On a class of semigroup graphs

Let $G=\Gamma(S)$ be a semigroup graph, i.e., a zero-divisor graph of a semigroup $S$ with zero element 0. For any adjacent vertices $x, y$ in $G$, denote $C(x,y)={z\in V(G) | N(z)={x,y}}$. Assume that in $G$ there exist two adjacent vertices $x,y$, a vertex $s\in C(x,y)$ and a vertex $z$ such that $d(s,z)=3$. In this paper, we study algebraic properties of $S$ with such graphs $G=\G(S)$, giving some sub-semigroups and ideals of $S$. We construct some classes of such semigroup graphs and classify all semigroup graphs with the property in two cases.