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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 xS). In this paper, the authors introduce and investigate the zero-divisor graph of S, denoted by Γ(S). In analogy with the recently studied zero-divisor graph of a commutative ring, the vertices of Γ(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 Γ(S). For example, Γ(S) is always connected and the diameter of Γ(S)3. The graphs without a cycle which can be realized by some Γ(S) are determined. If Γ(S) contains a cycle, then the core of Γ(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.

MSC:
20M14Commutative semigroups
05C25Graphs and abstract algebra