On graphs related to co-maximal ideals of a commutative ring

This paper studies the co-maximal graph $\text{Om}(R)$, the induced subgraph $\text{G}(R)$ of $\text{Om}(R)$ whose vertex set is $R\setminus (U(R)\cup J(R))$ and a retract ${\text{G}}_r(R)$ of $\text{G}(R)$, where $R$ is a commutative ring. We show that the core of $\text{G}(R)$ is a union of triangles and rectangles, while a vertex in $\text{G}(R)$ is either an end vertex or a vertex in the core. For a non-local ring $R$, we prove that both the chromatic number and clique number of $\text{G}(R)$ are identical with the number of maximal ideals of $R$. A graph ${\text{G}}_r(R)$ is also introduced on the vertex set $\{Rx\mathrm{\mid }x\in R\setminus (U(R)\cup J(R))\}$, and graph properties of ${\text{G}}_r(R)$ are studied.