The annihilating graph of a ring

Let A be a commutative ring with unity. The annihilating graph of A, denoted by G(A), is a graph whose vertices are all non-trivial ideals of A and two distinct vertices I and J are adjacent if and only if Ann(I)Ann(J) = 0. For every commutative ring A, we study the diameter and the girth of G(A). Also, we prove that if G(A) is a triangle-free graph, then G(A) is a bipartite graph. Among other results, we show that if G(A) is a tree, then G(A) is a star or a double star graph. Moreover, we prove that the annihilating graph of a commutative ring cannot be a cycle. Let n be a positive integer number. We classify all integer numbers n for which G(Z(n)) is a complete or a planar graph. Finally, we compute the domination number of G(Z(n)).