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arXiv · 2501.04245

A symmetric function approach to log-concavity of independence polynomials

Ethan Y. H. Li·Grace M. X. Li·Arthur L. B. Yang·Zhong-Xue Zhang
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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.

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Ethan Y. H. Li, Grace M. X. Li, Arthur L. B. Yang, Zhong-Xue Zhang. 2025-01-08. A symmetric function approach to log-concavity of independence polynomials. https://arxiv.org/abs/2501.04245

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