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

Mark Jerrum, Viresh Patel

TL;DR

This work analyzes zeros of the independence polynomial $Z_{G,\lambda}$ (the hard-core partition function) on restricted graph classes. It develops a cluster/decomposition framework and uses Asano contractions to establish zero-free regions, obtaining an open region $F$ containing $[0,\infty)$ for $S_{t,t,t}$-free graphs of bounded degree and a multivariate region $R(k)$ for claw-free (and near-claw-free) classes, with near-optimal sharpness demonstrated by rich examples. The results imply deterministic polynomial-time approximation schemes via Barvinok interpolation and provide insights into Lee–Yang phase transitions in restricted graphs. Sharpness results show the necessity of subdivided-claw structure and degree bounds, while the Appendix extends multivariate zero-free regions to line graphs of multigraphs. Overall, the paper advances understanding of phase behavior and computational tractability for the hard-core model on structured graphs.

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 $Δ$, 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.

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

TL;DR

This work analyzes zeros of the independence polynomial (the hard-core partition function) on restricted graph classes. It develops a cluster/decomposition framework and uses Asano contractions to establish zero-free regions, obtaining an open region containing for -free graphs of bounded degree and a multivariate region for claw-free (and near-claw-free) classes, with near-optimal sharpness demonstrated by rich examples. The results imply deterministic polynomial-time approximation schemes via Barvinok interpolation and provide insights into Lee–Yang phase transitions in restricted graphs. Sharpness results show the necessity of subdivided-claw structure and degree bounds, while the Appendix extends multivariate zero-free regions to line graphs of multigraphs. Overall, the paper advances understanding of phase behavior and computational tractability for the hard-core model on structured graphs.

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 and any , there is an open set containing such that the independence polynomial of any -free graph of maximum degree has all of its zeros outside of . We also show that no such result can hold when 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 -free graphs of bounded degree when is a subdivided claw. The statements of these results are more subtle, but are again best possible in various senses.

Paper Structure

This paper contains 13 sections, 16 theorems, 46 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Zero-free regions for graphs with no induced subdivided claw
  4. Claw-free graphs
  5. Examples
  6. Even cycles
  7. Blow-ups of cycles (dense)
  8. Paths with longer range interactions
  9. Blow-ups of cycles (sparse)
  10. Complete multipartite graphs
  11. The case when $H$ is neither a subdivided claw nor a path
  12. Concluding Remarks
  13. Appendix

Key Result

Theorem 1

If $G$ is a line graph (i.e., $G = L(H)$ for some $H$) then $Z_{G, \lambda} \not= 0$ for all $\lambda \in \mathbb{C} \setminus (- \infty, 0)$, i.e., all roots of $Z_{G, \lambda}$ are negative reals.

Figures (3)

  • Figure 1: The claw, $S_{1,1,1}$; the fork, $S_{1,1,2}$; and the E, $S_{1,2,2}$.
  • Figure 2: A derivation of admissible pair $(U,L)$ from admissible pair $(U',L')$.
  • Figure 3: The trees $T_1$ and $T_2$, when $k=1$.

Theorems & Definitions (30)

  • Theorem 1: Heilman-Lieb HeilmannLieb
  • Theorem 2: Chudnovsky-Seymour ChudnovskySeymour
  • Theorem 3
  • Theorem 4: Lebowitz-Ruelle-Speer LebowitzRuelleSpeer
  • Theorem 5
  • Theorem 6
  • Lemma 7
  • Lemma 8
  • proof
  • Lemma 9
  • ...and 20 more