Moore graphs and cycles are extremal graphs for convex cycles

TL;DR

This paper establishes an extremal property of Moore graphs and cycles related to convex cycles, providing a new characterization of Moore graphs through an inequality involving convex cycles.

Contribution

It proves an upper bound on the number of convex cycles in a graph and characterizes Moore graphs and cycles as extremal cases where equality holds.

Findings

Moore graphs and cycles maximize convex cycles among graphs of given parameters

The derived inequality characterizes Moore graphs uniquely

A new characterization of Moore graphs of diameter 2 and degree 57 is provided

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 $\rho(G) \leq \frac{n}{g}(m-n+1)$ 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.