This research article introduces the concept of the clear graph associated with a ring \({\mathcal {R}}\) with identity, denoted as \(Cr({\mathcal {R}})\). This graph comprises vertices of the form \(\{(x,u):\) x is a unit regular element of R and u is a unit of \({\mathcal {R}}\)} and two distinct vertices (x, u) and (y, v) are adjacent if and only if either \(xy=yx=0\) or \(uv=vu=1\). This research article also focuses on a specific subgraph of \(Cr({\mathcal {R}})\) denoted as \(Cr_2({\mathcal {R}})\), which is formed by vertices \(\{(x,u) :x\) is a nonzero unit regular element of \(R \}\). The significance of \(Cr_2({\mathcal {R}})\) within the context of \(Cr{({\mathcal {R}})}\) is explored in the article. Taken \(Cr_2({\mathcal {R}})\) into consideration, we found connectedness, regularity, planarity, and outer planarity. Moreover, we characterized the ring \({\mathcal {R}}\) for which \(Cr_2({\mathcal {R}})\) is unicyclic, a tree and a split graph. Finally, we have found genus one of \(Cr_2({\mathcal {R}})\).

本文介绍了与恒等环\({\mathcal {R}}\)相关的清晰图的概念，表示为\(Cr({\mathcal {R}})\)。该图由\(\{(x,u):\) 形式的顶点组成，x是R的单位正则元素，u是\({\mathcal {R}}\) }的单位，以及两个不同的顶点 ( x ,  u ) 和 ( y ,  v ) 相邻当且仅当\(xy=yx=0\)或\(uv=vu=1\)。本文还重点关注\(Cr({\mathcal {R}})\)的一个特定子图，表示为\(Cr_2({\mathcal {R}})\)，它由顶点\(\{ (x,u) :x\)是\(R \}\)的非零单位正则元素。本文探讨了\(Cr_2({\mathcal {R}})\)在\(Cr{({\mathcal {R}})}\)背景下的重要性。考虑到\(Cr_2({\mathcal {R}})\)，我们发现了连通性、规律性、平面性和外部平面性。此外，我们还描述了环\({\mathcal {R}}\)（其中\(Cr_2({\mathcal {R}})\)是单圈）、树和分裂图。最后，我们找到了\(Cr_2({\mathcal {R}})\)的属一。