The matching polynomials of hypergraphs and weighted hypergraphs

Abstract

Let [Formula: see text] be the set of the connected [Formula: see text]-uniform linear hypergraphs with [Formula: see text] vertices, where [Formula: see text]. The matching polynomial of a hypergraph [Formula: see text] is denoted by [Formula: see text], where [Formula: see text]. Several properties on the roots of [Formula: see text] are derived. We establish different expressions for [Formula: see text], such as a higher-order differential formula and an integral formula. A new concept is also introduced for the weighted matching polynomials of weighted hypergraphs, for which several basic calculating formulas are obtained. Based on the results we obtained, some formulas for counting the number of perfect matchings of [Formula: see text] are directly derived. Finally, for the weighted matching polynomials of the weighted loose paths and the weighted loose cycles, we not only obtain their recursive formulas, but also deduce their specific expression formulas.