This work is mainly devoted to study of the graph ????S(R) associated to a graded ring R and a multiplicatively closed subset S of R. Recall that the vertices of ????S(R) are all of the elements of R and two distinct vertices are adjacent if their sum belongs to S. In fact, we investigate some basic graph-theoretical properties of ????S(R), where R = L i Ri is a graded ring. Moreover, we deal with the relationship between the graph-theoretical properties of ????S(R) and ????S\R0 (R0).
