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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

TL;DR

This work develops a symmetric-function approach to log-concavity of independence polynomials by connecting $I_G(t)$ to chromatic symmetric functions and the 2-Schur-positivity of $Y_G= obreak extstyleigoplus_{oldsymbol{ ext{alpha}}} X_G^{oldsymbol{ ext{alpha}}}$. By Stanley’s framework, strong links between the coefficients of $s_{\lambda}$ in generating functions and the log-concavity of $I_G(t)$ are established, reducing the problem to proving 2-$s$-positivity of $Y_G$. The authors demonstrate the method on spiders, proving $I_{S(oldsymbol{eta})}(t)$ is strongly log-concave, and introduce symmetric-function analogues of a fundamental recurrence to handle pineapple graphs, proving their $I_G(t)$ is log-concave as well. Overall, the paper provides a robust tool to establish log-concavity (and related properties) for broader graph families, contributing evidence toward the conjecture that independence polynomials of all trees are unimodal and, conjecturally, log-concave. The approach also connects graph polynomials with the rich theory of symmetric functions, enabling combinatorial decompositions via stable partitions and clan graphs.

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 Erdős 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.

A symmetric function approach to log-concavity of independence polynomials

TL;DR

This work develops a symmetric-function approach to log-concavity of independence polynomials by connecting to chromatic symmetric functions and the 2-Schur-positivity of . By Stanley’s framework, strong links between the coefficients of in generating functions and the log-concavity of are established, reducing the problem to proving 2--positivity of . The authors demonstrate the method on spiders, proving is strongly log-concave, and introduce symmetric-function analogues of a fundamental recurrence to handle pineapple graphs, proving their is log-concave as well. Overall, the paper provides a robust tool to establish log-concavity (and related properties) for broader graph families, contributing evidence toward the conjecture that independence polynomials of all trees are unimodal and, conjecturally, log-concave. The approach also connects graph polynomials with the rich theory of symmetric functions, enabling combinatorial decompositions via stable partitions and clan graphs.

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 Erdős 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.
Paper Structure (5 sections, 11 theorems, 14 equations, 3 figures)

This paper contains 5 sections, 11 theorems, 14 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Schur-positivity and log-concavity
  3. Log-concavity of independence polynomials
  4. Log-concavity of independence polynomials of spiders
  5. Symmetric function analogues of the recurrence formula

Key Result

Theorem 1.2

Let $P(t)$ and $F_P(\mathbf{x})$ be defined as above. Then the following conditions are equivalent:

Figures (3)

  • Figure 1.1: Spider $S(3,2,2,1)$ and pineapple graph $Pi(6,(3,2,2,1))$
  • Figure 3.1: Spider $S(3,2,2,1)$
  • Figure 3.4: Pineapple graph $Pi(6,(3,2,2,1))$

Theorems & Definitions (12)

  • Conjecture 1.1: AMSE87
  • Theorem 1.2: Sta98
  • Theorem 1.3
  • Lemma 2.1
  • Corollary 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Corollary 2.5
  • Theorem 3.1
  • Proposition 3.3
  • ...and 2 more