Matching polynomial

In the mathematical fields of graph theory and combinatorics, a matching polynomial (sometimes called an acyclic polynomial) is a generating function of the numbers of matchings of various sizes in a graph. It is one of several graph polynomials studied in algebraic graph theory. Several different types of matching polynomials have been defined. Let G be a graph with n vertices and let mk be the number of k-edge matchings. One matching polynomial of G is Another definition gives the matching polynomial as A third definition is the polynomial Each type has its uses, and all are equivalent by simple transformations. For instance, and The first type of matching polynomial is a direct generalization of the rook polynomial. The second type of matching polynomial has remarkable connections with orthogonal polynomials. For instance, if G = Km,n, the complete bipartite graph, then the second type of matching polynomial is related to the generalized Laguerre polynomial Lnα(x) by the identity: If G is the complete graph Kn, then MG(x) is an Hermite polynomial: where Hn(x) is the "probabilist's Hermite polynomial" (1) in the definition of Hermite polynomials. These facts were observed by . If G is a forest, then its matching polynomial is equal to the characteristic polynomial of its adjacency matrix. If G is a path or a cycle, then MG(x) is a Chebyshev polynomial. In this case μG(1,x) is a Fibonacci polynomial or Lucas polynomial respectively. The matching polynomial of a graph G with n vertices is related to that of its complement by a pair of (equivalent) formulas. One of them is a simple combinatorial identity due to . The other is an integral identity due to . There is a similar relation for a subgraph G of Km,n and its complement in Km,n. This relation, due to Riordan (1958), was known in the context of non-attacking rook placements and rook polynomials. The Hosoya index of a graph G, its number of matchings, is used in chemoinformatics as a structural descriptor of a molecular graph. It may be evaluated as mG(1) .