Optimal zero-free regions for the independence polynomial of bounded degree hypergraphs

TL;DR

This paper establishes the optimal zero-free regions for the independence polynomial of bounded degree hypergraphs, confirming conjectures and extending known results from graphs to hypergraphs.

Contribution

It proves the conjecture that the largest zero-free disk for hypergraphs matches that of graphs and characterizes zero-free regions for specific hypergraph families.

Findings

Confirmed the zero-free disk radius for hypergraphs as conjectured.

Extended zero-free region results from graphs to hypergraphs.

Determined the zero-free disk radius for bounded degree k-uniform linear hypertrees.

Abstract

In this paper we investigate the distribution of zeros of the independence polynomial of hypergraphs of maximum degree $\Delta$. For graphs the largest zero-free disk around zero was described by Shearer as having radius $\lambda_s(\Delta)=(\Delta-1)^{\Delta-1}/\Delta^\Delta$. Recently it was shown by Galvin et al. that for hypergraphs the disk of radius $\lambda_s(\Delta+1)$ is zero-free; however, it was conjectured that the actual truth should be $\lambda_s(\Delta)$. We show that this is indeed the case. We also show that there exists an open region around the interval $[0,(\Delta-1)^{\Delta-1}/(\Delta-2)^\Delta)$ that is zero-free for hypergraphs of maximum degree $\Delta$, which extends the result of Peters and Regts from graphs to hypergraphs. Finally, we determine the radius of the largest zero-free disk for the family of bounded degree $k$-uniform linear hypertrees in terms of $k$ and $\Delta$.