The Regular Graph of a Non-Commutative Ring

Let R be a ring and Z(R) be the set of all zero-divisors of R. The total graph of R, denoted by T(Γ(R)) is a graph with all elements of R as vertices, and two distinct vertices x,y∈R are adjacent if and only if x+y∈Z(R). Let the regular graph of R,Reg(Γ(R)), be the induced subgraph of T(Γ(R)) on the regular elements of R. In 2008, Anderson and Badawi proved that the girth of total graph and regular graph of a commutative ring are contained in the set {3,4,∞}. In this paper, we extend this result to an arbitrary ring (not necessarily commutative). Also, we prove that if R is a reduced left Noetherian ring and 2∉Z(R), then the chromatic number and the clique number of Reg(Γ(R)) are the same and they are 2r, where r is the number of minimal prime ideals of R. Among other results we show that if R is a semiprime left Noetherian ring and Reg(R) is finite, then R is finite.