Stable multivariate generalizations of matching polynomials

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.