Independence Polynomials and the Unimodality Conjecture for Very Well-covered, Quasi-regularizable, and Perfect Graphs

Abstract

If s _κ denotes the number of stable sets of cardinality κ in the graph G, then $I(G;x) = \sum\limits_{k = 0}^\alpha {s_k x^k }$ is the independence polynomial of G ([Gutman, Harary, 1983)], where α = α(G) is the size of a maximum stable set in G. [Alavi, Malde, Schwenk and Erdös (1987)] conjectured that I(T, x) is unimodal for every tree T, while, in general, they proved that for each permutation π of 1, 2, ..., α there is a graph G with α(G) = α such that s _π(1) < s _π(2) < ... < s _π(α). [Brown, Dilcher and Nowakowski (2000)] conjectured that I(G; x) is unimodal for well-covered graphs. [Michael and Traves (2003)] provided examples of well-covered graphs with non-unimodal independence polynomials. They proposed the so-called “roller-coaster” conjecture: for a well-covered graph, the subsequence (s _⌈α/2⌉, s _⌈α/2⌉+1, ..., s _α) is unconstrained in the sense of Alavi et al. The conjecture of Brown et al. is still open for very well-covered graphs, and it is worth mentioning that, apart from K ₁ and the chordless cycle C ₇, connected well-covered graphs of girth ≥ 6 are very well covered [(Finbow, Hartnell and Nowakowski, 1993)].

In this paper we prove that s _{⌈(2α−1)/3⌉} ≥ ... ≥ s _α−1 ≥ s _α are valid for (a) bipartite graphs; (b) quasi-regularizable graphs on 2α vertices.

In particular, we infer that these inequalities are true for (a) trees, thus doing a step in an attempt to prove the conjecture of Alavi et al.; (b) very well-covered graphs. Consequently, for the latter case, the unconstrained subsequence appearing in the roller-coaster conjecture can be shortened to (s _⌈α/2⌉, s _⌈α/2⌉+1, ..., s _{⌈(2α−1/3⌍}). We also show that the independence polynomial of a very well-covered graph G is unimodal for α ≤ 9, and is logconcave whenever α ≤ 5.