On the total graph of a commutative ring without the zero element
type: Item
instance of: scholarly article
zbMATH Open document ID: 1247.13004
DOI: 10.1142/S0219498812500740
author: David F. Anderson
author: Ayman Badawi
publication date: 2012-09-06T00:00:00Z
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 05C99
zbMATH DE Number: 6078483
zbMATH Keywords: total graph
zbMATH Keywords: zero-divisor graph
MaRDI profile type: Publication
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published in: Journal of Algebra and its Applications
title: On the total graph of a commutative ring without the zero element
review text: 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.
reviewed by: Siamak Yassemi
description: scientific article; zbMATH DE number 6078483