  Let $R$ be a commutative ring with identity. Let $\Gamma(R)$ 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
$\Gamma_2(R)$ of $\Gamma(R)$ 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 $\Gamma_2(R)\setminus\J(R)$. In addition, it is shown
that for two finite semi-local rings $R$ and $S$, if $R$ is reduced, then
$\Gamma(R)\cong\Gamma(S)$ if and only if $R\cong S$.
