“School of Mathematics”
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Paper IPM / M / 8541  


Abstract:  
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 Γ_{2}(R) of Γ(R)
which consists of nonunit elements. We look at the connectedness
and the diameter of this graph. We completely characterize the
diameter of the graph Γ_{2}(R)\J(R). In addition,
it is shown that for two finite semilocal rings R and S, if
R is reduced, then Γ(R) ≅ Γ(S) if and only if R ≅ S.
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