Let $R$ be a commutative ring with 1. In a 1995 paper in J. Algebra, Sharma and Bhatwadekar defined a graph on $R$ , $\Gamma \left( R \right)$ , 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 ${{\Gamma }_{2}}\left( R \right)$ of $\Gamma \left( R \right)$ that consists of non-unit elements. We investigate the behavior of ${{\Gamma }_{2}}\left( R \right)$ and ${{\Gamma }_{2}}\left( R \right)\backslash \text{J}\left( R \right)$ , where $\text{J}\left( R \right)$ is the Jacobson radical of $R$ . We associate the ring properties of $R$ , the graph properties of ${{\Gamma }_{2}}\left( R \right)$ , and the topological properties of $\text{Max}\left( R \right)$ . Diameter, girth, cycles and dominating sets are investigated, and algebraic and topological characterizations are given for graphical properties of these graphs.