Let $G$ be a connected vertex-transitive graph and let $G_v$ denote the stabilizer of the vertex $v$. If $h$ is any automorphism of $G$ for which $d(v,h(v)) = 1$, and $G$ is symmetric, then $h$ and $G_v$ generate $Aut(G)$.
my Idea is to take an arbitrary member of $Aut(G)$ and show that it is equal to combination of members of $G_v$ and $h$,(of course it is the natural way)but I don't know how should I use the hypothesis to achieve,
any help or guidance will be great.