Digraph and Graph of Regular Rings with Some Properties

During this century, a new research trend has emerged wherein algebraic concepts are combined with graph theory, such as zero divisors, clean rings, ideals, and regular rings. Continuing this trajectory, the regular ring graph has been studied in two forms: the undirected graph and the directed graph. The vertex set consists of all regular elements in the ring ℜ, denoted by ℜ(ℜ), where the undirected graph is denoted by Γ(ℜ(ℜ)) and the directed graph by Ψ(ℜ(ℜ)). This study focuses on the rings Zn, where n = q, q², or 2q, and q is a prime number. In this paper, a general description of the graph is provided, and general formulas for calculating the number of vertices and edges are derived and algebraically proven for graphs. Furthermore, for the undirected graph, the number of triangles is calculated and proven. Therefore, some of its properties are studied, such as the domination number, Wiener number, average distance, and number of girths for the graph Γ(ℜ(Zn)) of the commutative ring Zn, where n = q,q2, 2q, and q is a prime number. In addition, the 𝕄-polynomial and the ℕ𝕄 -polynomial for both graphs Γ(ℜ(Zq)), Γ(ℜ(Zq2)) and Γ(ℜ(Z2q)), where q is a prime number, are computed.