Generalized Polygons and Moore Graphs

A graph with diameter d has girth at most 2d + 1, while a bipartite graph with diameter d has girth at most 2d. While these are very simple bounds, the graphs that arise when they are met are particularly interesting. Graphs with diameter d and girth 2d + 1 are known as Moore graphs. They were introduced by Hoffman and Singleton in a paper that can be viewed as one of the prime sources of algebraic graph theory. After considerable development, the tools they used in this paper led to a proof that a Moore graph has diameter at most two. They themselves proved that a Moore graph of diameter two must be regular, with valency 2, 3, 7, or 57. We will provide the machinery to prove this last result in our work on strongly regular graphs in Chapter 10.