Structural Properties of Graphs Defined by Bilinear Congruence on the Ring of Integers Modulo n

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Published: Sep 10, 2025
DOI: https://doi.org/10.26629/uzjns.2025.08
Keywords:
Bilinear Form, Zero-Divisor Graph, Modular Arithmetic, Hamiltonian Cycle, Maximum Clique, Supereulerian Graph

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Hamza A Daoub
University of Zawia
https://orcid.org/0009-0000-2999-3462
Omaema O Lasfar
جامعة الزاوية
https://orcid.org/0009-0007-1387-6780

Abstract

We investigate the structural properties of a graph defined on the ring . The Adjacency between two different vertices  and  is determined by the bilinear congruence . We analyze three fundamental cases,  and  for distinct odd primes . We describe the graph's breakdown into unit and non-unit vertex subsets. The unit subgraph forms disjoint cliques, with sizes depending on Euler's totient function. In contrast, the zero-divisor subgraph shows more complex behaviour governed by annihilation ideals. We establish general properties, including degree formulas, determination of maximum clique sizes in each component, determining the diameter, computing the girth, locating the graph centers, and finding the measures of vertex and edge connectivity. Additionally, we characterize independent sets and prove the existence of Hamiltonian cycles and supereulerian properties under certain connectivity conditions. Our results show how the prime factorization of  influences these properties.

Article Details

How to Cite
[1]
H. A. Daoub and O. O. Lasfar, “Structural Properties of Graphs Defined by Bilinear Congruence on the Ring of Integers Modulo n”, UZJNS, vol. 2, no. 2, pp. 56–70, month-09 2025.
Issue
Vol. 2 No. 2 (2025): Univ Zawia J Nat Sci
Section
Mathematics
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This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

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