Vertex Antimagic Total Labeling of Modified Fan Graphs

Abstract

Graph labeling is the process of assigning labels to the vertices and edges of a graph according to specific rules or conditions. A vertex antimagic total labeling (VATL) is a distinct assignment of positive integers to all vertices and edges of a graph such that the total weight obtained by adding the label of each vertex and the labels of all edges incident with it, is unique for every vertex in that graph. This paper investigates and proves the vertex antimagic total labeling of a modified fan graph, which is obtained by merging the isolated vertices K$_{1}$ from n copies of the disjoint union K$_{1}$ $\bigcup$ $\overline{(K_{2})}$ $\bigcup$ $\overline{(K_{m})}$ into a single apex vertex. This study extends it by proposing an algorithm and providing a computational complexity of O(nm) as evidence for verification of this labeling.