The ideal-based zero-divisor graph of commutative chained rings (Q2875777)

Let \(I\) be a proper ideal of a commutative ring \(R\) with \(1\neq 0\). The ideal-based zero-divisor graph of \(R\) with respect to \(I\) is the simple graph \(\Gamma_I(R)\), where \(V(\Gamma_I(R))=\{x\in R\setminus I \mid (\exists\;y\in R\setminus I) \;xy\in I\}\) and \(E(\Gamma_I(R))=\{\{x,y\} \mid x\neq y \wedge xy\in I\}\). The authors study \(\Gamma_I(R)\) for commutative rings \(R\) in which \(R/I\) is a chained ring. In particular they investigate some properties of the subgraphs of \(\Gamma_I(R)\). It allows them, among others, to characterize the cases when \(\Gamma_I(R)\) is a complete graph, to bound from above (by \(2\)) the diameter of \(\Gamma_I(R)\) and to give necessary and sufficient conditions for \(\Gamma_I(R)\) to have diameter \(0\), \(1\) or \(2\), respectively. They also characterize the cases when \(\operatorname{gr}(\Gamma_I(R))=3\) and \(\operatorname{gr}(\Gamma_I(R))=\infty\) and prove that these are the only possible cases.