@article{article_423751, title={Some results on the comaximal ideal graph of a commutative ring}, journal={Journal of Algebra Combinatorics Discrete Structures and Applications}, volume={5}, pages={85–99}, year={2018}, DOI={10.13069/jacodesmath.423751}, author={Visweswaran, Subramanian and Parejiya, Jaydeep}, keywords={Comaximal ideal graph,Special principal ideal ring,Planar graph,Split graph,Complement of a vertex in a graph}, abstract={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 $\mathscr{C}(R)$ is an undirected simple graph whose vertex set is the set of all proper ideals $I$ of $R$ such that $I\not\subseteq J(R)$, where $J(R)$ is the Jacobson radical of $R$ and distinct vertices $I_{1}$, $I_{2}$ are joined by an edge in $\mathscr{C}(R)$ if and only if $I_{1} + I_{2} = R$. In Section 2 of this article, we classify rings $R$ such that $\mathscr{C}(R)$ is planar. In Section 3 of this article, we classify rings $R$ such that $\mathscr{C}(R)$ is a split graph. In Section 4 of this article, we classify rings $R$ such that $\mathscr{C}(R)$ is complemented and moreover, we determine the $S$-vertices of $\mathscr{C}(R)$.}, number={2}, publisher={iPeak Academy}