Orbits in finite regular graphs

Given 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 = 1 ν = ϵ ≥ 1 when  k = 2 1 ≤ ν ≤ 2 ϵ ≤ 2 k ν when  k ≥ 3 . an arbitrary odd prime p , we construct countably many pairwise non-isomorphic p -regular graphs which are edge-transitive but not vertex-transitive.