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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)= i=1 n 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 𝒟-equivalent, written as GH, if D(G,x)=D(H,x). The 𝒟-equivalence class of G is [G]={H:HG}. A graph G is said to be 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))0 , - 2 ± 2 i , -3+3i 2·

Also, we study the 𝒟-equivalence classes of some certain graphs. It is shown that if n0,2(mod3), then C n is 𝒟-unique, and if n0(mod3), then [P n ] consists of exactly two graphs.

MSC:
05C31Graph polynomials
05C69Dominating sets, independent sets, cliques
References:
[1]S. Alikhani, Y.H. Peng, Introduction to domination polynomial of a graph, Ars Combin. (in press).
[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)
[6]Payan, C.; Xuong, N. H.: Domination-balanced graphs, J. graph theory 6, 23-32 (1982) · Zbl 0489.05049 · doi:10.1002/jgt.3190060104