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 2*k* + 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.

Read more about Moore Graph: Bounding Vertices By Degree and Diameter, Moore Graphs As Cages, Examples

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