Independence polynomials of well-covered graphs: Generic counterexamples for the unimodality conjecture

A graph G is well-covered if all its maximal stable sets have the same size, denoted by α(G) (M. D. Plummer, 1970). If sk denotes the number of stable sets of cardinality k in graph G, and α(G) is the size of a maximum stable set, then I(G;x) = α(G) ∑ k=0 skx k is the independence polynomial of G (I. Gutman and F. Harary, 1983). J. I. Brown, K. Dilcher and R. J. Nowakowski (2000) conjectured that I(G;x) is unimodal (i.e., there is some j ∈ {0, 1, ..., α(G)} such that s0 ≤ ... ≤ sj−1 ≤ sj ≥ sj+1 ≥ ... ≥ sα(G)) for any well-covered graph G. T. S. Michael and W. N. Traves (2002) proved that this assertion is true for α(G) ≤ 3, while for α(G) ∈ {4, 5, 6, 7} they provided counterexamples. In this paper we show that for any integer α ≥ 8, there exists a connected well-covered graph G with α = α(G), whose independence polynomial is not unimodal. In addition, we present a number of sufficient conditions for a graph G with α(G) ≤ 6 to have the unimodal independence polynomial. key words: stable set, independence polynomial, unimodal sequence, wellcovered graph.