On the number of generators of groups acting arc-transitively on graphs

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