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

• Authors: Vadim E. Levit;Eugen Mandrescu
• Venue: European Journal of Combinatorics
• Year: 2006

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 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) = Σk=0α(G) skxk 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 ∈ {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 [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.