On the matching polynomial of subdivision graphs

Let GG be a simple graph and let S(G)S(G) be the subdivision graph of GG, which is obtained from GG by replacing each edge of GG by a path of length two. In this paper, by the Principle of Inclusion and Exclusion we express the matching polynomial and Hosoya index of S(G)S(G) in terms of the matchings of GG. Particularly, if GG is a regular graph or a semi-regular bipartite graph, then the closed formulae of the matching polynomial and Hosoya index of S(G)S(G) are obtained. As an application, we prove a combinatorial identity.