Metric dimensions of the clean graph of a ring

Let [Formula: see text] be a commutative ring with unity. The vertices of the clean graph [Formula: see text] of the ring [Formula: see text] are of the form [Formula: see text], where [Formula: see text] is an idempotent and [Formula: see text] is a unit element of [Formula: see text]. In this paper, we primarily focused on [Formula: see text], a subgraph of [Formula: see text] induced by non-zero idempotent elements and units of R. Two vertices [Formula: see text], are adjacent if and only if [Formula: see text] or [Formula: see text]. The main objective of this study is to investigate the metric, upper, and fault-tolerant metric dimensions of [Formula: see text], based on the idempotent and unit elements of the rings.