A symmetric function approach to log-concavity of independence polynomials

TL;DR

This paper introduces a new symmetric function-based method to prove the log-concavity of independence polynomials in graphs, providing evidence for the conjecture that all trees have unimodal independence polynomials.

Contribution

The paper develops a novel approach using chromatic symmetric functions to establish log-concavity of independence polynomials, especially for trees with irregular structures.

Findings

All spiders have log-concave independence polynomials.

Pineapple graphs also have log-concave independence polynomials.

The method offers a new perspective for studying polynomial properties via symmetric functions.

Abstract

As introduced by Gutman and Harary, the independence polynomial of a graph serves as the generating polynomial of its independent sets. In 1987, Alavi, Malde, Schwenk and Erd\H{o}s conjectured that the independence polynomials of all trees are unimodal. In this paper we come up with a new way for proving log-concavity of independence polynomials of graphs by means of their chromatic symmetric functions, which is inspired by a result of Stanley connecting properties of polynomials to positivity of symmetric functions. This method turns out to be more suitable for treating trees with irregular structures, and as a simple application we show that all spiders have log-concave independence polynomials, which provides more evidence for the above conjecture. Moreover, we present two symmetric function analogues of a basic recurrence formula for independence polynomials, and show that all pineapple graphs also have log-concave independence polynomials.