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 ) .