On Unimodality of Independence Polynomials of some Well-Covered Trees

Abstract

The number of stable sets of cardinality $k$ in graph $G$ is the $k$-th coefficient of the independence polynomial of $G$ (I. Gutman and F. Harary, 1983). In 1990, Y. O. Hamidoune proved that for any claw-free graph, its independence polynomial is unimodal, i.e., there exists a coefficient $k$ such that the part of the sequence of coefficients from the first to $k$-th is non-decreasing while the second part of coefficients is non-increasing. Y. Alavi, P. J. Malde, A. J. Schwenk and P. Erdös (1987) asked whether for trees (or perhaps forests) the independence polynomial is unimodal. J. I. Brown, K. Dilcher and R. J. Nowakowski (2000) conjectured that it is true for any well-covered graph (a graph whose all maximal independent sets have the same size). V. E. Levit and E. Mandrescu (1999) demonstrated that every well-covered tree can be obtained as a join of a number of well-covered spiders, where a spider is a tree having at most one vertex of degree at least three. In this paper we show that the independence polynomial of any well-covered spider is unimodal. In addition, we introduce some graph transformations 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.