A regular graph cover is a graph cover 
whose deck transformations
act transitively on each fiber
of the graph projection. Equivalently, if 
and 
are two lifts of the same vertex
of 
,
then some deck transformation sends 
to 
. Here "regular" describes the covering action,
not the equality of vertex degrees in a regular graph.
Ordinary voltage graphs construct regular graph covers. More generally, permutation voltage assignments construct arbitrary graph covers (Gross and Tucker 1977, Gross and Tucker 1987).
