Strong VNMT Labelings on Graph Families and Their Applications in Disease Transmission Networks

Abstract

Graph labeling is a central concept in graph theory, serving as a mathematical language to represent various real-world phenomena. Among these, Vertex N-magic Total (VNMT) labeling has emerged as a robust method to encode structural properties through bijective mappings that assign unique integers to both vertices and edges. In VNMT labeling, each vertex’s weighted sum corre-
sponds to one of N distinct, strictly increasing magic constants, with values dependent on the vertex degree. This study investigates strong VNMT labelings for several graph families. For each graph type, explicit labeling schemes are constructed and proven to satisfy VNMT properties with three magic constant bounds. The dualities of graph labeling, particularly their applicability to modeling
epidemiological networks, are also examined. This work broadens the scope of magic labeling and its practical implications, such as tracking transmission patterns in contagious diseases, thereby show- casing its relevance in both theoretical and applied domains.