Journal of the
Korean Mathematical Society JKMS

Let $R$ be a ring with identity, $X$ the set of all nonzero, nonunits of $R$ and $G$ the group of all units of $R$. First, we investigate some connected conditions of the zero-divisor graph $\Gamma (R)$ of a noncommutative ring $R$ as follows: (1) if $\Gamma (R)$ has no sources and no sinks, then $\Gamma (R)$ is connected and diameter of $\Gamma (R)$, denoted by ${\rm diam}(\Gamma (R))$ (resp. girth of $\Gamma (R)$, denoted by $g(\Gamma(R))$) is equal to or less than 3; (2) if $X$ is a union of finite number of orbits under the left (resp. right) regular action on $X$ by $G$, then $\Gamma (R)$ is connected and ${\rm diam}(\Gamma (R))$ (resp. $g(\Gamma(R))$) is equal to or less than 3, in addition, if $R$ is local, then there is a vertex of $\Gamma (R)$ which is adjacent to every other vertices in $\Gamma (R)$; (3) if $R$ is unit-regular, then $\Gamma (R)$ is connected and ${\rm diam}(\Gamma (R))$ (resp. $g(\Gamma(R))$) is equal to or less than 3. Next, we investigate the graph automorphisms group of $\Gamma ({\rm Mat}_{2}(\mathbb Z_{p}))$ where ${\rm Mat}_{2}(\mathbb Z_{p})$ is the ring of 2 by 2 matrices over the galois field $\mathbb Z_{p}$ ($p$ is any prime).

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

MSC numbers: Primary 05C20; Secondary 16W22