In this thesis, we study Moore graphs, which attain the theoretical upper bound for the number of vertices given a specific maximal degree and diameter. We analyze their structural properties, which later serve as a basis for studying their existence. We explicitly construct the trivial Moore graphs and prove the uniqueness of the Hoffman-Singleton graph, which, together with the Petersen graph, is one of the two known nontrivial Moore graphs. We provide a detailed classification of all Moore graphs and, in the only remaining case, the Aschbacher graph of valency 57, show that if it exists, it is not distance-transitive.
We then shift our perspective and show that the Moore bound also provides a lower bound for the number of vertices in graphs with prescribed degree and girth. Sharpening this bound leads to the study of bipartite Moore graphs, which naturally correspond to regular generalized polygons. We analyze their constructions, encountering projective planes, orthogonal, symplectic, and Tits generalized quadrangles, as well as classical generalized hexagons. We conclude with a classification of the known regular generalized polygons, thereby tracing a path from a rigid extremal problem in graph theory to the rich geometry of rank 2.
|
