Enhanced Network Security Using Modular Multiplicative Divisor Labeling in Vertex Switched Graphs

This research investigates the characteristics of Modular Multiplicative Divisor labeling in a Jewel Graph under vertex switching when the jewel count is even. Its formally establish that the transformed graph, denoted as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{j}_{\boldsymbol{\eta}}^{\prime}$</tex>, admits an MMD labeling under specific vertex assignments. By defining a structured labeling function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mu: V\left(j_{\eta}^{\prime}\right) \rightarrow\{1,2,3, \ldots, \eta+4\}$</tex>, a systematic framework that ensures the total of the edge labels is equivalent to zero modulo the overall vertex count. 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 the broader understanding of MMD labeling in structured graph transformations, offering potential applications in network topology optimization and secure communication frameworks.