On the zero-divisor graph of Rickart *-rings

In this paper, we study the zero-divisor graph of Rickart *-rings. We determine the condition on Rickart *-ring so that its zero-divisor graph contains a cut vertex. It is proved that the set of cut vertices form a complete subgraph. We characterize Rickart *-rings for which the complement of the zero-divisor graph is connected. The diameter and girth of these graphs are characterized. Further, for a *-ring [Formula: see text] with unity we associate a graph, [Formula: see text], having all nonzero elements of [Formula: see text] as vertices and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if either [Formula: see text] or [Formula: see text] is a unit in [Formula: see text]. For a finite Rickart *-ring [Formula: see text] it is proved that [Formula: see text] is connected if and only if [Formula: see text] is not isomorphic to [Formula: see text] or [Formula: see text] (for any [Formula: see text].