zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Characterization of graphs using domination polynomials. (English) Zbl 1207.05092
Summary: Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $$D(G,x)=\sum^n_{i=1} d(G,i)^i,$$ where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. A root of $D(G,x)$ is called a domination root of $G$. We denote the set of distinct domination roots by $Z(D(G,x))$. Two graphs $G$ and $H$ are said to be ${\Cal D}$-equivalent, written as $G\sim H$, if $D(G,x)= D(H,x)$. The ${\Cal D}$-equivalence class of $G$ is $[G]= \{H: H\sim G\}$. A graph $G$ is said to be ${\Cal D}$-unique if $[G]= \{G\}$. In this paper, we show that if a graph $G$ has two distinct domination roots, then $Z(D(G,x))= \{-2, 0\}$. Also, if $G$ is a graph with no pendant vertex and has three distinct domination roots, then $$Z(D(G,x))\subseteq \Biggl\{0,-2\pm\sqrt{2}i,{-3+ \sqrt{3}i\over 2}\Biggr\}.$$ Also, we study the ${\Cal D}$-equivalence classes of some certain graphs. It is shown that if $n\equiv 0,2\pmod 3$, then $C_n$ is ${\Cal D}$-unique, and if $n\equiv 0\pmod 3$, then $[P_n]$ consists of exactly two graphs.

05C31Graph polynomials
05C69Dominating sets, independent sets, cliques
Full Text: DOI arXiv
[1] S. Alikhani, Y.H. Peng, Introduction to domination polynomial of a graph, Ars Combin. (in press). · Zbl 1324.05138
[2] Alikhani, S.; Peng, Y. H.: Dominating sets and domination polynomial of cycles, Glob. J. Pure appl. Math. 4, No. 2, 151-162 (2008)
[3] S. Alikhani, Y.H. Peng, Dominating sets and domination polynomials of paths, Int. J. Math. Math. Sci., 2009, Article ID 542040. · Zbl 1177.05081 · doi:10.1155/2009/542040
[4] Frucht, R.; Harary, F.: On the corona of two graphs, Aequationes math. 4, 322-324 (1970) · Zbl 0198.29302 · doi:10.1007/BF01844162
[5] Haynes, T. W.; Hedetniemi, S. T.; Slater, P. J.: Fundamentals of domination in graphs, (1998) · Zbl 0890.05002
[6] Payan, C.; Xuong, N. H.: Domination-balanced graphs, J. graph theory 6, 23-32 (1982) · Zbl 0489.05049 · doi:10.1002/jgt.3190060104