A hypergraph Heilmann--Lieb theorem

TL;DR

This paper extends the Heilmann--Lieb theorem to hypergraphs, revealing the rotational symmetry and bounds of zeros of hypergraph matching polynomials, generalizing classical graph results.

Contribution

It establishes a hypergraph Heilmann--Lieb theorem, showing symmetry and bounds of zeros of hypergraph matching polynomials, and generalizes Godsil's graph result to hypergraphs.

Findings

Zeros of hypergraph matching polynomials are rotationally symmetric.

Maximum modulus of zeros is a simple root and bounded by specific inequalities.

Matching polynomial divides the polynomial of the hypergraph's $k$-walk-tree.

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.