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. Another equivalent definition of a Moore graph G is that it has girth g = 2k+1 and precisely cycles of length g, where n,m is the number of vertices (resp. edges) of G. They are in fact extremal with respect to the number of cycles whose length is the girth of the graph (Azarija & Klavžar 2015).
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 (Erdős, 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.
Let G be any graph with maximum degree d and diameter k, and consider the tree formed by breadth first search starting from any vertex v. This tree has 1 vertex at level 0 (v itself), and at most d vertices at level 1 (the neighbors of v). In the next level, there are at most d(d-1) vertices: each neighbor of v uses one of its adjacencies to connect to v and so can have at most d-1 neighbors at level 2. In general, a similar argument shows that at any level 1 ≤ i ≤ k, there can be at most d(d-1)i vertices. Thus, the total number of vertices can be at most
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