Orbits in finite regular graphs

Given three integers k, v and c, we prove that there exists a finite k-regular graph whose automorphism group has exactly v orbits on the set of vertices and E orbits on the set of edges if and only if (v, is an element of) = (1, 0) when k = 0 (v, is an element of) = (1, 1) when k = 1 v = is an element of >= 1 when k = 2 1 <= v <= 2 is an element of <= 2kv when 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. (c) 2005 Elsevier Ltd. All rights reserved.