The Automorphism Group of $NU(3,q^2)$

TL;DR

This paper determines the automorphism group of a specific strongly regular graph derived from a Hermitian variety, showing it is either related to the projective unitary group or a wreath product, depending on the field size.

Contribution

It identifies the automorphism group of the graph $NU(3,q^2)$, revealing a dichotomy based on the value of q, which was previously unknown.

Findings

Automorphism group is isomorphic to $P ext{Gamma}U(3,q)$ for $q eq 2$.

Automorphism group is isomorphic to $S_3 ext{ wr } S_4$ for $q=2$.

The graph $NU(3,q^2)$ is strongly regular.

Abstract

Let $H(n, q^2)$ be a non-degenerate Hermitian variety of $PG(n,q^2)$, $n \geq 2$. Let $NU(n+1,q^2)$ be the graph whose vertices are the points of $PG(n,q^2) \setminus H(n,q^2)$ and two vertices $u,~v$ are adjacent if the line joining $u$ and $v$ is tangent to $H(n, q^2 )$. Then $NU(n + 1, q^2)$ is a strongly regular graph. In this paper we show that the automorphism group of the graph $NU(3,q^2)$ is isomorphic either to $P\Gamma U(3,q)$, the automorphism group of the projective unitary group $PGU(3,q)$, or to $S_{3} \wr S_{4}$, according as $q \neq 2$, or $q=2$.