距离正则图的最小欧氏失真常数
The least Euclidean distortion constant of a distance-regular graph
摘要：
在2008年，Vallentin提出了一个猜想，涉及将距离正则图嵌入到欧几里得空间的最小失真。Vallentin的猜想意味着，对于直径为$d$的距离正则图的最小失真欧氏嵌入，最收缩的顶点对是距离为$d$的顶点对。本文对几类距离正则图证明了Vallentin猜想。我们还提供了反例，这个猜想，其中最大的收缩发生在距离$d{-}1$的顶点对之间。我们提出了三种可供选择的图，并证明了他们的几个家庭的距离正则图。
英文摘要：
In 2008, Vallentin made a conjecture involving the least distortion of an embedding of a distance-regular graph into Euclidean space. Vallentin's conjecture implies that for a least distortion Euclidean embedding of a distance-regular graph of diameter $d$, the most contracted pairs of vertices are those at distance $d$. In this paper, we confirm Vallentin's conjecture for several families of distance-regular graphs. We also provide counterexamples to this conjecture, where the largest contraction occurs between pairs of vertices at distance $d{-}1$. We suggest three alternative conjectures and prove them for several families of distance-regular graphs.
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