  Distance-regular graphs are a class of regualr graphs with pretty
combinatorial symmetry. In 2007, Miklavi\v{c} and Poto\v{c}nik proposed the
problem of charaterizing distance-regular Cayley graphs, which can be viewed as
a natural extension of the problem of characterizing strongly-regular Cayley
graphs (or equivalently, regular partial difference sets). In this paper, we
provide a partial characterization for distance-regular Cayley graphs over
semi-dihedral groups and pseudo-semi-dihedral groups, both of which are
$2$-groups with a cyclic subgroup of index $2$.

Combinatorics