s-arc transitive graph is also (s-1)-arc transitive I read on a paper that an s-arc transitive graph is also (s-1)-arc transitive and thus (s-2)-transitive, which was stated as obvious. However, I was thinking that a path of 2 edges, $P_2$, is 2-arc transitive but not vertex-transitive, right?
Could anyone advise me where I got it wrong? Thanks!

The right answer is that the s-arc transitive graph must also be regular (all vertices have the same degree). with this extra condition it is easy to prove the claim. Note in some trivial cases the result can be concluded by resorting to trivial LOGICAL rules.
