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 non-zero zero-divisors of R, respectively. We denote by CAY(R) the Cayley graph Cay(R + , Z * (R)). In this paper, we study CAY(R). Among other results, it is shown that for every zero-dimensional non-local 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). We investigate rings R with perfect CAY(R) as well. We also study Reg(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(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. * Proof. Parts (i) and (iii) are obvious. Part (ii) follows from Z(R) = m. Part (iv) holds for every Cayley graph of a group. To prove the last part, note that under an automorphism of graph G, any component of G is isomorphically mapped to another component. Since CAY(R) is vertex-transitive, we conclude that the components of CAY(R) are isomorphic and so (v) is proved.Remark 2. CAY(R) is vertex transitive but it is not necessarily edge transitive. To see this, consider CAY(Z 6 ) ∼ = K 2 K 3 which is not edge transitive (see Figure 1).

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