Distance Antimagic Graphs

Abstract

Let $G=(V,E)$ be a graph of order $n$. Let $f:V\to \{1,2,\dots ,n\}$ be a bijection. For any vertex $v\in V$, the neighbor sum $\sum _{u\in N(v)}f(u)$ 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. In this paper, we present several results on distance antimagic graphs along with open problems and conjectures.