  In this paper we study sub-semigroups of a zero-divisor semigroup $S$
determined by properties of the zero-divisor graph $\Gamma(S)$. 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.
