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, Some covering concepts in graphs, Journal of Combinatorial Theory 8 (1970) 91-98]. If s_k 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 ) = ∑_{k = 0}^{α ( G )} s_k x^kis the independence polynomial of G [I. Gutman, F. Harary, Generalizations of the matching polynomial, Utilitas Mathematica 24 (1983) 97-106]. J.I. Brown, K. Dilcher and R.J. Nowakowski [Roots of independence polynomials of well-covered graphs, Journal of Algebraic Combinatorics 11 (2000) 197-210] conjectured that I ( G ; x ) is unimodal (i.e., there is some j ∈ { 0, 1, ... , α ( G ) } such that s₀ ≤ ... ≤ s_{j - 1} ≤ s_j≥ s_{j + 1} ≥...≥ s_{α ( G )}) for any well-covered graph G. T.S. Michael and W.N. Traves [Independence sequences of well-covered graphs: non-unimodality and the roller-coaster conjecture, Graphs and Combinatorics 19 (2003) 403-411] 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.