On the idempotent graph of a ring

TL;DR

This paper investigates the properties of the idempotent graph of a ring, providing conditions for planarity, and classifies certain finite rings based on the graph's structure, such as being a cograph, split, or threshold graph.

Contribution

It establishes a necessary and sufficient condition for the idempotent graph to be planar and classifies finite non-local commutative rings where the graph is a cograph, split, or threshold.

Findings

G_{Id}(R) cannot be outerplanar.

Characterization of rings with planar idempotent graphs.

Equivalence of split and threshold graph classes occurs only for R ≅ Z_2 × ... × Z_2.

Abstract

Let $R$ be a ring with unity. The \emph{idempotent graph} $G_{\text{Id}}(R)$ of a ring $R$ is an undirected simple graph whose vertices are the set of all the elements of ring $R$ and two vertices $x$ and $y$ are adjacent if and only if $x+y$ is an idempotent element of $R$. In this paper, we obtain a necessary and sufficient condition on the ring $R$ such that $G_{\text{Id}}(R)$ is planar. We prove that $G_{\text{Id}}(R)$ cannot be an outerplanar graph. Moreover, we classify all the finite non-local commutative rings $R$ such that $G_{\text{Id}}(R)$ is a cograph, split graph and threshold graph, respectively. We conclude that latter two graph classes of $G_{\text{Id}}(R)$ are equivalent if and only if $R \cong \mathbb{Z}_2 \times \mathbb{Z}_2 \times \cdots \times \mathbb{Z}_2$.