On the independent set polynomial of graphs and claw-free graphs

TL;DR

This paper improves the understanding of the independence polynomial of claw-free graphs by establishing a better zero-free region and introducing a new combinatorial expression inspired by statistical mechanics, linking multiple disciplines.

Contribution

It provides an improved zero-free region for the independence polynomial of claw-free graphs and introduces a new combinatorial expression inspired by polymer gas models.

Findings

Enhanced lower bound for zero-free region surpassing Shearer radius

New combinatorial expression for the independence polynomial

Strengthened connection between graph theory and statistical physics

Abstract

We present two new contributions to the study of the independence polynomial $Z_G(z)$ of a finite simple graph $G = (V,E)$. First, we provide an improved lower bound for the zero-free region of $Z_G(z)$ for the important class of claw-free graphs. Our bound exceeds the classical Shearer radius and it is derived through a refined application of the Fern\'andez-Procacci criterion using properties of the local neighborhood structure in claw-free graphs. Second, we establish a novel combinatorial expression for $Z_G(z)$, inspired by the connection with the abstract polymer gas models in statistical mechanics, which offers a new structural interpretation of the polynomial and may be of independent interest. These results strengthen the connection between statistical physics, combinatorics, and graph theory, and suggest new approaches for analytic exploration.