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The second type of matching polynomial has remarkable connections with orthogonal polynomials. For instance, if G = K m,n, the complete bipartite graph, then the second type of matching polynomial is related to the generalized Laguerre polynomial L n α (x) by the identity: , =!
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.
In a cubic graph with a perfect matching, the edges that are not in the perfect matching form a 2-factor. By orienting the 2-factor, the edges of the perfect matching can be extended to paths of length three, say by taking the outward-oriented edges. This shows that every cubic, bridgeless graph decomposes into edge-disjoint paths of length ...
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.
The case of exact graph matching is known as the graph isomorphism problem. [1] The problem of exact matching of a graph to a part of another graph is called subgraph isomorphism problem. Inexact graph matching refers to matching problems when exact matching is impossible, e.g., when the number of vertices in the two graphs are different. In ...
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 ...
Bivariegated graph; Cage (graph theory) Cayley graph; Circle graph; Clique graph; Cograph; Common graph; Complement of a graph; Complete graph; Cubic graph; Cycle graph; De Bruijn graph; Dense graph; Dipole graph; Directed acyclic graph; Directed graph; Distance regular graph; Distance-transitive graph; Edge-transitive graph; Interval graph ...