Matching polynomial

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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[edit]

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[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 = 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.

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 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[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 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[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 nO(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.