A Complete Characterization of all Magic Constants Arising from Distance Magic Graphs

A positive integer $k$ is called a magic constant if there is a graph $G$ along with a bijective function $f$ from $V(G)$ to first $\mathrm{\mid }V(G)\mathrm{\mid }$ natural numbers such that the weight of the vertex $w(v)={\sum }_{uv\in E}f(v)=k$ for all $v\in V$. It is known that all odd positive integers greater equal $3$ and the integer powers of $2$, $2^t$, $t\ge 6$ are magic constants. In this paper we characterise all positive integers which are magic constants.