Distance Antimagic Labelings of Graphs

Abstract

Let \(G=(V,E)\) be a graph of order n. Let \(f: V(G)\rightarrow \{1,2,\dots ,n\}\) be a bijection. For any vertex \(v \in V,\) the neighbor sum \(\sum \limits _{u\in N(v)}f(u)\) is called the weight of the vertex v and is denoted by w(v). If \(w(x) \ne w(y)\) for any two distinct vertices x and y, then f is called a distance antimagic labeling. A graph which admits a distance antimagic labeling is called a distance antimagic graph. If the weights form an arithmetic progression with first term a and common difference d, then the graph is called an (a, d)-distance antimagic graph.