Let $G$ be a connected vertex transitive graph and $G_v$ denote stabilizer of the vertex $v$. If $h$ is any automorphism of $G$ for which $d(v,h(v))=1$, and $G$ is symmetric,the $h$ and $G_v$ generate $\operatorname{Aut}(G)$.
We say that $G$ is symmetric if ,for all vertices $u,v,x,y$ of $G$ such that $u$ and $v$ adjacent, and $x$ and $y$ are adjacent there is an automorphism $g$ in $\operatorname{Aut}(G)$ for which $g(u)=x$ and $g(v)=y$
I think we should use this fact: Let $G$ be group acting transitively on set $X$, $H$ be a subgroup of $G$ and $G_a$ be stabilizer of G then $G=HG_a$ if and only if H is transitive.
Please advise me.