On the idempotent graph of a ring

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 $R \cong \mathbb{Z}_2 \times \mathbb{Z}_2 \times \cdots \times \mathbb{Z}_2$.