Exploring Modular Multiplicative Divisor Labeling with Minimum Spanning Tree Optimization using Kruskal’s Algorithm in Single Edge Deletion of the Join of Two Graphs

This research investigates the integration of Modular Multiplicative Divisor (MMD) labeling with Minimum Spanning Tree (MST) analysis by using Kruskals algorithum, focusing on the influence of single edge deletions in the join of two graphs, denoted as (𝐾1+𝐾1,𝜆)−𝜎1𝜌𝜆. The article presents a unique paradigm for preserving MMD labeling following edge deletions while simplifying complex graph structures, guaranteeing crucial connectivity and labeling attributes are retained. This study extends our theoretical understanding of graph labeling and provides practical strategies for optimizing network design, particularly in cases that need structural alterations. The findings demonstrate the proposed method's robustness in maintaining network integrity, making it a powerful tool for both theoretical insights and practical applications in network optimization. MSC : 05C30, 05C70, 05C76, 05C78, 05C99, 68R10.