On the matching polynomial of subdivision graphs

Let image be a simple graph and let image be the subdivision graph of image, which is obtained from image by replacing each edge of image 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 image in terms of the matchings of image. Particularly, if image is a regular graph or a semi-regular bipartite graph, then the closed formulae of the matching polynomial and Hosoya index of image are obtained. As an application, we prove a combinatorial identity.