Let R be a non-domain, commutative ring with nonzero identity. Let denote the set of all nonzero zero-divisors of a ring R. The zero-divisor graph of R, is an undirected graph with vertex set  and two distinct vertices x, y are adjacent if and only if  An ideal I of a ring R is an annihilating ideal if there exists a nonzero ideal J of a ring R such that  Let  be the set of all annihilating ideals of a ring R. Let  The annihilating ideal graph of a commutative ring R denoted by  is an undirected, simple graph with vertex set  and there is an edge between two distinct vertices I and J if For any positive integer n, a finite graph  with n number of vertices exists if R has only nonzero ideals of the type  For  we establish the relation between n and  We give some results on diameter, girth, chromatic number and prove that  is not bipartite. For a nonreduced ring R, we prove  is less than or equal to  where k is the number of minimal primes and t is the number of ideals contained in nil radical of R. Further, we settle the conjecture proposed in [9, Conjecture 0.1] by giving a counterexample of nonreduced ring  for  and also settle the conjecture [9, Conjecture 2.15].

domain, non-domain, annihilating ideals, zero-divisor graph.