Very well-covered graphs with log-concave independence polynomials

TL;DR

This paper investigates the unimodality of independence polynomials in well-covered graphs, providing new results that support conjectures for graphs with maximum stable set size less than 4 or specific graph classes.

Contribution

It proves that the independence polynomial of certain very well-covered graphs is log-concave, thus unimodal, when the maximum stable set size is less than 4 or for specific graph classes.

Findings

Independence polynomial is log-concave for graphs with maximum stable set size < 4.

Independence polynomial is log-concave for graphs in classes K_{1,n} and P_n.

Supports conjectures on unimodality of independence polynomials in well-covered graphs.

Abstract

If for any $k$ the $k$-th coefficient of a polynomial $I(G;x)$ is equal to the number of stable sets of cardinality $k$ in the graph $G$, then it is called the independence polynomial of $G$ (Gutman and Harary, 1983). Alavi, Malde, Schwenk and Erdos (1987) conjectured that $I(G;x)$ is unimodal, whenever $G$ is a forest, while Brown, Dilcher and Nowakowski (2000) conjectured that $I(G;x)$ is unimodal for any well-covered graph G. Michael and Traves (2003) showed that the assertion is false for well-covered graphs with $a(G)$ > 3 ($a(G)$ is the size of a maximum stable set of the graph $G$), while for very well-covered graphs the conjecture is still open. In this paper we give support to both conjectures by demonstrating that if $a(G)$ < 4, or $G$ belongs to ${K_{1,n}, P_{n}: n > 0}$, then $I(G*;x)$ is log-concave, and, hence, unimodal (where $G*$ is the very well-covered graph obtained from $G$ by appending a single pendant edge to each vertex).