Research Article Quasi-Regular Graphs Associated with Commutative Rings Nasr Zeyada , 1,2 Najat Muthana , 1 and Sultanah Al-Rashidi 1 1 Department of Mathematics, College of Science, University of Jeddah, Jeddah 23218, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt Correspondence should be addressed to Nasr Zeyada; nzeyada@gmail.com Received 4 May 2022; Revised 15 July 2022; Accepted 18 July 2022; Published 24 August 2022 Academic Editor: Francesca Tartarone Copyright © 2022 Nasr Zeyada et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. is study provides important results and paves the way for a lot of different inquiries and studies utilizing this novel approach. 1. Introduction e relationship between graph theory and rings has been extensively studied and has produced amazing results in algebraic graph theory. e investigation of the theoretical aspects of a graph over a ring graph was first thought of by Beck in 1988 [1]. He proposed the concept of a zero-divisor graph Γ(S) for a commutative ring S. He defined Γ(S) as an undirected graph with components at its vertices Z(S) *  Z(S)− 0 {}, the set of nonzero zero-divisors of S, where any two vertices w and z are adjacent if wz  0 [1]. e first main rearrangements of Beck’s zero-divisor graph were presented by Anderson and Livingston [2]. Grimaldi was the first to study the unit graph for Z n [3]. e unit graph of S, denoted by G(S), is the graph produced by making all elements of S vertices and defining different vertices w and z to be adjacent if and only if w + z ∈ U(S) where U(S) is the set of unit elements of S. One sort of graph related to rings, the unit graph was first introduced in 2010 by Ashrafi et al. [4]. Such graph extended the unit graph G(Z n ) to G(S) and derived numerous characterization findings for finite commutative rings in terms of connectedness, chromatic index, diameter, girth, and planarity of G(S). If we remove the phrase “distinct” from the definition, we get the closed unit graph G(S), which may contain loops. It is worth noting that if 2 ∉ U(S), then G(S) G(S). A graph’s diameter is an important invariant. Many articles in this field are devoted to the diameter of the generated graph; see, for example, [5, 6]. In 2014, Su and Zhou [7] proved that, for any arbitrary ring S, the girth of G(S) is 3, 4, 6, or ∞ and recently explored the diameter of G(S) with S/J(S) self-injective ring S and provided a thorough characterization of the diameter of G(S) 1, 2, 3 or ∞, respectively [8]. In this paper, we introduce a new concept that we call a quasiregular graph and symbolise it with Q(S), where S is a commutative ring with a nonzero identity. Furthermore, we determine when a quasiregular graph is isomorphic to several well-known graphs. We also provide some number theory approaches to prove some of the properties of a quasiregular graph over S. We refer to [9–12] for the undefined notions in this paper. 2. Quasiregular Graph Definition 1. A quasiregular Q(S) is a graph with vertex set V(Q(S))  S/0 {} in which any two distinct vertices w and z are adjacent if and only if 1 − wz is a unit in S. erefore, the Hindawi Journal of Mathematics Volume 2022, Article ID 6209466, 5 pages https://doi.org/10.1155/2022/6209466