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.

In this paper we prove that the hypercube $Q_n$ is an (a, d)-distance antimagic graph. Also, we present several families of disconnected distance antimagic graphs.