Summary: Let \(R\) be a commutative ring and \(\mathbb A(R)\) be the set of ideals with non-zero annihilators. The annihilating-ideal graph of \(R\) is defined as the graph \(\mathbb A\mathbb G(R)\) with the vertex \(\mathbb A(R)^{\ast}=\mathbb A(R)\setminus \{(0)\}\) and two distinct vertices \(I\) and \(J\) are adjacent if and only if \(IJ = (0)\). Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if \(R\) is an Artinian ring and \(\omega(\mathbb A\mathbb G(R)) = 2\), then \(R\) is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.