On the diameter and girth of zero-divisor graphs of inverse semigroups

Let $S$ be an inverse semigroup with zero and let $Z(S{)}^{\times }$ be its set of non-zero divisors with respect to the natural partial order $\le$ on $S$, that is, $a\in Z(S{)}^{\times }$ if there exists $b\in S\setminus \{0\}$ with $\omega (a,b)=\{c\in S:c\le aandc\le b\}=\{0\}$. The set $Z(S{)}^{\times }$ makes up the vertices of the corresponding {\it zero-divisor graph} $\mathrm{\Gamma }(S)$, with two distinct vertices $a,b$ forming an edge if $\omega (a,b)=\{0\}$. We characterize {\it zero-divisor graphs} of inverse semigroups in terms of their diameter and girth. We also classify inverse semigroups without zero by building a connection between the diameter (girth) and the least group congruence $\sigma$ on an inverse semigroup without zero. Finally, we give a description of the diameter and girth of graph inverse semigoups $I(G)$ in terms of the set of vertices and the set of edges of a graph $G$.