Properties of Clear Graph of Ring ℤp - Mohammad Ilham Maulana; Vira Hari Krisnawati; Ratno Bagus Edy Wibowo (Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang, 2026)
Let ℤp be the ring of integers modulo a prime p 3. The clear graph of ℤp, denoted by Cr2(ℤp), is a graph whose vertices are ordered pairs (x,u), where x is a nonzero regular unit and u is a unit of ℤp, 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 ℤp and analyze several fundamental graph-theoretic properties of Cr2(ℤp). 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 ℤp influence the combinatorial structure of its associated clear graph.