Zero divisor graphs of semigroups.
Let \(S\) be a commutative semigroup with \(0\). A simple graph \(G\) whose vertices are the nonzero zero divisors of \(S\) with two distinct vertices joined by an edge in case when their product in \(S\) is \(0\) is called the zero divisor graph of \(S\). In the paper some characterizations of graphs to be zero divisor graphs of a semigroup are given. An association with \(S\) of a simplicial complex \(K(S)\) with \(S\), where a simplex is a subset \(A\) of \(S-\{0\}\) with the property that \(x,y\in A\) with \(x\neq y\) implies \(xy=0\) is introduced to give some interesting examples of zero divisor graphs, such as a graph \(G\) with the property that every edge in \(G\) is the edge of a triangle and \(G\) is the zero divisor graph of a semilattice but not the zero divisor graph for any nilpotent semigroup.
scientific article; zbMATH DE number 2153359