On the Number of \(5\)-Matchings in Boron-Nitrogen Fullerene Graphs

Abstract

Given a graph \(G = (V,E)\), a matching \(M\) of \(G\) is a subset of \(E\), such that every vertex of \(V\) is incident to at most one edge of \(M\). A \(k\)-matching is a matching with \(k\) edges. The total number of matchings in \(G\) is used in chemoinformatics as a structural descriptor of a molecular graph. Recently, Vesalian and Asgari (MATCH Commun. Math. Comput. Chem. \(69 (2013) 33–46\)) gave a formula for the number of \(5\)-matchings in triangular-free and \(4\)-cycle-free graphs based on the degrees of vertices and the number of vertices, edges, and \(5\)-cycles. But, many chemical graphs are not triangular-free or \(4\)-cycle-free, e.g., boron-nitrogen fullerene graphs (or BN-fullerene graphs). In this paper, we take BN-fullerene graphs into consideration and obtain formulas for the number of \(5\)-matchings based on the number of hexagons.