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 Godsil (1981).
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 Zaslavsky (1981). The other is an integral identity due to Godsil (1981).
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) ( Gutman 1991 ).
The third type of matching polynomial was introduced by Farrell (1980) as a version of the "acyclic polynomial" used in chemistry.
On arbitrary graphs, or even planar graphs, computing the matching polynomial is #P-complete ( Jerrum 1987 ). However, it can be computed more efficiently when additional structure about the graph is known. In particular, computing the matching polynomial on n-vertex graphs of treewidth k is fixed-parameter tractable: there exists an algorithm whose running time, for any fixed constant k, is a polynomial in n with an exponent that does not depend on k( Courcelle, Makowsky & Rotics 2001 ). The matching polynomial of a graph with n vertices and clique-width k may be computed in time nO(k)( Makowsky et al. 2006 ).
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