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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 $\Gamma_{2}(R)\backslash J(R)$. 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 $R
\cong S$.
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