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

Abstract

Let $H(n,q^2)$ be a non-degenerate Hermitian variety of $PG(n,q^2)$, $n\ge 2$. 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 $q\ne 2$, or $q=2$.