Orbits on vertices and edges of finite graphs (Q1072562)

Let \(P\) be the set of all ordered pairs \((v,\varepsilon)\) of integers, \(v>0\), \(\varepsilon \geq 0\), for which there exists a finite graph whose automorphism group has exactly \(v\) orbits on the set of vertices and \(\varepsilon\) orbits on the set of edges. Let \(P_c\) be the analogous set for connected graphs. Theorem 1. \((v,\varepsilon)\in P\) iff \(v\leq 2\varepsilon +1\). Theorem 2. \((v,\varepsilon)\in P_c\) iff \(v\leq \varepsilon +1\).