In this thesis, the relalationship between algebric properties and graph properties of the annihilating ideal graph of a commutative ring is expressed. Suppose that $ R $ is a commutative ring with identitiy . Let $ \\mathbb{AG}(R) $ be the set of all ideals of $ R $ with non-zero annihilators. The annihilating ideal graph of $ R $ is defined as the graph $ \\mathbb{AG}(R) $ with the vertex set $ \\mathbb{A} ^{ \\star }(R) = \\mathbb{AG}(R) \\setminus \\lbrace (0) \\rbrace $ and two distinc vetices $ I $ and $ J $ are adjacent if and only if $ IJ = (0) $ . This thesis contains four chapters. Chapter 1 include introduction. Chapter 2 include prerequisites. In Chapter 3, it has been dir=rtl In Chapter 4, it has been studied the vertex coloring of the annihilating-ideal graph of some rings of fractions. Next was prepared some formulas for the cliqe number and chromatic numbers of the annihilating-ideal graphs of a direct product of rings. First, it has been identified commutative rings whose annihilating-ideal graph are complete, path, bipartite or complete bipartite. In this regard, it has been found a positive integer number $ n $ such that $ \\mathbb{AG}(R) \\cong K _{n} $ , where $ K _{n} $ is the complete graph of order $ n $ . Also it has been proved that the annihilating ideal graph of a commutative ring $ R $ is bipartite if and only if the annihilating ideal graph of $ R $ is traingle-free . Furthermore, it has been obtained some results on rings whose all non-trivial ideals are annihilating . For example, it has been shown that if $ R $ is an artinian ring and the chromatic number of its annihilating ideal graph is $ 2 $ , then $ R $ is Gorensteins. Also, in this thesis, it has been present some results on the cliqe number and chromatic number of the annihilating -ideal graph of a commutative ring. Among other results, it has been shown that if the chromatice number of the zero-divisor graph is finite, then the chromatic number of the annihilating-ideal graph is finite, too. In Behboodi and Rakeei { \\cite{p14} it has been conjectured that for a reduced ring $ R $ with more than two minimal prime ideals, the girth of annihilating-ideal graph is 3. Here, it has bee proved that for every (not necessarily reduced) commutative ring $ R $ , the chromatic number of annihilatin-ideal graph is greater than or equal to the number of minimal prime ideals of $ R $, which shows that the conjecture is true in general.
