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It has 104 edges and 52 vertices and is currently the smallest known example of a 4-regular matchstick graph. [3] It is a rigid graph. [4] Every 4-regular matchstick graph contains at least 20 vertices. [5] Examples of 4-regular matchstick graphs are currently known for all number of vertices ≥ 52 except for 53, 55, 56, 58, 59, 61 and 62.
207 is a Wedderburn-Etherington number. [1] There are exactly 207 different matchstick graphs with eight edges. [2] [3] 207 is a deficient number, as 207's proper divisors (divisors not including the number itself) only add up to 105: + + + + = <.
Every matchstick graph is a planar graph, [14] but some otherwise-planar unit distance graphs (such as the Moser spindle) have a crossing in every representation as a unit distance graph. Additionally, in the context of unit distance graphs, the term 'planar' should be used with care, as some authors use it to refer to the plane in which the ...
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 ...
The arboricity of a graph is a measure of how dense the graph is: graphs with many edges have high arboricity, and graphs with high arboricity must have a dense subgraph. In more detail, as any n-vertex forest has at most n-1 edges, the arboricity of a graph with n vertices and m edges is at least ⌈ / ⌉. Additionally, the subgraphs of any ...
Therefore, penny graphs have also been called minimum-distance graphs, [3] smallest-distance graphs, [4] or closest-pairs graphs. [5] Similarly, in a mutual nearest neighbor graph that links pairs of points in the plane that are each other's nearest neighbors , each connected component is a penny graph, although edges in different components ...
A parity graph (the unique smallest cubic, matchstick graph) that is neither distance-hereditary nor bipartite. In graph theory, a parity graph is a graph in which every two induced paths between the same two vertices have the same parity: either both paths have odd length, or both have even length. [1]
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 ...