Metric dimensions of the clean graph of a ring

Let R be a commutative ring with unity. The vertices of the clean graph Cl(R) of the ring R are of the form (e, u), where e is an idempotent and u is a unit element of R. In this article, we primarily focused on Cl 2 (R), a subgraph of Cl(R) induced by non-zero idempotent elements and units of R. Two vertices (e, u), (f, v), are adjacent if and only if ef = fe = 0 or uv = 1. The main objective of this study is to investigate the metric, upper, and fault-tolerant metric dimensions of Cl 2 (R), based on the idempotent and unit elements of the rings.