On the Cayley Graph of a Commutative Ring with Respect to its Zero-divisors

Abstract

Let R be a commutative ring with unity and R+ and Z*(R) be the additive group and the set of all nonzero zero-divisors of R, respectively. We denote by ℂ𝔸𝕐(R) the Cayley graph Cay(R+, Z*(R)). In this article, we study ℂ𝔸𝕐(R). Among other results, it is shown that for every zero-dimensional nonlocal ring R, ℂ𝔸𝕐(R) is a connected graph of diameter 2. Moreover, for a finite ring R, we obtain the vertex connectivity and the edge connectivity of ℂ𝔸𝕐(R). As a result, ℂ𝔸𝕐(R) gives an algebraic construction for vertex transitive graphs of maximum connectivity. In addition, we characterize all zero-dimensional semilocal rings R whose ℂ𝔸𝕐(R) is perfect. We also study Reg(ℂ𝔸𝕐(R)) the induced subgraph on the regular elements of R. This graph gives a family of vertex transitive graphs as well. We show that if R is a Noetherian ring and Reg(ℂ𝔸𝕐(R)) has no infinite clique, then R is finite. Furthermore, for every finite ring R, the clique number and the chromatic number of Reg(ℂ𝔸𝕐(R)) are determined.