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 alpha(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 alpha(G) is the size of a maximum stable set, then I (G; x) = Sigma(alpha(G))(k=0) s(k)x(k) is 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 is an element of {0, 1,..,, alpha(G)} such that s(o) <= ...<= s(j-1) <= s(j) >= s(j+1) >=...>= s(alpha(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 alpha(G) <= 3, while for alpha(G) is an element of {4, 5, 6, 7} they provided counterexamples. In this paper we show that for any integer alpha >= 8, there exists a connected well-covered graph G with alpha = alpha(G), whose independence polynomial is not unimodal, In addition, we present a number of sufficient conditions for a graph G with alpha(G) <= 6 to have the unimodal independence polynomial. (C) 2005 Elsevier Ltd. All rights reserved.