Let \(R\) be a ring (not necessarily commutative) with identity. The clean graph \(Cl(R)\) of a ring \(R\) is a graph with vertices in form \((e, u)\), where \(e\) is an idempotent and \(u\) is a unit of \(R\); and two distinct vertices \((e, u)\) and \((f, v)\) are adjacent if and only if \(ef = fe = 0\) or \(uv = vu = 1\). This paper introduced and studied some basic properties of the clean graph \(Cl(R)\) and its subgraphs \(Cl_{1}(R)\) and \(Cl_{2}(R)\). In Section 2, the authors investigated the graph properties of \(Cl_{2}(R)\) such as diameter, maximum and minimum degree, regularity, etc. Among many results, they gave a necessary and sufficient condition for the connectedness of \(Cl_{2}(R)\), that is, \(Cl_{2}(R)\) is connected if and only if \(R\) has a nontrivial idempotent. Moreover, if \(Cl_{2}(R)\) is connected, then diam\((Cl_{2}(R))\leq 3\). Also, they proved that \(Cl_{2}(R)\) is regular if and only if \(|U(R)| = 1\) or \(R\) has no nontrivial idempotent and \(U''(R) =\emptyset\). In Section 3, They showed that the parameters of clean graph of every commutative Artinian ring depend to the number of maximal ideals of it. It is proved that the clique number \(\omega(Cl_{2}(R)) = \chi(Cl_{2}(R))\), for every commutative Artinian ring \(R\). Moreover, the exact value of \(\omega(Cl_{2}(R))\) is given. Among other results, the independence number and domination number of \(Cl_{2}(R)\) are given.