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 $\Delta$. 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 $\Delta$.

34 pages, 4 figures