Quasi-Regular Graphs Associated with Commutative Rings

permits reproduction any provided original properly One of the most important branches of mathematics is algebraic graph theory, which solves graph problems with algebraic methods. In graph theory, several algebraic properties of a ring can be represented. In this paper, we define an innovative graph on rings, explore its characteristics, and examine how it relates to other notions in the field. Let S be a ring; the quasi-regular graph of S is a graph with a vertex set of S - {0} and any two different vertices w and z are adjacent if 1- wz is a unit in S. We study this graph by providing different examples and proving some crucial characteristics. This study provides important results and paves the way for a lot of different inquiries and studies utilizing this novel approach.