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The zero-divisor graph of a commutative semigroup. (English) Zbl 1011.20056
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 ${\Gamma }\left(S\right)$. In analogy with the recently studied zero-divisor graph of a commutative ring, the vertices of ${\Gamma }\left(S\right)$ 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 ${\Gamma }\left(S\right)$. For example, ${\Gamma }\left(S\right)$ is always connected and the diameter of ${\Gamma }\left(S\right)\le 3$. The graphs without a cycle which can be realized by some ${\Gamma }\left(S\right)$ are determined. If ${\Gamma }\left(S\right)$ contains a cycle, then the core of ${\Gamma }\left(S\right)$ 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.

##### MSC:
 20M14 Commutative semigroups 05C25 Graphs and abstract algebra