A graph is distance transitive, if
$\forall x,y,u,v \in V \ \text{so that}\ \ d(x,y)=d(u,v)$
there exists $g \in Aut(G)$ so that $g(x)=u$ and $g(y)=v$.
$G$ is vertex transitive if $\forall x,y\in V$ the is some $g \in Aut(G)$ so that $g(x)=y$
I want to prove that:
$G$ is distance transitive $\iff$ $G$ is vertex transitive and a stabilizer of a point over $Aut(G)$ has $diam(G)+1$ orbits.
Could you help me with where to begin?