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Concept
Moore graph
Summary
In graph theory, a Moore graph is a regular graph whose girth (the shortest cycle length) is more than twice its diameter (the distance between the farthest two vertices). If the degree of such a graph is d and its diameter is k, its girth must equal 2k + 1. This is true, for a graph of degree d and diameter k, if and only if its number of vertices equals
an upper bound on the largest possible number of vertices in any graph with this degree and diameter. Therefore, these graphs solve the degree diameter problem for their parameters.
Another equivalent definition of a Moore graph G is that it has girth g = 2k + 1 and precisely n/g(m – n + 1) cycles of length g, where n and m are, respectively, the numbers of vertices and edges of G. They are in fact extremal with respect to the number of cycles whose length is the girth of the graph.
Moore graphs were named by 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. 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−1 vertices. Thus, the total number of vertices can be at most
originally defined a Moore graph as a graph for which this bound on the number of vertices is met exactly.
Official source
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