A hypergraph Heilmann--Lieb theorem
Jiang-Chao Wan, Yi Wang, Yi-zheng Fan
TL;DR
The paper generalizes the Heilmann-Lieb framework to connected $k$-graphs by introducing the matching polynomial $μ(\mathcal{H},x)$ and proving that its zeros exhibit $k$-fold rotational symmetry with a simple largest zero $λ(\mathcal{H})$ satisfying $Δ^{1/k}\leq λ(\mathcal{H})< \frac{k}{k-1}((k-1)(Δ-1))^{1/k}$. A central method is the $k$-walk-tree construction, which yields the divisibility $μ(\mathcal{H},x) \mid μ(\mathcal{T}(\mathcal{H},\prec,u),x)$, linking hypergraph matchings to the spectrum of an associated $k$-tree. The authors prove that the cyclic index of $μ(\mathcal{H},x)$ is exactly $k$ and that the largest zero is a simple root, with the nonreal zeros appearing as $λ(\mathcal{H}) e^{2\pi i j/k}$ for $j=0,\dots,k-1$. These results extend Ramanujan-type eigenvalue phenomena to hypergraphs and provide new tools for studying second eigenvalue bounds and spectral hypergraph theory.
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 $Δ\geq 2$ and let $μ(\h, x)$ be its matching polynomial. We show that the zeros (with multiplicities) of $μ(\h, x)$ are invariant under a rotation of an angle $2π/{\ell}$ 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 $λ(\h)$ of all the zeros of $μ(\h, x)$ is a simple root of $μ(\h, x)$ and satisfies $$Δ^{\frac{1}{ k}} \leq λ(\h)< \frac{k}{k-1}\big((k-1)(Δ-1)\big)^{\frac{1}{ k}}.$$ To achieve these, we prove that $μ(\h, x)$ divides the matching polynomial of the $k$-walk-tree of $\h$, which generalizes a classical result due to Godsil from graphs to hypergraphs.