  This paper represents a significant leap forward in the problem of
enumerating vertex-transitive graphs. Recent breakthroughs on symmetry of
Cayley (di)graphs show that almost all finite Cayley (di)graphs have the
smallest possible automorphism group. Extending the scope of these results, we
enumerate (di)graphs admitting a fixed semiregular group of automorphisms with
m orbits. Moreover, we consider the more intricate inquiry of prohibiting arcs
within each orbit, where the special case m = 2 is known as the problem of
finding Haar graphical representations (HGRs). We significantly advance the
understanding of HGRs by proving that the proportion of HGRs among Haar graphs
of a finite nonabelian group approaches 1 as the group order grows. As a
corollary, we obtain an improved bound on the proportion of DRRs among Cayley
digraphs in the solution of Morris and the second author to the Babai-Godsil
conjecture.
