AI 中文总结
研究针对Rao等人提出的预测两个特定圈笛卡尔积\(C_m\Box C_m\)（\(m\equiv 2\mod 4\)）所有距离幻标注的问题给出部分解，还证明距离幻图的距离幻标注数量是其自同构群阶数的倍数。
AI 中文摘要
设\(G=(V,E)\)是一个阶为\(n\)的图。双射\(f:V\rightarrow\{1,2,\cdots,n\}\)是\(G\)的距离幻标注，若存在正整数\(k\)，使得对所有\(v\in V\)，有\(\sum_{u\in N(v)}f(u)=k\)，其中\(N(v)\)是\(v\)的邻域。任何具有距离幻标注的图称为距离幻图。本文对Rao等人提出的预测两个圈\(C_m\Box C_m\)（\(m\equiv 2\mod 4\)）笛卡尔积的所有距离幻标注问题给出部分解决方案。此外，我们证明距离幻图的距离幻标注数量是其自同构群\(Aut(G)\)阶数的倍数。
英文摘要
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 $G$.
Comments10 pages
