(e\in \mathbb{G}_l\), for some \(l\), \(l=1,2,\ldots,p\) is an edge-covering of \(\mathbb{G} \). If \(\mathbb{G}_l\cong\mathbb{H}\), \(\forall l \), then \(\mathbb{G}\) has an \(\mathbb{H} \)-covering. Graph \(\mathbb{G}\) with \(\mathbb{H} \)-covering is an \(\left( a_d, d\right)\)-\(\mathbb{H} \)-antimagic if \(\psi:V\left( \mathbb{G}\right)\cup E\left( \mathbb{G}\right)\longrightarrow\left\{ 1,2, \ldots, v + e\right\}\) a bijection exists and the sum over all vertex-weights and edge-weights of \(\mathbb{H}\) forms a set \(\left\{ a_d, a_d + d, \ldots, a_d + \left( p - 1\right) d\right\}\). The labeling \(\psi\) is super for \(\psi\left( V \left( \mathbb{G}\right)\right)=\left\{ 1,2,3, \ldots, v\right\}\) and graph \(\mathbb{G}\) is \(\mathbb{H} \)-supermagic for \(d=0\). This manuscript proves results about super \(\mathbb{H} \)-antimagic labeling of path amalgamation of ladders and fans for several differences.A graph polynomial for independent sets of Fibonacci trees.https://zbmath.org/1459.051302021-05-28T16:06:00+00:00"Sreeja, K. U."https://zbmath.org/authors/?q=ai:sreeja.k-u"Vinodkumar, P. B."https://zbmath.org/authors/?q=ai:vinodkumar.p-b"Ramkumar, P. B."https://zbmath.org/authors/?q=ai:ramkumar.p-bA set \(S \subseteq V(G)\) of vertices of a graph \(G\) is called an independent set if no two vertices in \(S\) are adjacent. The independence polynomial of a graph \(G\) is then defined as
\[I(G,x) = \sum_{k \geq 0} s_k(G),\]
where the number \(s_k(G)\) denotes the number of independent sets of size \(k\) (note that \(s_0(G)=1\)).
The Fibonacci trees \(F_n\), \(n \geq 0\), were introduced in [\textit{S. G. Wagner}, Fibonacci Q. 45, No. 3, 247--253 (2007; Zbl 1155.05005)] as follows: \(F_0 = K_1\) (with root \(r_0\)) and \(F_1 = K_2\) (with root \(r_1\)). Then, for \(n \geq 2\), the Fibonacci tree \(F_n\) is obtained by adding a new vertex \(r_n\) (root of \(F_n\)) to the disjoint union of graphs \(F_{n-1}\) and \(F_{n-2}\) and by adding two edges between \(r_n\) and the roots of \(F_{n-1}\) and \(F_{n-2}\).
In the present paper, the authors describe a recursive formula for computing the independence polynomial for Fibonacci trees. Next, the roots of independence polynomials of Fibonacci trees are investigated. It is shown that \(-1\) is one of the roots of the independence polynomial of Fibonacci tree \(F_n\), \(n \geq 2\). Moreover, the authors investigate some independence polynomials of Fibonacci trees that are stable (which means that all its roots lie in the open left half-plane) and also describe some rational roots of independence polynomials. In addition, some other questions are discussed.
