Python based Approach for Modular Labeling in Switched Jewel Graphs

This research investigates the characteristics of
Modular Multiplicative Divisor (MMD) labeling in a Jewel
Graph under vertex switching when the number of jewels is odd.
We formally establish that the transformed graph admits an
MMD labeling under specific vertex assignments. By defining a
structured labeling function that maps each vertex of the
transformed graph to a sequential set of positive integers, we
develop a systematic framework that ensures the total sum of
edge labels is congruent to zero modulo the total number of
vertices. The analysis is rigorously verified through algebraic
formulations of edge sums and a Python pseudocode program,
confirming the existence of MMD labeling in such cases. These
findings contribute to a deeper understanding of MMD labeling
in structured graph transformations, offering potential
applications in network topology optimization and secure
communication frameworks.