Some domination parameters in Cayley graphs of a commutative ring

Let H be a group and S subset of H be a set of group elements such that the identity element e is not an element of S and S = S-1. The Cayley graph Cay(H,S) associated with (H,S) is an undirected graph with a vertex set equal to H and two vertices g, h is an element of H are adjacent, whenever gh(-1) is an element of S. Let R be a commutative ring with unity. R+ and Z*(R) are the additive group and the set of all non-zero zero-divisors of R, respectively. The symbol CAY(R) denotes the Cayley graph Cay(R+, Z*(R)) and GR represents the unitary Cayley graph Cay(R+, U(R)). In this study, the total domination number, perfect domination number and bondage number for CAY(R) and G(R) have been found. Moreover, we establish the relationship between the total domination number of G(R) and the number of prime ideals in R. Additionally, we identify all finite commutative rings with unity R where the perfect domination number of CAY(R) is equal to |R|.