Zero-free regions for the independence polynomial on restricted graph classes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910013733208064 |
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| author | Jerrum, Mark Patel, Viresh |
| author_facet | Jerrum, Mark Patel, Viresh |
| contents | 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 $Δ$, 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 $Δ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_01466 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zero-free regions for the independence polynomial on restricted graph classes Jerrum, Mark Patel, Viresh Combinatorics Mathematical Physics 05, 68, 82 G.2.1; G.2.2 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 $Δ$, 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 $Δ$ 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. |
| title | Zero-free regions for the independence polynomial on restricted graph classes |
| topic | Combinatorics Mathematical Physics 05, 68, 82 G.2.1; G.2.2 |
| url | https://arxiv.org/abs/2510.01466 |