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.
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
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
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" = "K""m","n", the complete bipartite graph, then the second type of matching polynomial is related to the generalized Laguerre polynomial "L""n""α"("x") by the identity:
If "G" is the complete graph "K""n", then "M""G"("x") is an Hermite polynomial:
where "H""n"("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 "M""G"("x") is a Chebyshev polynomial. In this case
μ"G"(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 . The other is an integral identity due to .
There is a similar relation for a subgraph "G" of "K""m","n" and its complement in "K""m","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 "m""G"(1) .
The third type of matching polynomial was introduced by 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 . 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" .
The matching polynomial of a graph with "n" vertices and clique-width "k" may be computed in time "n"O("k") .