Orbits in finite regular graphs

Given three integers kk, νν and ϵϵ, we prove that there exists a finite kk-regular graph whose automorphism group has exactly νν orbits on the set of vertices and ϵϵ orbits on the set of edges if and only if {(ν,ϵ)=(1,0)when k=0(ν,ϵ)=(1,1)when k=1ν=ϵ≥1when k=21≤ν≤2ϵ≤2kνwhen k≥3.Given an arbitrary odd prime pp, we construct countably many pairwise non-isomorphic pp-regular graphs which are edge-transitive but not vertex-transitive.