Orbits on vertices and edges of finite graphs

Abstract

Given two integers ν > 0 and ϵ >/ 0, we prove that there exists a finite graph (resp. a finite connected graph) whose automorphism group has exactly ν orbits on the sets of vertices and ϵ orbits on the set of edges if and only if ν ⩽ 2 ϵ + 1 (resp. ν ⩽ ϵ + 1).