The Nil-Graph of Ideals of a Commutative Ring

Let R be a commutative ring with identity and Nil(R) be the set of nilpotent elements of R. The nil- graph of ideals of R is defined as the graph AGN (R) whose vertex set is {I : (0) not equal I R, and there exists a nontrivial ideal J such that I J subset of Nil(R)} and two distinct vertices I and J are adjacent if and only if I J subset of Nil(R). Here, some graph properties of AG(N) (R) are studied. For instance, some bounds for the diameter, girth, and radius of AG(N) (R) are given. In case that AG(N) (R) is a finite graph, it is proved that the center and median of AG(N) (R) coincide. Furthermore, we determine when the edge chromatic number of AG(N) (R) equals its maximum degree. Also, for every ring R, it is shown that both the clique number and vertex chromatic number of AG(N) (R) equal n + t, where n is the number of minimal prime ideals of R and t is the number of nonzero ideals of R which are contained in Nil(R).