Topological indices and structural properties of ideal-based unit graphs in commutative rings
Abstract
Introduction/Purpose: This study introduced the concept of a prime ideal-based unit graph associated with a commutative ring . In this graph, the vertices consisted of units of that were not contained in a chosen prime ideal , and two such vertices were considered adjacent if their difference belonged to . The aim was to investigate the structural, algebraic, and topological properties of this graph and examine the algebraic implications of various graph-theoretic invariants.
Methods: The construction of ideal-based unit graphs was carried out using the ring , where units excluded from the chosen prime ideal formed the vertex set. The adjacency between two vertices was determined by whether their difference lay in the ideal. The analysis involved computing several topological indices including the Zagreb indices, Wiener index, Arithmetic-Geometric index, Harmonic index, Estrada index, and graph energy. Adjacency matrices and graphical visualizations were employed to understand structural complexity and connectivity.
Results: It was observed that the structure of the resulting graph depended significantly on both the modulus and the nature of the selected ideal. Smaller ideals produced graphs with higher connectivity, while larger ideals led to sparser or disconnected graphs. The calculated indices reflected patterns in symmetry, degree distribution, and distances, revealing deeper algebraic characteristics.
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