Zero-free regions for the independence polynomial on restricted graph classes

Mark Jerrum; Viresh Patel

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arXiv · 2510.01466

Zero-free regions for the independence polynomial on restricted graph classes

Mark Jerrum·Viresh Patel

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Abstract

Generalising the Heilman-Lieb Theorem from statistical physics, Chudnovsky and Seymour [J. Combin. Theory Ser. B, 97(3):350--357] showed that the univariate independence polynomial of any claw-free graph has all of its zeros on the negative real line. In this paper, we show that for any fixed subdivded claw $H$ and any $\Delta$, there is an open set $F \subseteq \mathbb{C}$ containing $[0, \infty)$ such that the independence polynomial of any $H$-free graph of maximum degree $\Delta$ has all of its zeros outside of $F$. We also show that no such result can hold when $H$ is any graph other than a subdivided claw or if we drop the maximum degree condition. We also establish zero-free regions for the multivariate independence polynomial of $H$-free graphs of bounded degree when $H$ is a subdivided claw. The statements of these results are more subtle, but are again best possible in various senses.

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Mark Jerrum, Viresh Patel. 2025-10-01. Zero-free regions for the independence polynomial on restricted graph classes. https://arxiv.org/abs/2510.01466

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