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 , R e g ( 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 R e g ( Γ ( R ) ) are the same and they are 2 r , 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 R e g ( R ) is finite, then R is finite.