Study on the new graph constructed by a commutative ring

Let R be a commutative ring and G(R) be a graph with vertices as proper and non-trivial ideals of R. Two distinct vertices I and J are said to be adjacent if and only if I + J = R. In this paper we study a graph constructed from a subgraph G(R)/∆(R) of G(R) which consists of all ideals I of R such that I 5 J(R), where J(R) denotes the Jacobson radical of R. In this paper we study about the relation between the number of maximal ideal of R and the number of partite of graph G(R)/4(R). Also we study on the structure of ring R by some properties of vertices of subgraph G(R)/4(R). In another section, it is shown that under some conditions on the G(R), the ring R is Noetherian or Artinian. Finally we characterize clean rings and then study on diameter of this constructed graph.