2 9 Se p 20 13 Operations of graphs and unimodality of independence polynomials ∗
نویسنده
چکیده
Given two graphs G and H, assume that C = {C1, C2, . . . , Cq} is a clique cover of G and U is a subset of V (H). We introduce a new graph operation called the clique cover product, denoted by G ⋆ HU , as follows: for each clique Ci ∈ C , add a copy of the graph H and join every vertex of Ci to every vertex of U . We prove that the independence polynomial of G ⋆ HU I(G ⋆ H ;x) = I(H;x)I(G; xI(H − U ;x) I(H;x) ), which generalizes some known results on independence polynomials of corona and rooted products of graphs obtained by Gutman and Rosenfeld, respectively. Based on this formula, we show that the clique cover product of some special graphs preserves symmetry, unimodality, log-concavity or reality of zeros of independence polynomials. As applications we derive several known facts in a unified manner and solve some unimodality conjectures and problems. MSC: 05C69; 05A20
منابع مشابه
The unimodality of independence polynomials of some graphs
In this paper we study unimodality problems for the independence polynomial of a graph, including unimodality, log-concavity and reality of zeros. We establish recurrence relations and give factorizations of independence polynomials for certain classes of graphs. As applications we settle some unimodality conjectures and problems. © 2010 Elsevier Ltd. All rights reserved.
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