Some domination parameters in Cayley graphs of a commutative ring

Let [Formula: see text] be a group and [Formula: see text] be a set of group elements such that the identity element [Formula: see text] and [Formula: see text]. The Cayley graph [Formula: see text] associated with [Formula: see text] is an undirected graph with a vertex set equal to [Formula: see text] and two vertices [Formula: see text] are adjacent, whenever [Formula: see text]. Let [Formula: see text] be a commutative ring with unity. [Formula: see text] and [Formula: see text] are the additive group and the set of all non-zero zero-divisors of [Formula: see text], respectively. The symbol [Formula: see text] denotes the Cayley graph [Formula: see text] and [Formula: see text] represents the unitary Cayley graph [Formula: see text]. In this study, the total domination number, perfect domination number and bondage number for [Formula: see text] and [Formula: see text] have been found. Moreover, we establish the relationship between the total domination number of [Formula: see text] and the number of prime ideals in [Formula: see text]. Additionally, we identify all finite commutative rings with unity [Formula: see text] where the perfect domination number of [Formula: see text] is equal to [Formula: see text].