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Maximum weight matching of 2 graphs. The first is also a perfect matching, while the second is far from it with 4 vertices unaccounted for, but has high value weights compared to the other edges in the graph. In computer science and graph theory, the maximum weight matching problem is the problem of finding, in a weighted graph, a matching in ...
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.
In graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph G with edges E and vertices V, a perfect matching in G is a subset M of E, such that every vertex in V is adjacent to exactly one edge in M. The adjacency matrix of a perfect matching is a symmetric permutation matrix.
A graph can only contain a perfect matching when the graph has an even number of vertices. A near-perfect matching is one in which exactly one vertex is unmatched. Clearly, a graph can only contain a near-perfect matching when the graph has an odd number of vertices, and near-perfect matchings are maximum matchings. In the above figure, part (c ...
The problem of finding a matching with maximum weight in a weighted graph is called the maximum weight matching problem, and its restriction to bipartite graphs is called the assignment problem. If each vertex can be matched to several vertices at once, then this is a generalized assignment problem .
In graph theory, the Tutte matrix A of a graph G = (V, E) is a matrix used to determine the existence of a perfect matching: that is, a set of edges which is incident with each vertex exactly once. If the set of vertices is V = { 1 , 2 , … , n } {\displaystyle V=\{1,2,\dots ,n\}} then the Tutte matrix is an n -by- n matrix A with entries
In the mathematical discipline of graph theory the Tutte–Berge formula is a characterization of the size of a maximum matching in a graph. It is a generalization of Tutte theorem on perfect matchings , and is named after W. T. Tutte (who proved Tutte's theorem) and Claude Berge (who proved its generalization).
Unlike bipartite matching, the key new idea is that an odd-length cycle in the graph (blossom) is contracted to a single vertex, with the search continuing iteratively in the contracted graph. The algorithm runs in time O (| E || V | 2 ) , where | E | is the number of edges of the graph and | V | is its number of vertices .