Orbits in finite regular graphs

AbstractGiven three integers k, ν and ϵ, we prove that there exists a finite k-regular graph whose automorphism group has exactly ν orbits on the set of vertices and ϵ 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 p, we construct countably many pairwise non-isomorphic p-regular graphs which are edge-transitive but not vertex-transitive