In this paper, we introduce a series of necessary conditions namely the concept of $p$-distance magic labeling and give a relation between labeling and graph-theoretic properties of a graph. We also prove that a graph which is $p$-distance magic, for all $p$ is also distance magic. As a result we have a new necessary and sufficient condition for a distance magic graph. We also obtain conditions for when the ``group distance magic constant will be unique".

翻译：本文引入了一系列必要条件，即 $p$-距离幻标号的概念，并给出了标号与图论性质之间的关系。我们还证明，对于所有 $p$ 均为 $p$-距离幻的图也是距离幻的。由此，我们得到了距离幻图的一个新的充要条件。此外，我们还得到了“群距离幻常数唯一”的条件。