On Super \((a, d)\)-Edge-Antimagic Total Labeling of Subdivided Stars

Abstract

Let \(G = (V,E)\) be a graph with \(v = |V(G)|\) vertices and \(e = |E(G)|\) edges. An \((a, d)\)-edge-antimagic total labeling of the graph \(G\) is a one-to-one map \(A\) from \(V(G) \cup E(G)\) onto the integers \(\{1,2,\ldots,v+e\}\) such that the set of edge weights of the graph \(G\), \(W = \{w(xy) : xy \in E(G)\}\) form an arithmetic progression with the initial term \(a\) and common difference \(d\), where \(w(xy) =\lambda(x) + \lambda(y) + \lambda(xy)\) for any \(xy \in E(G)\). If \(\lambda(V(G)) = \{1,2,\ldots,v\}\) then \(G\) is super \((a, d)\)-edge-antimagic total, i.e., \((a,d)\)-EAT. In this paper, for different values of \(d\), we formulate super \((a, d)\)-edge-antimagic total labeling on subdivision of stars \(K_{1,p}\) for \(p \geq 5\).