THE REGULAR GRAPH OF A NONCOMMUTATIVE 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(Gamma(R)) is a graph with all elements of R as vertices, and two distinct vertices x, y is an element of R are adjacent if and only if x + y is an element of Z(R). Let the regular graph of R, Reg(Gamma(R)), be the induced subgraph of T(Gamma(R)) on the regular elements of R. In 2008, Anderson and Badawi proved that the girth of the total graph and the regular graph of a commutative ring are contained in the set {3, 4, infinity}. In this paper, we extend this result to an arbitrary ring (not necessarily commutative). We also prove that if R is a reduced left Noetherian ring and 2 is not an element of Z(R), then the chromatic number and the clique number of Reg(Gamma(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 Reg(R) is finite, then R is finite.