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All Journal CAUCHY: Jurnal Matematika Murni dan Aplikasi
Vira Hari Krisnawati
University of Brawijaya
Mathematics

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Properties of Clear Graph of Ring ℤp Mohammad Ilham Maulana; Vira Hari Krisnawati; Ratno Bagus Edy Wibowo
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 11, No 1 (2026): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Universitas Islam Negeri Maulana Malik Ibrahim Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/cauchy.v11i1.41856

Abstract

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.
Total Edge Irregularity Strength of Cycle Snake Graphs Stenly Pranata; Vira Hari Krisnawati; Darmajid Darmajid
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 11, No 1 (2026): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Universitas Islam Negeri Maulana Malik Ibrahim Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/cauchy.v11i1.41552

Abstract

Let G be a simple undirected graph with vertex and edge sets. A total labeling that assigns integers from 1 to k to the union of the vertex and edge sets is called a k-total labeling. The weight of an edge uv, denoted by w(uv), is defined as the sum of the labels of the two vertices u, v, and the label of edge uv itself. A k-total labeling is called an edge irregular total k-labeling of G if the weights of all distinct edges are different. The minimum k for which every edge of G has a distinct weight is called the total edge irregularity strength of G, denoted by tes(G). A cycle snake graph CSm,n is obtained from the path graph Pn with n + 1 vertices and n edges by replacing each edge with a cycle graph Cm, where m 3 and n 2. In this paper, we study the graphs CS3,n and CSm,n and determine their total edge irregularity strength. The case m = 3 is considered separately because the structure of CS3,n gives a different representation of the vertex and edge sets than for m 4, requiring a different labeling construction.
Co-Authors Darmajid Darmajid Mohammad Ilham Maulana Ratno Bagus Edy Wibowo Stenly Pranata