关于价为6的距离魔幻循环图
On distance magic circulants of valency 6
AI总结：
本文研究价为6的距离魔幻循环图，给出其成为距离魔幻图的若干必要与充分条件，并完整分类阶不被12整除的情形。
摘要：
一个阶为 n 的图 Γ=(V,E) 被称为距离魔幻图，如果它允许对其顶点进行双射标号 ℓ:V→{1,2,…,n}，且存在正整数 κ，使得对于所有顶点 v∈V，都有 ∑_{u∈N(v)} ℓ(u)=κ，其中 N(v) 是 v 的邻域。循环图是指允许一个自同构对其顶点进行循环置换的图。本文研究价为6的距离魔幻循环图。我们得到了价为6的循环图成为距离魔幻图的若干必要条件和充分条件，从而找到了若干无限族示例。本文的综合结果对所有价为6的距离魔幻循环图给出了部分分类。特别地，我们对阶不被12整除的价为6的距离魔幻循环图进行了分类。
英文摘要：
A graph $Γ= (V,E)$ of order $n$ is {\em distance magic} if it admits a bijective labeling $\ell \colon V \to \{1,2, \ldots, n\}$ of its vertices for which there exists a positive integer $κ$ such that $\sum_{u \in N(v)} \ell(u) = κ$ for all vertices $v \in V$, where $N(v)$ is the neighborhood of $v$. %It is well known that a regular distance magic graph is necessarily of even valency. A {\em circulant} is a graph admitting an automorphism cyclically permuting its vertices. In this paper we study distance magic circulants of valency $6$. We obtain some necessary and some sufficient conditions for a circulant of valency $6$ to be distance magic, thereby finding several infinite families of examples. The combined results of this paper provide a partial classification of all distance magic circulants of valency $6$. In particular, we classify distance magic circulants of valency $6$, whose order is not divisible by $12$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
