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The '''Hoffman–Singleton theorem''' states that any Moore graph with girth 5 must have degree 2, 3, 7, or 57. The Moore graphs are:{{sfnp|Bollobás|1998|loc=Theorem 19, p. 276}}
* The [[complete graph]]s <math> K_n </math> on n > 2 nodes
* The odd [[Cycle graph|cycles]] <math> C_{2n+1} </math>
* The [[Petersen graph]]
* The [[Hoffman–Singleton graph]]
* A hypothetical graph of diameter 2, girth 5, degree 57 and order 3250
Unlike all other Moore graphs, [[Graham Higman|Higman]] proved that the unknown Moore graph cannot be [[Vertex-transitive graph|vertex-transitive]]. Mačaj and Širáň further proved that the order of the automorphism group of such a graph is at most 375.{{sfnp|Mačaj|Širáň|2010}}
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