  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.
