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Abstract. 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 Γ(R) of a noncommutative ring R as follows: (1) if Γ(R) has no sources and no sinks, then Γ(R) is connected and diameter of Γ(R), denoted by diam(Γ(R)) (resp. girth of Γ(R), denoted by g(Γ(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 Γ(R) is connected and diam(Γ(R)) (resp. g(Γ(R))) is equal to or less than 3, in addition, if R is local, then there is a vertex of Γ(R) which is adjacent to every other vertices in Γ(R); (3) if R is unitregular, then Γ(R) is connected and diam(Γ(R)) (resp. g(Γ(R))) is equal to or less than 3. Next, we investigate the graph automorphisms group of Γ(Mat 2 (Zp)) where Mat 2 (Zp) is the ring of 2 by 2 matrices over the galois field Zp (p is any prime).

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Abstract. We study the structure of right duo ring property when it is restricted within the group of units, and introduce the concept of right unit-duo. This newly introduced property is first observed to be not left-right symmetric, and we examine several conditions to ensure the symmetry. Right unit-duo rings are next proved to be Abelian, by help of which the class of noncommutative right unit-duo rings of minimal order is completely determined up to isomorphism. We also investigate some properties of right unit-duo rings which are concerned with annihilating conditions.Throughout this paper all rings are associative with identity unless otherwise stated. Let R be a ring. X(R) denotes the set of all nonzero nonunits in R, and G(R) denotes the group of all units in R., and N (R) denote the Jacobson radical, the set of all idempotents, and the set of all nilpotent elements in R, respectively. |S| denotes the cardinality of a subset S of R. Write R * = R\{0}. Z (Z n ) denotes the ring of integers (modulo n). Q denotes the field of rational numbers. GF (p n ) denotes the Galois field of order p n .Denote the n by n full (resp., upper triangular) matrix ring over R by Mat n (R) (resp., U n (R)) and use e ij for the matrix with (i, j)-entry 1 and elsewhere 0. Following the literature, we write D n (R) = {(a ij ) ∈ U n (R) | all diagonal entries are equal } and V n (R) = {(a ij ) ∈ D n (R) | a 1k = a 2(k+1) = · · · = a hn for h = 1, 2, . . . , n − 1 and k = 2, . . . , n}.Due to Feller [7], a ring is called right (resp. left) duo if every right (resp. left) ideal is an ideal; a ring is called duo if it is both right and left duo. It is easily shown that idempotents of right duo rings are central. There are very useful results for duo rings in [3,16,24].