Clique cover products and unimodality of independence polynomials

Abstract

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 GC⋆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 GC⋆HUI(GC⋆HU;x)=[I(H;x)]qI(G;xI(H−U;x)I(H;x)), which generalizes some known results on independence polynomials of the compound graph introduced by Song, Staton and Wei, the 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 graphs preserves symmetry, unimodality, log-concavity or reality of zeros of independence polynomials. As applications we derive several known facts and solve some open unimodality conjectures and problems in a simple and unified manner.