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Zbl 1011.20056
DeMeyer, F.R.; McKenzie, T.; Schneider, K.
The zero-divisor graph of a commutative semigroup.
(English)
[J] Semigroup Forum 65, No.2, 206-214 (2002). ISSN 0037-1912

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 $\Gamma(S)\le 3$. 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.
[David F.Anderson (Knoxville)]
MSC 2000:
*20M14 Commutative semigroups
05C25 Graphs and groups

Keywords: commutative semigroups; clique numbers; zero-divisor graphs

Cited in: Zbl 1250.13006 Zbl 1115.20054

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