On unimodality of independence polynomials of some well-covered trees

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. © Springer-Verlag Berlin Heidelberg 2003.