A hypergraph Heilmann--Lieb theorem

Abstract

The Heilmann--Lieb theorem is a fundamental theorem in algebraic combinatorics which provides a characterization of the distribution of the zeros of matching polynomials of graphs. In this paper, we establish a hypergraph Heilmann--Lieb theorem as follows. Let $\h$ be a connected $k$-graph with maximum degree ${\Delta}\geq 2$ and let $\mu(\h, x)$ be its matching polynomial. We show that the zeros (with multiplicities) of $\mu(\h, x)$ are invariant under a rotation of an angle $2\pi/{\ell}$ in the complex plane for some positive integer $\ell$ and $k$ is the maximum integer with this property. We further prove that the maximum modulus $\lambda(\h)$ of all the zeros of $\mu(\h, x)$ is a simple root of $\mu(\h, x)$ and satisfies $$\Delta^{\frac{1}{ k}} \leq \lambda(\h)< \frac{k}{k-1}\big((k-1)(\Delta-1)\big)^{\frac{1}{ k}}.$$ To achieve these, we prove that $\mu(\h, x)$ divides the matching polynomial of the $k$-walk-tree of $\h$, which generalizes a classical result due to Godsil from graphs to hypergraphs.