Unimodality of the Independence Polynomials of Some Composite Graphs

Let I(G;x) denote the independence polynomial of a graph G. In this paper we study the unimodality properties of I(G;x) for some composite graphs G. Given two graphs G1 and G2, let G1(G2) denote the lexicographic product of G1 and G2. Assume I ( G 1 ; x ) = ∑ i ≥ 0 a i x i and I ( G 2 ; x ) = ∑ i ≥ 0 b i x i where I(G2; x) is log-concave. Then we prove (i) if I(G1; x) is log-concave and ( a i 2 − a i − 1 a i + 1 ) b 1 2 ≥ a i a i − 1 b 2 for all 1 ≤ i ≤ α(G1), then I(G1[G2]; x) is log-concave; (ii) if ai-1≤ b1aifor 1 ≤ i ≤ α(G1), then I(G1[G2]; x) is unimodal. In particular, if aiis increasing in i, then I(G1[G2]; x) is unimodal. We also give two sufficient conditions when the independence polynomial of a complete multipartite graph is unimodal or log-concave. Finally, for every odd positive integer α > 3, we find a connected graph G not a tree, such that α(G) = α, and I(G;x) is symmetric and has only real zeros. This answers a problem of Mandrescu and Mirică.