Unitary graphs

Abstract

Unitary graphs are arc-transitive graphs with vertices the flags of Hermitian unitals and edges defined by certain elements of the underlying finite fields. They played a significant role in a recent classification of a class of arc-transitive graphs that admit an automorphism group acting imprimitively on the vertices. In this paper we prove that all unitary graphs are connected of diameter two and girth three. Based on this we obtain, for any prime power $q > 2$, a lower bound of order $O(\Delta^{5/3})$ on the maximum number of vertices in an arc-transitive graph of degree $\Delta = q(q^2-1)$ and diameter two.