Given a finite connected graph $Γ$ and a group $G$ acting transitively on the vertices of $Γ$, we prove that the number of vertices of $Γ$ and the cardinality of $G$ are bounded above by a function depending only on the cardinality of $Γ$ and on the exponent of $G$. We also prove that the number of generators of a group $G$ acting transitively on the arcs of a finite graph $Γ$ cannot be bounded by a function of the valency alone.