Automorphism related parameters of a graph associated to a finite vector space

In this paper, we discuss automorphism related parameters of a graph associated to a finite vector space. The fixing neighborhood of a pair $(u,v)$ of vertices of a graph $G$ is the set of all those vertices $w$ of $G$, such that the orbits of $u$ and $v$ under the action of stabilizer of $w$ are not equal. The fixed number of a graph is the minimum number $k$ such that every subset of vertices of $G$ of cardinality $k$ is a fixing set of $G$. We study some properties of automorphisms of a graph associated to finite vector space and find the fixing neighborhood of pair of vertices of the graph. We also find the fixed number of the graph. It is shown that, for every positive integer $N$, there exists a graph $G$ with $fxd(G)-fix(G)\geq N$, where $fxd(G)$ is the fixed number and $fix(G)$ is the fixing number of $G$.
Comment: 12 pages