Properties of graphs with polynomial growth

Let X be a connected locally finite transitive graph with polynomial growth. We prove that groups with intermediate growth cannot act transitively on X. Furthermore, it follows from this result that the automorphism group AUT(X) is uncountable if and only if it contains a finitely generated subgroup with exponential growth which acts transitively on X. If X has valency at least three, we prove that X cannot be 8-transitive.