Algorithmic Verification of Modular Multiplicative Divisor Labeling in Vertex Switched Jewel Graphs with Odd/Even Jewel Count for Secure Networks Using Python Pseudocode

This research investigates the characteristics of Modular Multiplicative Divisor MMD labeling in a Jewel Graph under vertex switching when the jewel count 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 ensuring that the total sum of edge labels is congruent to zero when divided by the total number of vertices. The analysis is rigorously verified through algebraic formulations of edge sums, confirming the existence of an 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.