With the aid of a computer. we give a regular graph of girth 6 and valency 7, which has 90 vertices and show that this is the unique smallest graph with these properties.

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Letf(v,5)be the number of vertices of a(v,5)-cage(v≥3). We give an upper bound forf(v,5)which is considerably better than the previously known upper bounds. In particular, whenv=7, it coincides with the well-known Hoffman- Singleton graph.

Coordination sequences have been determined for tilings {