Strong zero-divisor graph of p.q.-Baer $\ast$-rings

Abstract

In this paper, we study the strong zero-divisor graph of a p.q.-Baer $\ast$-ring. We determine the condition on a p.q.-Baer $\ast$-ring (in terms of the smallest central projection in a lattice of central projections of a $\ast$-ring), so that its strong zero-divisor graph contains a cut vertex. It is proved that the set of cut vertices of a strong zero-divisor graph of a p.q.-Baer $\ast$-ring forms a complete subgraph. We prove that the complement of the strong zero-divisor graph of a p.q.-Baer $\ast$-ring is connected if and only if the $\ast$-ring contains at least six central projections. We characterize the diameter and girth of the complement of a strong zero-divisor graph of a p.q.-Baer $\ast$-ring. Also, we characterize p.q.-Baer $\ast$-rings whose strong zero-divisor graph is complemented.