  We prove that a distance-regular graph with a dominant distance is a spectral
expander. The key ingredient of the proof is a new inequality on the
intersection numbers. We use the spectral gap bound to study the structure of
the automorphism group.
  The minimal degree of a permutation group $G$ is the minimum number of points
not fixed by non-identity elements of $G$. Lower bounds on the minimal degree
have strong structural consequences on $G$. In 2014 Babai proved that the
automorphism group of a strongly regular graph with $n$ vertices has minimal
degree $\geq c n$, with known exceptions. Strongly regular graphs correspond to
distance-regular graphs of diameter 2. Babai conjectured that Hamming and
Johnson graphs are the only primitive distance-regular graphs of diameter
$d\geq 3$ whose automorphism group has sublinear minimal degree. We confirm
this conjecture for non-geometric primitive distance-regular graphs of bounded
diameter. We also show if the primitivity assumption is removed, then only one
additional family of exceptions arises, the cocktail-party graphs. We settle
the geometric case in a companion paper.
