Is there another infinite family of Moore graphs besides the sequence of cycle graphs $C_{2d+1}$?
(By definition a Moore graph must contain a cycle of length $2d+1$ where $d$ is its diameter, so complete graphs are ruled out by this reason.)
Is there another infinite family of Moore graphs besides the sequence of cycle graphs $C_{2d+1}$?
(By definition a Moore graph must contain a cycle of length $2d+1$ where $d$ is its diameter, so complete graphs are ruled out by this reason.)
The set $\text{{Moore graphs}} \setminus\left(\text{{odd cycles}}\cup \text{{complete graphs}}\right)$ is finite. It consists of the trivial graph with one vertex, the Petersen graph, the Hoffman–Singleton graph, and some finite (possibly zero) number of graphs with diameter $2$ and degree $57$. See, e.g., The Hoffman–Singleton graph and outer automorphisms by Markus Junker or Moore graphs and beyond: A survey of the degree/diameter problem.