Let $G$ be a transitive group on $\Omega$. Every orbits of $G$ on its natural action on $\Omega\times\Omega$ is called an orbital of $G$ on $\Omega$. For each orbital $\Delta$ of $G$ on $\Omega$, the orbital digraph $Graph(\Delta)$ is a digraph with vertex set $\Omega$ and edge set $\Delta$. Clearly, $G$ is a subgroup of automorphism group of $Graph(\Delta)$. Is there any paper or book that determine for which groups has $G$, there is an orbital graph $\Delta$ that $G$ is equal to the automorphism group of $Graph(\Delta)$? Clearly, the action of $S_5$ on 2-subets of ${1,\ldots,5}$ is an example.
Let $G$ be a transitive group on $\Omega$. Every orbits of $G$ on its natural action on $\Omega\times\Omega$ is called an orbital of $G$ on $\Omega$. For each orbital $\Delta$ of $G$ on $\Omega$, the orbital digraph $Graph(\Delta)$ is a digraph with vertex set $\Omega$ and edge set $\Delta$. Clearly, $G$ is a subgroup of automorphism group of $Graph(\Delta)$. Is there any paper or book that determine for which groups has an orbital graph $\Delta$ that $G$ is equal to the automorphism group of $Graph(\Delta)$? Clearly, the action of $S_5$ on 2-subets of ${1,\ldots,5}$ is an example.