Clear graph of a ring

Abstract

This research article introduces the concept of the clear graph associated with a ring R with identity, denoted as Cr(R). This graph comprises vertices of the form {(x,u):x is a unit regular element of R and u is a unit of 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(R) denoted as Cr2(R), which is formed by vertices {(x,u):x is a nonzero unit regular element of R}. The significance of Cr2(R) within the context of Cr(R) is explored in the article. Taken Cr2(R) into consideration, we found connectedness, regularity, planarity, and outer planarity. Moreover, we characterized the ring R for which Cr2(R) is unicyclic, a tree and a split graph. Finally, we have found genus one of Cr2(R). © The Indian National Science Academy 2024.