The zero-divisor graph of a commutative semigroup
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.
scientific article; zbMATH DE number 1837299
scientific article