Clear graph of a ring

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 \(Cr_2({\mathcal {R}})\).