Moore Graphs and Cycles Are Extremal Graphs for Convex Cycles

Let ρ ( 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 ρ ( G ) ≤ 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.