A Novel Computational Technique based on Python and MATLAB for Structural Analysis of Zero Divisor Graphs in Modular Rings

Résumé

This paper presents advanced computational methods for visualizing zero-divisor graphs (ZDGs) of the ring of integers modulo  (denoted as  using MATLAB and Python. In a finite commutative ring  with unity, elements  and  are zero divisors if The set of zero divisors forms the basis for constructing ZDGs, which are pivotal for uncovering the algebraic characteristics of . Our study focuses on developing and implementing novel algorithms in MATLAB 2020 and Python 3.12.3 for constructing and analyzing the ZDGs for ​ efficiently. We provide a comprehensive methodology, theoretical foundation, and proofs of correctness for these algorithms. The paper also includes a detailed complexity analysis, demonstrating the computational efficiency and validity of our methods. By benchmarking our algorithms against existing approaches, we show their superior performance in terms of both speed and accuracy. This research not only enhances the understanding of algebraic structures through effective visualization but also offers a significant improvement over previous methods, paving the way for further exploration and application of ZDGs in algebraic research.