Hypergraph independence polynomials with a zero close to the origin

TL;DR

This paper constructs specific hypergraphs with large degrees whose independence polynomials have roots very close to zero, challenging a recent conjecture in hypergraph theory.

Contribution

It provides a counterexample to a conjecture by explicitly constructing hypergraphs with roots near zero, showing the conjecture does not hold universally.

Findings

Hypergraphs with large maximum degree can have roots of their independence polynomial close to zero.

Disproves the conjecture that roots of independence polynomials are bounded away from zero.

Constructs explicit examples of hypergraphs with roots at O(log Δ / Δ).

Abstract

For each uniformity $k \geq 3$, we construct $k$-uniform linear hypergraphs $G$ with arbitrarily large maximum degree $\Delta$ whose independence polynomial $Z_G$ has a root $\lambda$ with $\lvert\lambda\rvert = O\left(\frac{\log \Delta}{\Delta}\right)$. This disproves a recent conjecture of Galvin, McKinley, Perkins, Sarantis, and Tetali.