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In this paper, we characterize the perfect zero-divisor graphs and chordal zero-divisor graphs (its complement) of ordered sets. These results are applied to the zero-divisor graphs of finite reduced rings, the comaximal ideal graphs of rings, the annihilating ideal graphs of rings, the intersection graphs of ideals of rings, and the intersection graphs of subgroups of groups. In fact, it is shown that these graphs associated with a commutative ring R with identity can be effectively studied via the zero-divisor graph of a specially constructed poset from R. [ABSTRACT FROM AUTHOR]

Let be a commutative ring with identity. The comaximal ideal graph of is a simple graph with its vertices are the proper ideals of R which are not contained in the Jacobson radical of , and two vertices and are adjacent if and only if . In this paper, a dominating set of is constructed using elements of the center when is a commutative Artinian ring. Also we prove that the domination number of is equal to the number of factors in the Artinian decomposition of . Also, we characterize all commutative Artinian rings(non local rings) with identity for which is planar.

Let $R$ be a commutative ring with unity. The comaximal ideal graph of $R$, denoted by $mathcal{C}(R)$, is a graph whose vertices are the proper ideals of $R$ which are not contained in the Jacobson radical of $R$, and two vertices $I_1$ and $I_2$ are adjacent if and only if $I_1 +I_2 = R$. In this paper, we classify all comaximal ideal graphs with finite independence number and present a formula to calculate this number. Also, the domination number of $mathcal{C}(R)$ for a ring $R$ is determined. In the last section, we introduce all planar and toroidal comaximal ideal graphs. Moreover, the commutative rings with isomorphic comaximal ideal graphs are characterized. In particular we show that every finite comaximal ideal graph is isomorphic to some $mathcal{C}(mathbb{Z}_n)$.

The rings considered in this article are commutative with identity which admit at least two maximal ideals. Let R be a ring such that R admits at least two maximal ideals. Recall from Ye and Wu (J. Algebra Appl. 11(6): 1250114, 2012) that the comaximal ideal graph of R, denoted by C (R) is an undirected simple graph whose vertex set is the set of all proper ideals I of R such that I 6... J(R), where J(R) is the Jacobson radical of R and distinct vertices I1, I2 are joined by an edge in C (R) if and only if I1 + I2 = R. In Section 2 of this article, we classify rings R such that C (R) is planar. In Section 3 of this article, we classify rings R such that C (R) is a split graph. In Section 4 of this article, we classify rings R such that C (R) is complemented and moreover, we determine the S-vertices of C (R). [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with unity. The comaximal ideal graph of R, denoted by C(R), is a graph whose vertices are the proper ideals of R which are not contained in the Jacobson radical of R, and two vertices I1 and I2 are adjacent if and only if I1 + I2 = R. In this paper, we classify all comaximal ideal graphs with finite independence number and present a formula to calculate this number. Also, the domination number of C(R) for a ring R is determined. In the last section, we introduce all planar and toroidal comaximal ideal graphs. Moreover, the commutative rings with isomorphic comaximal ideal graphs are characterized. In particular we show that every finite comaximal ideal graph is isomorphic to some C(Zn). [ABSTRACT FROM AUTHOR]