On the zeroes of hypergraph independence polynomials

TL;DR

This paper investigates the complex zeroes of hypergraph independence polynomials, establishing near-optimal zero-free regions for bounded-degree hypergraphs and proposing conjectures for larger zero-free disks in uniform hypergraphs.

Contribution

It extends the understanding of zero-free regions from graphs to hypergraphs, providing bounds close to optimal and analyzing special cases like linear hypertrees.

Findings

Hypergraphs of maximum degree Δ have a zero-free disk nearly as large as the optimal for graphs.

The zero-free disk size is optimal up to logarithmic factors in Δ for hypergraphs with edges larger than size 2.

Conjecture: k-uniform linear hypergraphs have larger zero-free disks of radius Ω(Δ^{-1/(k-1)})

Abstract

We study the locations of complex zeroes of independence polynomials of bounded degree hypergraphs. For graphs, this is a long-studied subject with applications to statistical physics, algorithms, and combinatorics. Results on zero-free regions for bounded-degree graphs include Shearer's result on the optimal zero-free disk, along with several recent results on other zero-free regions. Much less is known for hypergraphs. We make some steps towards an understanding of zero-free regions for bounded-degree hypergaphs by proving that all hypergraphs of maximum degree $\Delta$ have a zero-free disk almost as large as the optimal disk for graphs of maximum degree $\Delta$ established by Shearer (of radius $\sim 1/(e \Delta)$). Up to logarithmic factors in $\Delta$ this is optimal, even for hypergraphs with all edge-sizes strictly greater than $2$. We conjecture that for $k\ge 3$, $k$-uniform linear hypergraphs have a much larger zero-free disk of radius $\Omega(\Delta^{- \frac{1}{k-1}} )$. We establish this in the case of linear hypertrees.