Reviewer: Niko Tratnik (Maribor)The 3-good-neighbor connectivity of modified bubble-sort graphs.https://zbmath.org/1459.051432021-05-28T16:06:00+00:00"Wang, Yanling"https://zbmath.org/authors/?q=ai:wang.yanling"Wang, Shiying"https://zbmath.org/authors/?q=ai:wang.shiyingSummary: Let \(G=\left( V \left( G\right), E \left( G\right)\right)\) be a connected graph. A subset \(F\subseteq V\left( G\right)\) is called a \(g\)-good-neighbor cut if \(G-F\) is disconnected and each vertex of \(G-F\) has at least \(g\) neighbors. The \(g\)-good-neighbor connectivity of \(G\) is the minimum cardinality of \(g\)-good-neighbor cuts. The \(n\)-dimensional modified bubble-sort graph \(\text{MB}_n\) is a special Cayley graph. It has many good properties. In this paper, we prove that the 3-good-neighbor connectivity of \(\mathrm{MB}_n\) is \(8n-24\) for \(n\geq6\).An existence criterion of a non-crossing spanning tree in the geometric complement of a convex spanning tree.https://zbmath.org/1459.050412021-05-28T16:06:00+00:00"Benediktovich, V. I."https://zbmath.org/authors/?q=ai:benediktovich.vladimir-iSummary: In this article an existence criterion of a non-crossing spanning tree in the geometric complement of a convex spanning tree has been obtained.Decision-making analysis based on fuzzy graph structures.https://zbmath.org/1459.901172021-05-28T16:06:00+00:00"Koam, Ali N. A."https://zbmath.org/authors/?q=ai:koam.ali-n-a"Akram, Muhammad"https://zbmath.org/authors/?q=ai:akram.muhammad"Liu, Peide"https://zbmath.org/authors/?q=ai:liu.peide|liu.peide.1Summary: A graph structure is a useful framework to solve the combinatorial problems in various fields of computational intelligence systems and computer science. In this research article, the concept of fuzzy sets is applied to the graph structure to define certain notions of fuzzy graph structures. Fuzzy graph structures can be very useful in the study of various structures, including fuzzy graphs, signed graphs, and the graphs having labeled or colored edges. The notions of the fuzzy graph structure, lexicographic-max product, and degree and total degree of a vertex in the lexicographic-max product are introduced. Further, the proposed concepts are explained through several numerical examples. In particular, applications of the fuzzy graph structures in decision-making process, regarding detection of marine crimes and detection of the road crimes, are presented. Finally, the general procedure of these applications is described by an algorithm.Regularity lemma for distal structures.https://zbmath.org/1459.030412021-05-28T16:06:00+00:00"Chernikov, Artem"https://zbmath.org/authors/?q=ai:chernikov.artem"Starchenko, Sergei"https://zbmath.org/authors/?q=ai:starchenko.sergeiSummary: It is known that families of graphs with a semialgebraic edge relation of bounded complexity satisfy much stronger regularity properties than arbitrary graphs, and can be decomposed into very homogeneous semialgebraic pieces up to a small error (see e.g. [\textit{J. Pach} and \textit{J. Solymosi}, J. Comb. Theory, Ser. A 96, No. 2, 316--325 (2001; Zbl 0989.05031); \textit{N. Alon} et al., J. Comb. Theory, Ser. A 111, No. 2, 310--326 (2005; Zbl 1099.14048); \textit{J. Fox} et al., J. Reine Angew. Math. 671, 49--83 (2012; Zbl 1306.05171); \textit{J. Fox} et al., SIAM J. Comput. 45, No. 6, 2199--2223 (2016; Zbl 1353.05090)]). We show that similar results can be obtained for families of graphs with the edge relation uniformly definable in a structure satisfying a certain model-theoretic property called distality, with respect to a large class of generically stable measures. Moreover, distality characterizes these strong regularity properties. This applies in particular to graphs definable in arbitrary o-minimal structures and in \(p\)-adics.Robustness analysis of urban road networks from topological and operational perspectives.https://zbmath.org/1459.900672021-05-28T16:06:00+00:00"Shang, Wen-Long"https://zbmath.org/authors/?q=ai:shang.wen-long"Chen, Yanyan"https://zbmath.org/authors/?q=ai:chen.yanyan"Song, Chengcheng"https://zbmath.org/authors/?q=ai:song.chengcheng"Ochieng, Washington Y."https://zbmath.org/authors/?q=ai:ochieng.washington-ySummary: This study comprehensively analyses the robustness of urban road networks through topological indices based on the complex network theory and operational indices based on traffic assignment theory: User Equilibrium (UE), System Optimum (SO), and Price of Anarchy (POA). Analysing topological indices may pin down the most important nodes for URNs from the perspective