Unitary graphs

TL;DR

This paper studies the properties of unitary graphs, showing they are connected with diameter two and girth three, and derives bounds on the size of certain arc-transitive graphs based on these properties.

Contribution

It proves that all unitary graphs are connected with diameter two and girth three, and establishes a lower bound on the size of arc-transitive graphs with diameter two.

Findings

All unitary graphs are connected with diameter two.

All unitary graphs have girth three.

A lower bound of O(Δ^{5/3}) on the number of vertices in certain arc-transitive 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.