The higher-order matching polynomial of a graph

Given a graph G with n vertices, let p ( G , j ) denote the number of ways j mutually nonincident edges can be selected in G . The polynomial M ( x ) = ∑ j = 0 [ n / 2 ] ( − 1 ) j p ( G , j ) x n − 2 j , called the matching polynomial of G , is closely related to the Hosoya index introduced in applications in physics and chemistry. In this work we generalize this polynomial by introducing the number of disjoint paths of length t , denoted by p t ( G , j ) . We compare this higher-order matching polynomial with the usual one, establishing similarities and differences. Some interesting examples are given. Finally, connections between our generalized matching polynomial and hypergeometric functions are found.