Properties of Clear Graph of Ring ℤp

Abstract

Let ℤ_p be the ring of integers modulo a prime p > 3. The clear graph of ℤ_p, denoted by Cr₂(ℤ_p), is a graph whose vertices are ordered pairs (x,u), where x is a nonzero regular unit and u is a unit of ℤ_p, and two vertices (x,u) and (y,v) are adjacent if either xy = yx = 0 or uv = vu = 1. This work extends previous research on clear graphs, which established the basic structure of Cr₂(R) for certain rings, including aspects of isomorphism, connectedness, and other structural properties. In this paper, we focus on the prime ring ℤ_p and analyze several fundamental graph-theoretic properties of Cr₂(ℤ_p). Specifically, we show that this graph has order (p−1)², size ½(p²−2p−1)(p−1), diameter ∞, radius at most 2, independence number ½(p²−4p+7), and clique, chromatic, and domination numbers each equal to p−1. The results provide a deeper understanding of how algebraic properties of ℤ_p influence the combinatorial structure of its associated clear graph.