Examples
The Hoffman–Singleton theorem states that any Moore graph with girth 5 must have degree 2, 3, 7, or 57. The Moore graphs are:
- The complete graphs on n > 2 nodes. (diameter 1, girth 3, degree n-1, order n)
- The odd cycles . (diameter n, girth 2n+1, degree 2, order 2n+1)
- The Petersen graph. (diameter 2, girth 5, degree 3, order 10)
- The Hoffman–Singleton graph. (diameter 2, girth 5, degree 7, order 50)
- A graph of diameter 2, girth 5, degree 57 and order 3250, whose existence is not known. It is currently unknown whether such graph (if exists) is unique.
Unlike all other Moore graphs, Higman proved that the unknown Moore graph cannot be vertex-transitive.
If the generalized definition of Moore graphs that allows even girth graphs is used, the Moore graphs also include the even cycles, the complete bipartite graphs with girth four, the Heawood graph with degree 3 and girth 6, and the Tutte–Coxeter graph with degree 3 and girth 8. More generally, it is known (Bannai & Ito 1973; Damerell 1973) that, other than the graphs listed above, all Moore graphs must have girth 5, 6, 8, or 12.
Read more about this topic: Moore Graph
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