  Recently, several works by a number of authors have studied integrality,
distance integrality, and distance powers of Cayley graphs over some finite
groups, such as dicyclic groups and (generalized) dihedral groups. Our aim is
to generalize and/or to give analogues of these results for generalized
dicyclic groups. For example, we give a necessary and sufficient condition for
a Cayley graph over a generalized dicyclic group to be integral (i.e., all
eigenvalues of its adjacency matrix are in $\mathbb{Z}$). We also obtain
sufficient conditions for the integrality of all distance powers of a Cayley
graph over a given generalized dicyclic group. These results extend works on
dicyclic groups by Cheng--Feng--Huang and Cheng--Feng--Liu--Lu--Stevanovic,
respectively.
