The least Euclidean distortion constant of a distance-regular graph

In 2008, Vallentin made a conjecture involving the least distortion of an embedding distance-regular graph into Euclidean space. Vallentin’s implies that for diameter d, most contracted pairs vertices are those at distance d. this paper, we confirm several families graphs. We also provide counterexamples to conjecture, where largest contraction occurs between d−1. suggest three alternative conjectures and prove them various graphs 3.