# Stable multivariate generalizations of matching polynomials

## TL;DR

This paper proves the stability and real-rootedness of multivariate matching polynomials, extending classical results to graphs with loops and hypergraphs, with implications for graph coverings and spectral graph theory.

## Contribution

It introduces a multivariate stability result for the $d$-matching polynomial, including graphs with loops, and generalizes the Heilmann-Lieb theorem to hypergraphs.

## Key findings

- Multivariate $d$-matching polynomial is stable.
- Real-rootedness of the $d$-matching polynomial is established.
- Hypergraphic generalization of Heilmann-Lieb theorem is provided.

## Abstract

The first part of this note concerns stable averages of multivariate matching polynomials. In proving the existence of infinite families of bipartite Ramanujan $d$-coverings, Hall, Puder and Sawin introduced the $d$-matching polynomial of a graph $G$, defined as the uniform average of matching polynomials over the set of $d$-sheeted covering graphs of $G$. We prove that a natural multivariate version of the $d$-matching polynomial is stable, consequently giving a short direct proof of the real-rootedness of the $d$-matching polynomial. Our theorem also includes graphs with loops, thus answering a question of said authors. Furthermore we define a weaker notion of matchings for hypergraphs and prove that a family of natural polynomials associated to such matchings are stable. In particular this provides a hypergraphic generalization of the classical Heilmann-Lieb theorem.