Co-maximal graph of non-commutative rings

Let R be a ring (not necessarily commutative) with 1. Following Sharma and Bhatwadekar [P.K. Sharma, S.M. Bhatwadekar, A note on graphical representation of rings, J. Algebra 176 (1995) 124–127], we define a graph on R, Γ(R), with vertices as elements of non-units of R, where two distinct vertices a and b are adjacent if and only if Ra+Rb=R. In this paper, we investigate the behavior of Γ(R). We are able to prove that if R is left Artinian then Γ(R)-J(R) is connected and if Γ(R)-J(R) is a forest then Γ(R)-J(R) is a star graph, where J(R) is the Jacobson radical of R. For any finite field Fq, we obtain the minimal degree, the maximal degree, the connectivity, the clique number and the chromatic number of Γ(Mn(Fq)). Finally, for any finite field and any integer n⩾2, we prove that if R is a ring with identity and , then . We also prove that if R and S are two finite commutative rings with identity, and n,m⩾2 such that , then n=m and provided that R is reduced.