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

By Vadim E. Levit and Eugen Mandrescu

Abstract

AbstractA 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)=∑k=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

Publisher: Elsevier Ltd.
Year: 2006
DOI identifier: 10.1016/j.ejc.2005.04.007
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