Question

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 $G$, there is an orbital $\Delta$ 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.

Answer 1

The property of $G$ you are looking at is called 2-closure, i.e., you ask for a classification of 2-closed permutation groups. See the paper by Liebeck, Praeger, Saxl, On the 2-closures of finite permutation groups. J. London Math. Soc. (2) 37 (1988), no. 2, 241–252, where this question is investigated for a particular class of primitive permutation groups.

More precisely, this is not 100% equivalent to your question (it would be, if you allowed graphs have coloured arcs), as there exist 2-closed permutation groups $G$ such that for each orbital the automorphism group of the underlying (di)graph is strictly bigger than $G$.

In general such a classification is not known, and the problem is hopelessly hard, I think.