of connectivity, while more sophisticated operational indices are helpful to examine the importance of nodes for URNs by taking into account link capacity, travel demand, and drivers' behaviour. The previous way is calculated in a static way, which reduces the computation times and increases the efficiency for quick assessment of the robustness of URNs, while the latter is in a dynamic way, namely, calculating is based on removal of individual nodes, although this way is more likely to capture realistic meanings but consumes huge amount of time. The efforts made in this study try to find the relationship between topological and operational indices so as to assist the assessment of robustness of URNs to local disruptions. Seven realistic urban road networks such as Sioux Falls and Anaheim are used as network examples, and results show that different indices reflect robustness characteristics of urban road networks from different ways, and rank correlations between any two indices are poor although small network such as Sioux Falls have better correlations than others.Several asymptotic bounds on the Balaban indices of trees.https://zbmath.org/1459.050492021-05-28T16:06:00+00:00"Deng, Bo"https://zbmath.org/authors/?q=ai:deng.bo"Ye, Chengfu"https://zbmath.org/authors/?q=ai:ye.chengfu"Liang, Weilin"https://zbmath.org/authors/?q=ai:liang.weilin"Li, Yalan"https://zbmath.org/authors/?q=ai:li.yalan"Su, Xueli"https://zbmath.org/authors/?q=ai:su.xueliSummary: The Balaban index (also called the \(J\) index) of a connected graph \(G\) is a distance-based topological index, which has been successfully used in various QSAR and QSPR modeling. Although the index was introduced 30 years ago, there are few results on the asymptotic relations. In this paper, several asymptotic bounds on the Balaban indices of trees with diameters 3 and 4 are shown, respectively.Vertex-disjoint paths in a 3-ary \(n\)-cube with faulty vertices.https://zbmath.org/1459.051372021-05-28T16:06:00+00:00"Ma, Xiaolei"https://zbmath.org/authors/?q=ai:ma.xiaolei"Wang, Shiying"https://zbmath.org/authors/?q=ai:wang.shiyingSummary: The construction of vertex-disjoint paths (disjoint paths) is an important research topic in various kinds of interconnection networks, which can improve the transmission rate and reliability. The \(k\)-ary \(n\)-cube is a family of popular networks. In this paper, we determine that there are \(m\) (\( 2 \leq m \leq n\)) disjoint paths in 3-ary \(n\)-cube covering \(Q_n^3-F\) from \(S\) to \(T\) (many-to-many) with \(\left| F\right|\leq2n-2m\) and from \(s\) to \(T\) (one-to-many) with \(\left| F\right|\leq2n-m-1\) where \(s\) is in a fault-free cycle of length three.Fault-tolerant metric dimension of generalized wheels and convex polytopes.https://zbmath.org/1459.050702021-05-28T16:06:00+00:00"Zheng, Zhi-Bo"https://zbmath.org/authors/?q=ai:zheng.zhibo"Ahmad, Ashfaq"https://zbmath.org/authors/?q=ai:ahmad.ashfaq"Hussain, Zaffar"https://zbmath.org/authors/?q=ai:hussain.zaffar"Munir, Mobeen"https://zbmath.org/authors/?q=ai:munir.mobeen"Qureshi, Muhammad Imran"https://zbmath.org/authors/?q=ai:qureshi.muhammad-imran.1"Ali, Imtiaz"https://zbmath.org/authors/?q=ai:ali.imtiaz"Liu, Jia-Bao"https://zbmath.org/authors/?q=ai:liu.jia-baoSummary: For a graph \(G\), an ordered set \(S\subseteq V\left( G\right)\) is called the resolving set of \(G\), if the vector of distances to the vertices in \(S\) is distinct for every \(v\in V\left( G\right)\). The minimum cardinality of \(S\) is termed as the metric dimension of \(G\). \(S\) is called a fault-tolerant resolving set (FTRS) for \(G\), if \(S\backslash\left\{ v\right\}\) is still the resolving set \(\forall v\in V\left( G\right)\). The minimum cardinality of such a set is the fault-tolerant metric dimension (FTMD) of \(G\). Due to enormous application in science such as mathematics and computer, the notion of the resolving set is being widely studied. In the present article, we focus on determining the FTMD of a generalized wheel graph. Moreover, a formula is developed for FTMD of a wheel and generalized wheels. Recently, some bounds of the FTMD of some of the convex polytopes have been computed, but here we come up with the exact values of the FTMD of two families of convex polytopes denoted as \(D_k\) for \(k\geq4\) and \(Q_k\) for \(k\geq6\). We prove that these families of convex polytopes have constant FTMD. This brings us to pose a natural open problem about the existence of a polytope having nonconstant FTMD.Concepts on coloring of cluster hypergraphs with