Clear graph of a ring

Abstract

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