Some Criteria for Graphs to be Annihilating-Ideal Graphs

The annihilating-ideal graph of a commutative ring R, denoted by AG(R), is a graph whose vertex set consists of all non-zero annihilating ideals and two distinct vertices are adjacent if and only if their product is zero. Here, some criteria for a graph to be isomorphic with an annihilating-ideal graph of a ring, is given. Annihilating-ideal graphs which are trees are specified. Some results about the vertices with maximum degree are stated, too. Finally, it is shown that the induced subgraph of AG(R) on vertices with maximum degree is either complete or a discrete graph.