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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.
A maximal matching is a matching M of a graph G that is not a subset of any other matching. A matching M of a graph G is maximal if every edge in G has a non-empty intersection with at least one edge in M. The following figure shows examples of maximal matchings (red) in three graphs. A maximum matching (also known as maximum-cardinality ...
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.
Maximum cardinality matching is a fundamental problem in graph theory. [1] We are given a graph G, and the goal is to find a matching containing as many edges as possible; that is, a maximum cardinality subset of the edges such that each vertex is adjacent to at most one edge of the subset. As each edge will cover exactly two vertices, this ...
A matching M is called perfect if every vertex v in V is contained in exactly one hyperedge of M. This is the natural extension of the notion of perfect matching in a graph. A fractional matching M is called perfect if for every vertex v in V, the sum of fractions of hyperedges in M containing v is exactly 1.
The minimum number of colors (induced matchings) needed to cover the graph is the graph's strong chromatic index. In graph theory , an induced matching or strong matching is a subset of the edges of an undirected graph that do not share any vertices (it is a matching ) and these are the only edges connecting any two vertices which are endpoints ...
An graph (or a component) with an odd number of vertices cannot have a perfect matching, since there will always be a vertex left alone. The goal is to characterize all graphs that do not have a perfect matching. Start with the most obvious case of a graph without a perfect matching: a graph with an odd number of vertices.
Matching (graph theory) – matching between different vertices of the graph; usually unrelated to preference-ordering. Envy-free matching – a relaxation of stable matching for many-to-one matching problems; Rainbow matching for edge colored graphs; Stable matching polytope; Lattice of stable matchings