Sub-semigroups determined by the zero-divisor graph

In this paper we study sub-semigroups of a zero-divisor semigroup $S$ determined by properties of the zero-divisor graph $\mathrm{\Gamma }(S)$. We use these sub-semigroups to study the correspondence between zero- divisor semigroups and zero-divisor graphs. We study properties of sub- semigroups of Boolean semigroups via the zero-divisor graph. As an application, we provide a characterization of the graphs which are zero- divisor graphs of Boolean rings.