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arXiv · 2109.09708

The least Euclidean distortion constant of a distance-regular graph

Sebastian M. Cioabă·Himanshu Gupta·Ferdinand Ihringer·Hirotake Kurihara
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Abstract

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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Sebastian M. Cioabă, Himanshu Gupta, Ferdinand Ihringer, Hirotake Kurihara. 2021-09-20. The least Euclidean distortion constant of a distance-regular graph. https://arxiv.org/abs/2109.09708

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