A graph $G=(V,E)$ is said to be distance magic if there is a bijection $f$ from a vertex set of $G$ to the first $|V(G)|$ natural numbers such that for each vertex $v$, its weight given by $\sum_{u \in N(v)}f(u)$ is constant, where $N(v)$ is an open neighborhood of a vertex $v$. In this paper, we introduce the concept of $p$-distance magic labeling and establish the necessary and sufficient condition for a graph to be distance magic. Additionally, we introduce necessary and sufficient conditions for a connected regular graph to exhibit distance magic properties in terms of the eigenvalues of its adjacency and Laplacian matrices. Furthermore, we study the spectra of distance magic graphs, focusing on singular distance magic graphs. Also, we show that the number of distance magic labelings of a graph is, at most, the size of its automorphism group.

翻译：图$G=(V,E)$称为距离魔幻图，若存在一个从顶点集$V(G)$到前$|V(G)|$个自然数的双射$f$，使得对每个顶点$v$，其权重$\sum_{u \in N(v)}f(u)$为常数，其中$N(v)$表示顶点$v$的开邻域。本文引入$p$-距离魔幻标号的概念，并建立了图为距离魔幻的充要条件。此外，我们基于邻接矩阵和拉普拉斯矩阵的特征值，给出了连通正则图具有距离魔幻性质的充要条件。进一步，我们研究了距离魔幻图的谱，重点关注奇异距离魔幻图。同时，我们证明了图的距离魔幻标号数量至多等于其自同构群的阶数。