# 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.

## Contents

## Definition[edit]

Several different types of matching polynomials have been defined. Let *G* be a graph with *n* vertices and let *m _{k}* 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[edit]

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 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 *M*_{G}(*x*) is a Chebyshev polynomial. In this case
μ_{G}(1,*x*) is a Fibonacci polynomial or Lucas polynomial respectively.

## Complementation[edit]

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 *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[edit]

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) (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[edit]

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 *n*^{O(k)} (Makowsky et al. 2006).

## References[edit]

- Courcelle, B.; Makowsky, J. A.; Rotics, U. (2001), "On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic" (PDF),
*Discrete Applied Mathematics*,**108**(1–2): 23–52, doi:10.1016/S0166-218X(00)00221-3. - Farrell, E. J. (1980), "The matching polynomial and its relation to the acyclic polynomial of a graph",
*Ars Combinatoria*,**9**: 221–228. - Godsil, C.D. (1981), "Hermite polynomials and a duality relation for matchings polynomials",
*Combinatorica*,**1**(3): 257–262, doi:10.1007/BF02579331. - Gutman, Ivan (1991), "Polynomials in graph theory", in Bonchev, D.; Rouvray, D. H. (eds.),
*Chemical Graph Theory: Introduction and Fundamentals*, Mathematical Chemistry,**1**, Taylor & Francis, pp. 133–176, ISBN 978-0-85626-454-2. - Jerrum, Mark (1987), "Two-dimensional monomer-dimer systems are computationally intractable",
*Journal of Statistical Physics*,**48**(1): 121–134, doi:10.1007/BF01010403. - Makowsky, J. A.; Rotics, Udi; Averbouch, Ilya; Godlin, Benny (2006), "Computing graph polynomials on graphs of bounded clique-width",
*Proc. 32nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG '06)*(PDF), Lecture Notes in Computer Science,**4271**, Springer-Verlag, pp. 191–204, doi:10.1007/11917496_18. - Riordan, John (1958),
*An Introduction to Combinatorial Analysis*, New York: Wiley. - Zaslavsky, Thomas (1981), "Complementary matching vectors and the uniform matching extension property",
*European Journal of Combinatorics*,**2**: 91–103, doi:10.1016/s0195-6698(81)80025-x.