Narrow Results

For a *-ring $A$, we associate a simple undirected graph ${\mathrm{\Gamma }}^{*}(A)$ having all nonzero left zero-divisors of $A$ as vertices and, two vertices $x$ and $y$ are adjacent if $x{y}^{*}=0$. In case of Artinian *-rings and Rickart *-rings, characterizations are obtained for those *-rings having ${\mathrm{\Gamma }}^{*}(A)$ a complete graph or a star graph, and sufficient conditions are obtained for ${\mathrm{\Gamma }}^{*}(A)$ to be connected and also for ${\mathrm{\Gamma }}^{*}(A)$ to be disconnected. For a Rickart *-ring $A$, we characterize the girth of $\mathrm{\text{gr}}({\mathrm{\Gamma }}^{*}(A))$ and prove a sort of Beck’s conjecture.

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 $A$ with unity we associate a graph, ${\mathrm{\Gamma }}_1^{*}(A)$, having all nonzero elements of $A$ as vertices and two distinct vertices $x$ and $y$ are adjacent if and only if either $x{y}^{*}=0$ or $x+y$ is a unit in $A$. For a finite Rickart *-ring $A$ it is proved that ${\mathrm{\Gamma }}_1^{*}(A)$ is connected if and only if $A$ is not isomorphic to ${\mathbb{Z}}_3$ or ${\mathbb{Z}}_2^k$ (for any $k\in \mathbb{N}\setminus \{1\})$.