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The number of perfect matchings in a complete graph K n (with n even) is given by the double factorial (n − 1)!!. [13] The numbers of matchings in complete graphs, without constraining the matchings to be perfect, are given by the telephone numbers. [14] The number of perfect matchings in a graph is also known as the hafnian of its adjacency ...
Kőnig had announced in 1914 and published in 1916 the results that every regular bipartite graph has a perfect matching, [11] and more generally that the chromatic index of any bipartite graph (that is, the minimum number of matchings into which it can be partitioned) equals its maximum degree [12] – the latter statement is known as Kőnig's ...
Since any 0–1 matrix is the biadjacency matrix of some bipartite graph, Valiant's theorem implies [9] that the problem of counting the number of perfect matchings in a bipartite graph is #P-complete, and in conjunction with Toda's theorem this implies that it is hard for the entire polynomial hierarchy. [10] [11]
However, counting the number of perfect matchings, even in bipartite graphs, is #P-complete. This is because computing the permanent of an arbitrary 0–1 matrix (another #P-complete problem) is the same as computing the number of perfect matchings in the bipartite graph having the given matrix as its biadjacency matrix.
For every subset F of edges, the dot product 1 E(v) · 1 F represents the number of edges in F that are adjacent to v. Therefore, the following statements are equivalent: A subset F of edges represents a matching in G; For every node v in V: 1 E(v) · 1 F ≤ 1. A G · 1 F ≤ 1 V. The cardinality of a set F of edges is the dot product 1 E · 1 F.
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 .
Hall's condition is that for any group of sets from the collection, the total unique elements they contain is at least as large as the number of sets in the group. The graph theoretic formulation answers whether a finite bipartite graph has a perfect matching —that is, a way to match each vertex from one group uniquely to an adjacent vertex ...
One application of the Edmonds matrix of a bipartite graph is that the graph admits a perfect matching if and only if the polynomial det(A ij) in the x ij is not identically zero. Furthermore, the number of perfect matchings is equal to the number of monomials in the polynomial det(A), and is also equal to the permanent of .