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Let R be a commutative ring with identity. Let Γ(R) be a graph 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) which consists of non-unit elements. We look at the connectedness and the diameter of this graph. We completely characterize the diameter of the graph Γ₂(R)∖J(R). In addition, it is shown that for two finite semi-local rings R and S, if R is reduced, then Γ(R)≅Γ(S) if and only if R≅S.