Operations of graphs and unimodality of independence polynomials

الأصل · EN

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⋆ Hᵘ I(GC⋆ Hᵘ;x)=Iq(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.