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.
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,
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:
If G is the complete graph Kn, then MG(x) is an Hermite polynomial:
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
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)
- Courcelle, B.; Makowsky, J. A.; Rotics, U. (2001), "On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic", 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., 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), 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.