The Sylvester graph and Moore graphs

Abstract

The combinatorial structure of a famous graph with large girth, namely the Sylvester graph, is studied. Simple techniques, such as two-way counting, partitions, circuit chasing and covers are used to identify smaller structures and to show that there are no other graphs that share a small number of regularity properties with it. As a consequence, we show the same for the Hoffman–Singleton graph. We notice that some much larger graphs with large girth have similar properties, and could be studied using the same techniques. In particular, we show that just as the Hoffman–Singleton graph contains the Sylvester graph, a Moore graph of valency 57, whose existence is a famous open problem, must contain a subgraph with a structure that is similar to the one we derived for the Sylvester graph.