元恋人2-距离图的极限与周期性
Limits and Periodicity of Metamour $2$-Distance Graphs
AI总结：
本文研究有限简单图在2-距离运算迭代下的周期行为，分类了周期为3的连通图并部分刻画周期为2的情形，同时证明广义Petersen图和完全m叉树的2-距离序列最终周期为2，并给出前周期与极限图。
摘要：
给定一个有限简单图$G$，令$\operatorname{M}(G)$表示其2-距离图，其中两个顶点相邻当且仅当它们在$G$中的距离为2。本文考虑通过迭代2-距离运算得到的序列$G, \operatorname{M}(G), \operatorname{M}^2(G), \operatorname{M}^3(G), \ldots$的周期行为。特别地，我们对周期为3的连通图进行了分类，并部分刻画了周期为2的连通图。随后，我们研究了两类2-距离序列最终呈周期性的图族：即广义Petersen图和完全$m$叉树。对于每个图族，我们证明其最终周期为2，并确定了该序列的前周期和两个极限图。
英文摘要：
Given a finite simple graph $G$, let $\operatorname{M}(G)$ denote its 2-distance graph, in which two vertices are adjacent if and only if they have distance 2 in $G$. In this paper, we consider the periodic behavior of the sequence $G, \operatorname{M}(G), \operatorname{M}^2(G), \operatorname{M}^3(G), \ldots$ obtained by iterating the 2-distance operation. In particular, we classify the connected graphs with period 3, and we partially characterize those with period 2. We then study two families of graphs whose 2-distance sequence is eventually periodic: namely, generalized Petersen graphs and complete $m$-ary trees. For each family, we show that the eventual period is 2, and we determine the pre-period and the two limit graphs of the sequence.
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