Radical-Depended Graph of a Commutative Ring

Let R be a commutative ring with identity and root I be the radical of an ideal I of R. We introduce the radical-depended graph G(I)(R) whose vertex set is {x is an element of R \root I vertical bar xy is an element of I for some y is an element of R\ root I} and distinct vertices x and y are adjacent if and only if xy is an element of I. In this paper, several properties of G(I)(R) are investigated and some results on the parameters of this graph are given. It follows that if I is a quasi primary ideal, then G(I)(R) = theta. It is shown that if I is a 2-absorbing ideal of R which is not quasi primary, then G(I)(R) is the complete bipartite graph K-1,K-1 or K-m,K- n for some m, n >= 2. Moreover, it is proved that G(I)(R) is a connected graph with diameter at most 3, and if it has a cycle, then its girth is at most 4. Also, it is shown that if R is a Noetherian ring, then the clique number of G(I)(R) is equal to vertical bar Min(R/I)vertical bar for every ideal I of R.