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Let $R$ be a commutative ring with unity and $R^{+}$ be $Z^*(R)$ be theadditive group and the set of all non-zero zero-divisors of $R$, respectively.We denote by $\mathbb{CAY}(R)$ the Cayley graph $Cay(R^+,Z^*(R))$. In thispaper, we study $\mathbb{CAY}(R)$. Among other results, it is shown that forevery zero-dimensional non-local ring $R$, $\mathbb{CAY}(R)$ is a connectedgraph of diameter 2. Moreover, for a finite ring $R$, we obtain the vertexconnectivity and the edge connectivity of $\mathbb{CAY}(R)$. We investigaterings $R$ 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 aNoetherian ring and $Reg(\mathbb{CAY}(R))$ has no infinite clique, then $R$ isfinite. Furthermore, for every finite ring $R$, the clique number and thechromatic number of $Reg(\mathbb{CAY}(R))$ are determined.

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