Comaximal graph of commutative rings

Abstract

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 $R\cong S$.