On unimodality of independence polynomials of some well-covered trees

Vadim E. Levit, Eugen Mandrescu

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

14 Scopus citations

Abstract

The stability number α(G) of the graph G is the size of a maximum stable set of G. If sk denotes the number of stable sets of cardinality k in graph G, then I(G; x) = Σk=0α(G) skxk is the independence polynomial of G (I. Gutman and F. Harary 1983). In 1990, Y.O. Hamidoune proved that for any claw-free graph G (a graph having no induced subgraph isomorphic to K1,3), I(G; x) is unimodal, i.e., there exists some k ∈ {0, 1, ..., α(G)} such that s0 ≤ s1 ≤ ... ≤ sk-1 ≤ sk ≥ sk+1 ≥ ... ≥ sα(G). Y. Alavi, P.J. Malde, A.J. Schwenk, and P. Erdös (1987) asked whether for trees the independence polynomial is unimodal. J. I. Brown, K. Dilcher and R.J. Nowakowski (2000) conjectured that I(G; x) is unimodal for any well-covered graph G (a graph whose all maximal independent sets have the same size). Michael and Traves (2002) showed that this conjecture is true for well-covered graphs with α(G) ≤ 3, and provided counterexamples for α(G) ∈ {4, 5, 6, 7}. In this paper we show that the independence polynomial of any well-covered spider is unimodal and locate its mode, where a spider is a tree having at most one vertex of degree at least three. In addition, we extend some graph transformations, first introduced in [14], respecting independence polynomials. They allow us to reduce several types of well-covered trees to claw-free graphs, and, consequently, to prove that their independence polynomials are unimodal.

Original languageEnglish
Title of host publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
EditorsCristian S. Calude, Michael J. Dinneen, Vincent Vajnovszki
PublisherSpringer Verlag
Pages237-256
Number of pages20
ISBN (Print)3540405054, 9783540405054
DOIs
StatePublished - 2003
Externally publishedYes

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume2731
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

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