application.https://zbmath.org/1459.050912021-05-28T16:06:00+00:00"Samanta, Sovan"https://zbmath.org/authors/?q=ai:samanta.sovan"Lee, Jeong Gon"https://zbmath.org/authors/?q=ai:lee.jeong-gon"Naseem, Usman"https://zbmath.org/authors/?q=ai:naseem.usman"Khan, Shah Khalid"https://zbmath.org/authors/?q=ai:khan.shah-khalid"Das, Kousik"https://zbmath.org/authors/?q=ai:das.kousikSummary: Coloring of graph theory is widely used in different fields like the map coloring, traffic light problems, etc. Hypergraphs are an extension of graph theory where edges contain single or multiple vertices. This study analyzes cluster hypergraphs where cluster vertices too contain simple vertices. Coloring of cluster networks where composite/cluster vertices exist is done using the concept of coloring of cluster hypergraphs. Proper coloring and strong coloring of cluster hypergraphs have been defined. Along with these, local coloring in cluster hypergraphs is also provided. Such a cluster network, COVID19 affected network, is assumed and colored to visualize the affected regions properly.Hyperbolic and parabolic unimodular random maps.https://zbmath.org/1459.600182021-05-28T16:06:00+00:00"Angel, Omer"https://zbmath.org/authors/?q=ai:angel.omer"Hutchcroft, Tom"https://zbmath.org/authors/?q=ai:hutchcroft.tom"Nachmias, Asaf"https://zbmath.org/authors/?q=ai:nachmias.asaf"Ray, Gourab"https://zbmath.org/authors/?q=ai:ray.gourabSummary: We show that for infinite planar unimodular random rooted maps. many global geometric and probabilistic properties are equivalent, and are determined by a natural, local notion of average curvature. This dichotomy includes properties relating to amenability, conformal geometry, random walks, uniform and minimal spanning forests, and Bernoulli bond percolation. We also prove that every simply connected unimodular random rooted map is sofic, that is, a Benjamini-Schramm limit of finite maps.Non-equilibrium opinion spreading on 2D small-world networks.https://zbmath.org/1459.911312021-05-28T16:06:00+00:00"Candia, Julián"https://zbmath.org/authors/?q=ai:candia.julianCounting near-perfect matchings on \(C_m \times C_n\) tori of odd order in the Maple system.https://zbmath.org/1459.051272021-05-28T16:06:00+00:00"Perepechko, S. N."https://zbmath.org/authors/?q=ai:perepechko.s-nSummary: In the Maple computer algebra system, a set of recurrence relations and associated generating functions is derived for the number of near-perfect matchings on \({{C}_m} \times{{C}_n}\) tori of odd order at fixed values of the parameter \(m\) \((3 \leqslant m \leqslant 11)\). The identity of the recurrence relations for the number of perfect and near-perfect matchings is revealed for the same value of \(m\). An estimate for the number of near-perfect matchings is obtained at large odd \(m\) when \(n \to \infty \).New lower bound on the modularity of Johnson graphs.https://zbmath.org/1459.052642021-05-28T16:06:00+00:00"Koshelev, Mikhail"https://zbmath.org/authors/?q=ai:koshelev.mikhailSummary: The modularity of a graph is a value that shows how well the graph can be split into clusters. It is a key part in many clustering algorithms. In this paper, we improve the lower bound on the modularity of Johnson graphs significantly.Exact modularity of line graphs of complete graphs.https://zbmath.org/1459.052812021-05-28T16:06:00+00:00"Ipatov, Mikhail"https://zbmath.org/authors/?q=ai:ipatov.mikhailSummary: Modularity is designed to measure the strength of a division of a network into clusters. For \(n \in \mathbb{N} \), consider a set \(S\) of \(n\) elements. Let \(V\) be the set of all subsets of \(S\) of size \(r\). Consider the graph \(G(n, r, s)\) with vertices \(V\) and edges that connect two vertices if and only if their intersection has size \(s\). In this article, we find the exact modularity of \(G(n, 2, 1)\) for \(n \geq 5\).Shattered matchings in intersecting hypergraphs.https://zbmath.org/1459.052232021-05-28T16:06:00+00:00"Frankl, Peter"https://zbmath.org/authors/?q=ai:frankl.peter"Pach, János"https://zbmath.org/authors/?q=ai:pach.janosSummary: Let \(X\) be an \(n\)-element set, where \(n\) is even. We refute a conjecture of J. Gordon and Y. Teplitskaya, according to which, for every maximal intersecting family \(\mathcal{F}\) of \(\frac{n}{2}\)-element subsets of \(X\), one can partition \(X\) into \(\frac{n}{2}\) disjoint pairs in such a way that no matter how we pick one element from each of the first \(\frac{n}{2} - 1\) pairs, the set formed by them can always be completed to a member of \(\mathcal{F}\) by adding an element of the last pair.
