Strongly Edge Perfectly Antimagic Total Graphs: Properties, Constraints

Abstract

Specialized graph labeling known as Strongly Edge Perfectly Antimagic Total (SEPAT) labeling guarantees different assignments throughout a graph's structure by making sure that all edge weights are greater than vertex weights. The existence or nonexistence of SEPAT labeling is examined in this study across several graph families, such as path, cycle, star, and disjoint unions of cycles. Aside from path and cycle graphs, star graphs only display SEPAT characteristics for restricted orders, and $r$-regular graphs with $r>2$ cannot be labeled with SEPAT, according to the research. In addition, we show the scope and limits of SEPAT labeling by providing thorough constructs and demonstrations. This study furthers our knowledge of antimagic total labelings and paves the way for future investigations into labeling limitations in intricate graph structures.