Generalized strong zero-divisor graph of rings with involution

Abstract

We introduce and examine the generalized strong zero-divisor graph Γ'_s(R) of a *-ring, focusing on generalized p.q.-Baer *-rings. We show that Γ'_s(R) is connected with diameter at most three and determine its girth. Beck’s conjecture holds for this graph in the considered setting. We identify conditions under which Γ'_s(R) is uniquely complemented or contains a cut vertex. For p.q.-Baer *-rings, connectedness of the complement of Γ'_s(R) ensures the existence of at least six central projections. We also describe the diameter and girth of the complement and provide examples where the generalized strong zero-divisor graph is complemented.