Enumerating the distance magic labelings of a distance magic graph

Let $G = (V,E)$ be a graph of order $n$. A bijection $f : V \rightarrow \{1,2,\cdots,n\}$ is a distance magic labeling of $G$ if there exists a positive integer $k$ such that $\sum_{u \in N(v)}f(u) = k$ for all $v \in V$, where $N(v)$ is the neighborhood of $v$. Any graph which admits a distance magic labeling is called a distance magic graph. In this article, we give a partial solution to the problem by Rao et al.[10] to predict all distance magic labelings of cartesian product of two cycles, $C_m \Box C_m$, where $m\equiv 2 \mod 4$. Further, we prove that the number of distance magic labelings of a distance magic graph is a multiple $| Aut(G)|$ where $Aut(G)$ is the automorphism group of the distance magic graph $G$.