Graphs associated to co-maximal ideals of commutative rings

Let R be a commutative ring R with 1. In [P.K. Sharma, S.M. Bhatwadekar, A note on graphical representation of rings, J. Algebra 176 (1995) 124–127], Sharma and Bhatwadekar defined a graph on R, Γ(R), with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if aR+bR=R. Let Γ2(R) be the subgraph of Γ(R) induced by non-unit elements. In this paper, we derive several properties of Γ2(R). We are able to characterize those rings R for which Γ2(R)−J(R) is a forest and those rings R for which Γ2(R)−J(R) is Eulerian, where J(R) is the Jacobson radical of R. We are also able to show that ω(Γ2(R))=Max(R), where ω(G) is the clique number of a graph G and Max(R) is the set of maximal ideals of R. Moreover, we find all finite rings R such that the genus of Γ2(R) (resp. Γ(R)) is at most one.