  This paper studies whether there are distance-transitive graphs arising from
the coset actions of $G_2(q)$ on the subfield subgroup $G_2(\sqrt{q})$ or
$G_2(q)$ on the Ree subgroup $^2G_2(q)$. It is found that there are no such
graphs, even if the groups are extended by outer automorphisms of $G_2(q)$.
