# 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.

## Definition

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

$m_G(x) := \sum_{k\geq0} m_k x^k.$

Another definition gives the matching polynomial as

$M_G(x) := \sum_{k\geq0} (-1)^k m_k x^{n-2k}.$

A third definition is the polynomial

$\mu_G(x,y) := \sum_{k\geq0} m_k x^k y^{n-2k}.$

Each type has its uses, and all are equivalent by simple transformations. For instance,

$M_G(x) = x^n m_G(-x^{-2})$

and

$\mu_G(x,y) = y^n m_G(x/y^2).$

## Connections to other polynomials

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:

$M_{K_{m,n}}(x) = n! L_n^{(m-n)}(x^2). \,$

If G is the complete graph Kn, then MG(x) is an Hermite polynomial:

$M_{K_n}(x) = H_n(x), \,$

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 its characteristic polynomial.

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.

## Complementation

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.

## Applications in chemical informatics

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.

## Computational complexity

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)