The least Euclidean distortion constant of a distance-regular graph

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 various families of distance-regular graphs and for distance-regular graphs of diameter 3. (c) 2022 Elsevier B.V. All rights reserved.