Vertex N-magic total labelings in graph families: Existence, constructions, and constraints

Abstract

 In this manuscript, we are going to analyze this novel labeling technique and methods for the detection of vertex $N$-magic graphs and pursue the types of graphs under scrutiny. The study then focuses on the complicated field of vertex $N$-magic total labeling in graph theory. Theoretical derives the existence of distinct magic constants for several graphs and their viability in triangle, snake, and H-tree graphs. The approach employs vertex $N$-magic labeling to optimize infrastructure requirements and reduce interference among communication towers.  Potential areas for future investigation include the unresolved issues surrounding irregular graph behavior and the dependability of generalized fan graphs.