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

Abstract

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 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 ∈ { 0 , 1 , … , α ( G ) } such that s 0 ≤ ⋯ ≤ 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.