The zero-divisor graph of a commutative semigroup
type: Item
instance of: scholarly article
title: The zero-divisor graph of a commutative semigroup
zbMATH Open document ID: 1011.20056
author: K. Schneider
author: Frank R. DeMeyer
author: Thomas McKenzie
published in: Semigroup Forum
publication date: 2002-12-01T00:00:00Z
review text: Let \(S\) be a commutative multiplicative semigroup with \(0\) (\(0x=0\) for all \(x\in S\)). In this paper, the authors introduce and investigate the zero-divisor graph of \(S\), denoted by \(\Gamma(S)\). In analogy with the recently studied zero-divisor graph of a commutative ring, the vertices of \(\Gamma(S)\) are the nonzero zero-divisors of \(S\), and two distinct vertices \(x\) and \(y\) are connected by an edge if \(xy=0\). They give several results about the shape of \(\Gamma(S)\). For example, \(\Gamma(S)\) is always connected and the diameter of \(\Gamma(S)\leq 3\). The graphs without a cycle which can be realized by some \(\Gamma(S)\) are determined. If \(\Gamma(S)\) contains a cycle, then the core of \(\Gamma(S)\) is a union of squares and triangles, and any vertex not in the core is an end which is connected to the core by a single edge.
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 1837299
zbMATH Keywords: commutative semigroups
zbMATH Keywords: clique numbers
zbMATH Keywords: zero-divisor graphs
reviewed by: David F. Anderson
MaRDI profile type: Publication
full work available at URL: s002330010128
OpenAlex ID: W2010754252
DOI: 10.1007/S002330010128
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description: scientific article; zbMATH DE number 1837299