The above problem is related to classical questions in extremal set theory. For any \(t\ge 2\), we call a family of sets \(\mathcal{F}\subset 2^X\) \(t\)-separable if there is a \(t\)-element subset \(T\subseteq X\) such that for every ordered pair of elements \((x,y)\) of \(T\), there exists \(F\in\mathcal{F}\) such that \(F\cap\{x,y\}=\{x\} \). For a fixed \(t\), \(2\le t\le 5\), and \(n\rightarrow\infty \), we establish asymptotically tight estimates for the smallest integer \(s=s(n,t)\) such that every family \(\mathcal{F}\) with \(|\mathcal{F}|\ge s\) is \(t\)-separable.Dense subgraphs of power-law random graphs.https://zbmath.org/1459.051452021-05-28T16:06:00+00:00"Lazarev, Denis O."https://zbmath.org/authors/?q=ai:lazarev.denis-o"Kuzyurin, Nikolay N."https://zbmath.org/authors/?q=ai:kuzyurin.nikolai-nikolaevichSummary: The problem of finding a maximal dense subgraph of a power-law random graph \(G(n,\alpha)\) is considered for every value of density \(c \in (0,1)\) and for every \(\alpha \in (0, +\infty)\). It is shown that in case \(\alpha < 2\) a maximal \(c\)-dense subgraph has size \(\Theta_p ( n^{1-\alpha/2} ) \), in case \(\alpha = 2\) it is limited whp, and in case \(\alpha > 2\) it is whp less than \(\frac{4}{c} \).On the component graphs of finitely generated free semimodules.https://zbmath.org/1459.051202021-05-28T16:06:00+00:00"Maity, Sushobhan"https://zbmath.org/authors/?q=ai:maity.sushobhan"Bhuniya, Anjan Kumar"https://zbmath.org/authors/?q=ai:bhuniya.anjan-kumarSummary: A semiring \(S\) is said to have invariant basis number property if any two bases of a finitely generated free semimodule over \(S\) have the same cardinality. Here we characterize reduced zero and reduced non-zero component graphs of every finitely generated free semimodule \(\mathcal{V}\) over such semirings. It is shown that if \(|S|\ge\aleph_0\), those two graphs of a semimodule \(\mathcal{V}\) over \(S\) are isomorphic.Panchromatic colorings of random hypergraphs.https://zbmath.org/1459.053042021-05-28T16:06:00+00:00"Kravtsov, D. A."https://zbmath.org/authors/?q=ai:kravtsov.d-a"Krokhmal, N. E."https://zbmath.org/authors/?q=ai:krokhmal.n-e"Shabanov, D. A."https://zbmath.org/authors/?q=ai:shabanov.dmitry-aThis paper studies panchromatic colorings of random hypergraphs. Let \(H(n,k,p)\) be the binomial model of a random \(k\)-homogeneous hypergraph over \(n\) vertices with edge probability \(p\). It is shown that there exists a natural number \(k_0\) such that for each \(k>k_0\), \(3\le r<0.1\sqrt{k}\) and \(c<[(\ln r)/r][r/(r-1)]^k-[(\ln r)/2]-20k^2\ln r\big\{[r/(r-1)]\max(r^{-1/(r-1)},(r/(r+1))^2)\big\}^k\) the random hypergraph \(H(n,k,cn/\binom{n}{k})\) may be panchromatically colored with \(r\) colors almost surely as \(n\) tends to infinity. There also exists a constant \(C>0\) such that if under the same conditions \(c>[(\ln r)/r][r/(r-1)]^k-[(\ln r)/2]+C\ln r\{r(r-2)/(r-1)^2\}^k\), there does not exist a panchromatic \(r\)-coloring of the graph \(H(n,k,cn/\binom{n}{k})\) almost surely.
Reviewer: Yilun Shang (Newcastle)