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 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 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 diam($Gamma(R)$) (resp. g($Gamma(R)$)) is equal to or less than 3. Next, we investigate the graph automorphisms group of $Gamma(Mat_2(mathbb{Z}_p))$ where $Mat_2(mathbb{Z}_p)$ is the ring of 2 by 2 matrices over the galois field $mathbb{Z}_p$ (p is any prime).