# Moore Graph

In graph theory, a Moore graph is a regular graph of degree d and diameter k whose number of vertices equals the upper bound

An equivalent definition of a Moore graph is that it is a graph of diameter k with girth 2k + 1. Moore graphs were named by Hoffman & Singleton (1960) after Edward F. Moore, who posed the question of describing and classifying these graphs.

As well as having the maximum possible number of vertices for a given combination of degree and diameter, Moore graphs have the minimum possible number of vertices for a regular graph with given degree and girth. That is, any Moore graph is a cage (Erdõs, Rényi & Sós 1966). The formula for the number of vertices in a Moore graph can be generalized to allow a definition of Moore graphs with even girth as well as odd girth, and again these graphs are cages.

### Other articles related to "moore graph, moore graphs, graphs, graph":

Moore Graph - Examples
... The Hoffman–Singleton theorem states that any Moore graph with girth 5 must have degree 2, 3, 7, or 57 ... The Moore graphs are The complete graphs on n > 2 nodes ... n, girth 2n+1, degree 2, order 2n+1) The Petersen graph ...

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