Planarity and genus of co-unit graphs in direct product of local rings

Abstract

This research article focuses on the co-unit graph, denoted as $G_{nu}(R)$, which is associated with a commutative ring $R$. The vertex set of $G_{nu}(R)$ is $U(R)$, which represents the set of units of $R$, and two distinct vertices $x$ and $y$ of $U(R)$ are considered to be adjacent if and only if $x + y \notin U(R)$. The objective of the research article is to characterize the ring $R$, where $R$ is taken as the direct product of local rings $\mathbb{Z}_n$, and determine whether $G_{nu}(R)$ is planar, outerplanar, a tree, or a split graph. Moreover, the study aims to identify the rings for which $G_{nu}(R)$ has a genus of one. Overall, this research article investigates the properties of $G_{nu}(R)$ and explores the characteristics of rings that lead to specific graph structures.