Abstract
Let R be a commutative ring with unity and R +, U(R), and Z*(R) be the additive group, the set of unit elements, and the set of all nonzero zero-divisors of R, respectively. We denote by ℂ𝔸𝕐(R) and G R , the Cayley graph Cay(R +, Z*(R)) and the unitary Cayley graph Cay(R +, U(R)), respectively. For an Artinian ring R, Akhtar et al. (2009) studied G R . In this article, we study ℂ𝔸𝕐(R) and determine the clique number, chromatic number, edge chromatic number, domination number, and the girth of ℂ𝔸𝕐(R). We also characterize all rings R whose ℂ𝔸𝕐(R) is planar. Moreover, we determine all finite rings R whose ℂ𝔸𝕐(R) is strongly regular. We prove that ℂ𝔸𝕐(R) is strongly regular if and only if it is edge transitive. As a consequence, we characterize all finite rings R for which G R is a strongly regular graph.
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