Orbits in finite regular graphs (Q854845)

Given three integers \(k\), \(v\), and \(e\), the author shows that there is a finite \(k\)-regular graph whose automorphism group has exactly \(v\) orbits on the vertex set and \(e\) orbits on the edge set, if and only if: when \(k= 0\), \((v, e)= (1, 0)\); when \(k= 1\), \((v, e)= (1, 1)\); when \(k= 2\), \(v= e\geq 1\); and when \(k\geq 3\), \(1\leq v\leq 2e\leq 2kv\). Also, given an arbitrary odd prime \(p\), he constructs countably many nonisomorphic \(p\)-regular graphs which are edge-transitive, but not vertex-transitive.