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

TL;DR

This paper investigates the properties of the strong zero-divisor graph of p.q.-Baer *-rings, establishing conditions for cut vertices, connectivity, diameter, girth, and complementarity, thus advancing the understanding of their algebraic and graph-theoretic structure.

Contribution

It provides new characterizations of the strong zero-divisor graph of p.q.-Baer *-rings, including conditions for cut vertices, connectivity of the complement, and properties of the graph's diameter and girth.

Findings

The set of cut vertices forms a complete subgraph.

The complement is connected iff the ring has at least six central projections.

Characterizations of diameter, girth, and when the graph is complemented.

Abstract

In this paper, we study the strong zero-divisor graph of a p.q.-Baer $*$-ring. We determine the condition on a p.q.-Baer $*$-ring (in terms of the smallest central projection in a lattice of central projections of a $*$-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 $*$-ring forms a complete subgraph. We prove that the complement of the strong zero-divisor graph of a p.q.-Baer $*$-ring is connected if and only if the $*$-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 $*$-ring. Also, we characterize p.q.-Baer $*$-rings whose strong zero-divisor graph is complemented.