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

of a noncommutative ring R as follows: (1) if

has no sources and no sinks, then

is connected and diameter of

, denoted by diam(

) (resp. girth of

, denoted by g(

)) 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

is connected and diam(

) (resp. g(

)) is equal to or less than 3, in addition, if R is local, then there is a vertex of

which is adjacent to every other vertices in

; (3) if R is unit-regular, then

is connected and diam(

) (resp. g(

)) is equal to or less than 3. Next, we investigate the graph automorphisms group of

where

is the ring of 2 by 2 matrices over the galois field

(p is any prime).