Moore graphs and cycles are extremal graphs for convex cycles

Abstract

Let $\rho (G)$ denote the number of convex cycles of a simple graph G of order n, size m, and girth 3 <= g <=n. It is proved that $\rho (G)\le \frac{n}{g}(m-n+1)$ and that equality holds if and only if G is an even cycle or a Moore graph. The equality also holds for a possible Moore graph of diameter 2 and degree 57 thus giving a new characterization of Moore graphs.