On the Cayley graph of a commutative ring with respect to its zero-divisors

Abstract

Let $R$ be a commutative ring with unity and $R^{+}$ be $Z^{\ast }(R)$ be the additive group and the set of all non-zero zero-divisors of $R$, respectively. We denote by $\mathbb{CAY}(R)$ the Cayley graph $Cay(R^{+},Z^{\ast }(R))$. In this paper, we study $\mathbb{CAY}(R)$. Among other results, it is shown that for every zero-dimensional non-local ring $R$, $\mathbb{CAY}(R)$ is a connected graph of diameter 2. Moreover, for a finite ring $R$, we obtain the vertex connectivity and the edge connectivity of $\mathbb{CAY}(R)$. We investigate rings $R$ with perfect $\mathbb{CAY}(R)$ as well. We also study $Reg(\mathbb{CAY}(R))$ the induced subgraph on the regular elements of $R$. This graph gives a family of vertex transitive graphs. We show that if $R$ is a Noetherian ring and $Reg(\mathbb{CAY}(R))$ has no infinite clique, then $R$ is finite. Furthermore, for every finite ring $R$, the clique number and the chromatic number of $Reg(\mathbb{CAY}(R))$ are determined.

21 pages, 1 figure