A Proof of the Conjecture on complemented zero-divisor graphs of semigroups

Abstract

In this paper, we are motivated by the conjectures proposed by C.~Bender \textit{et al.}, \cite{C} in 2024. We have settled the first two conjectures negatively by providing a counter example in \cite{KTJ}, whereas in this paper, we prove the third conjecture positively, which has remained an open question until now. The third conjecture is stated as if $G(S)$ is uniquely complemented with the clique number $3$ or greater and has the property that every vertex has a unique complement, then the graph $G(S)$ is isomorphic to the graph $G(\mathcal{P}(n))$, where $n$ is the clique number of $G(S)$.