On the Comaximal Graph of a Commutative Ring

摘要

Let $R$ be a commutative ring with $1$. In [P. K. Sharma, S. M. Bhatwadekar, A note on graphical representation of rings, J.Algebra 176(1995) 124-127], Sharma and Bhatwadekar defined agraph on $R$, $Gamma(R)$, with vertices as elements of $R$, wheretwo 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 investigatethe behavior of $Gamma_2(R)$ and $Gamma_2(R) setminus operatorname{J}(R)$,where $operatorname{J}(R)$ is the Jacobson radical of $R$. We associate thering properties of $R$, the graph properties of $Gamma_2(R)$ andthe topological properties of $operatorname{Max}(R)$. Diameter, girth, cyclesand dominating sets are investigated and the algebraic and thetopological characterizations are given for graphical properties of these graphs.