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

Abstract

In this paper we investigate the distribution of zeros of the independence polynomial of hypergraphs of maximum degree $Δ$. For graphs the largest zero-free disk around zero was described by Shearer as having radius $λ_s(Δ)=(Δ-1)^{Δ-1}/Δ^Δ$. Recently it was shown by Galvin et al. that for hypergraphs the disk of radius $λ_s(Δ+1)$ is zero-free; however, it was conjectured that the actual truth should be $λ_s(Δ)$. We show that this is indeed the case. We also show that there exists an open region around the interval $[0,(Δ-1)^{Δ-1}/(Δ-2)^Δ)$ that is zero-free for hypergraphs of maximum degree $Δ$, 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 $Δ$.