TY - JOUR

T1 - Independence polynomials of well-covered graphs

T2 - Generic counterexamples for the unimodality conjecture

AU - Levit, Vadim E.

AU - Mandrescu, Eugen

PY - 2006/8

Y1 - 2006/8

N2 - 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 ) = ∑k = 0α ( G ) sk xkis 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.

AB - 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 ) = ∑k = 0α ( G ) sk xkis 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.

UR - http://www.scopus.com/inward/record.url?scp=33748702498&partnerID=8YFLogxK

U2 - 10.1016/j.ejc.2005.04.007

DO - 10.1016/j.ejc.2005.04.007

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AN - SCOPUS:33748702498

SN - 0195-6698

VL - 27

SP - 931

EP - 939

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

IS - 6

ER -