On the diameter and girth of ideal-based zero-divisor graphs (Q2898758)

For a given ideal \(I\) of a ring \(R\) the zero-divisor graph \(\Gamma_I (R)\) of \(R\) with respect to \(I\) is the undirected graph with vertices \(T(I) = \{x\in R\setminus I : xy \in I\;\text{for some}\;y \in R\setminus I\}\), where distinct vertices \(x\) and \(y\) are adjacent if and only if \(xy \in I\). These graphs provide a generalization of zero-divisor graphs of rings.NEWLINENEWLINEThe authors describe the diameter and the girth of the zero-divisor graph of a commutative ring \(R\) with respect to a nonzero ideal \(I\).