Journal of the
Korean Mathematical Society JKMS

Let $R = {\rm Mat}_{2}(F)$ be the ring of all 2 by 2 matrices over a finite field $F$, $X$ the set of all nonzero, nonunits of $R$ and $G$ the group of all units of $R$. After investigating some properties of orbits under the left (and right) regular action on $X$ by $G$, we show that the graph automorphisms group of $\Gamma (R)$ (the zero-divisor graph of $R$) is isomorphic to the symmetric group $S_{|F|+1}$ of degree $|F| + 1$.

Keywords: zero-divisor graph, left (resp. right) regular action, orbit, graph automorphisms group