A generalization of the unit and unitary Cayley graphs of a commutative ring

Let R be a commutative ring with non-zero identity and G be a multiplicative subgroup of U(R), where U(R) is the multiplicative group of unit elements of R. Also, suppose that S is a non-empty subset of G such that S−1={s−1∣s∈S}⫅S. Then we define Γ(R,G,S) to be the graph with vertex set R and two distinct elements x,y∈R are adjacent if and only if there exists s∈S such that x+sy∈G. This graph provides a generalization of the unit and unitary Cayley graphs. In fact, Γ(R,U(R),S) is the unit graph or the unitary Cayley graph, whenever S={1} or S={−1}, respectively. In this paper, we study the properties of the graph Γ(R,G,S) and extend some results in the unit and unitary Cayley graphs.