Nilpotent Graphs of Rings of Integer Modulo: Structural Properties and Topological Indices - Deny Putra Malik; Gusti Yogananda Karang; Qurratul Aini; I Gede Adhitya Wisnu Wardhana (Contemporary Mathematics and Applications (ConMathA), 2026)
Nilpotent elements in modular rings play a fundamental role in understanding the algebraic structure of rings and their applications in various mathematical domains. Motivated by the need to explore the interplay between algebraic and combinatorial representations, this study introduces and investigates nilpotent graphs constructed from rings of integers modulo prime powers. We begin by characterizing nilpotent sets and establishing theorems that describe their distribution and algebraic behavior. Using these characterizations, we construct nilpotent graphs, where vertices represent nilpotent elements and edges reflect their interactions. The structural properties of these graphs are examined, and several well known topological indices, such as the Zagreb, Harary, Hyper Wiener, Randić, Harmonic, Sombor, and Schultz indices, are computed to quantify connectivity, complexity, and centrality. The results reveal meaningful patterns that bridge ring theory and graph theory.