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

2023/05/31 by Ferenc Bencs, Bencs, Ferenc, Pjotr Buys +1 · 3 citations
Computer Science · Mathematics · #05C31 #05C65 #05C69 #37F44 #37F46 #Advanced Differential Equations and Dynamical Systems #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2306.00122

openalex publication_date 2023/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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