Let A be a commutative RING with unity. The annihilating graph of A, denoted by Gð AÞ , 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 Annð IÞ Annð JÞ ¼ 0. For every commutative RING A, we study the diameter and the girth of Gð AÞ . Also, we prove that if Gð AÞ is a triangle-free graph, then Gð AÞ is a bipartite graph. Among other results, we show that if Gð AÞ is a tree, then Gð AÞ 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 Gð ZnÞ is a complete or a planar graph. Finally, we compute the domination number of Gð ZnÞ .