Let R be a commutative ring with 1. In a 1995 paper in J. Algebra, 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 Ra + Rb = R. In this paper, we consider a subgraph Γ2(R) of Γ(R) that consists of non-unit elements. We investigate the behavior of Γ2(R) and Γ2(R)\J(R), where J(R) is the Jacobson radical of R. We associate the ring properties of R, the graph properties of Γ(R), and the topological properties of Max(R). Diameter, girth, cycles and dominating sets are investigated, and algebraic and topological characterizations are given for graphical properties of these graphs.
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