Bounding Vertices By Degree and Diameter
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
Hoffman & Singleton (1960) originally defined a Moore graph as a graph for which this bound on the number of vertices is met exactly. Therefore, any Moore graph has the maximum number of vertices possible among all graphs with maximum degree d and diameter k.
Later, Singleton (1968) showed that Moore graphs can equivalently be defined as having diameter k and girth 2k+1; these two requirements combine to force the graph to be d-regular for some d and to satisfy the vertex-counting formula.
Read more about this topic: Moore Graph
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