Strong metric dimension of clean graphs of commutative rings

Let R be a ring with unity. The clean graph Cl(R) of a ring R is the simple undirected graph whose vertices are of the form (e,u), where e is an idempotent element and u is a unit of the ring R, and two vertices (e,u), (f,v) of Cl(R) are adjacent if and only if ef = fe = 0 or uv = vu = 1. In this paper, for a commutative ring R, first we obtain the strong resolving graph of Cl(R) and its independence number. Using them, we determine the strong metric dimension of the clean graph of an arbitrary commutative ring. As an application, we compute the strong metric dimension of Cl(R), where R is a commutative Artinian ring.