On the total graph of a commutative ring without the zero element
Let \(R\) be a commutative ring with nonzero identity, and let \(Z(R)\) be its set of zero-divisors. The total graph of \(R\) is the (undirected) graph \(\mathrm{T}(\Gamma(R))\) with vertices all elements of \(R\), and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(x +y\in \mathrm{Z}(R)\). The total graph of a commutative ring has been studied by several authors.NEWLINENEWLINEIn the paper under review the authors study the two (induced) subgraphs \(\mathrm{Z}_0(\Gamma(R))\) and \(\mathrm{T}_0(\Gamma(R))\) of \(\mathrm{T}(\Gamma(R))\), with vertices \(\mathrm{Z}(R)\setminus\{0\}\) and \(R\setminus\{0\}\), respectively. They determine when \(\mathrm{Z}_0(\Gamma(R))\) and \(\mathrm{T}_0(\Gamma(R))\) are connected and compute their diameter and girth.
scientific article; zbMATH DE number 6078483