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) vertical bar s is an element of S} subset of S. Then we define Gamma(R, G, S) to be the graph with vertex set R and two distinct elements x, y is an element of R are adjacent if and only if there exists s is an element of S such that x + sy is an element of G. This graph provides a generalization of the unit and unitary Cayley graphs. In fact, Gamma(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 Gamma(R, G, S) and extend some results in the unit and unitary Cayley graphs.