Let $G$ be a primitive group acting on $\Omega$. Let $\alpha \in \Omega$. Then we have a natural bijection between the orbitals of $G$ and the orbits of $G_{\alpha}$: $$\Delta \to \lbrace \beta \in \Omega \mid (\alpha , \beta) \in \Delta \rbrace $$ Let $\gamma \in \Omega$, then we can consider the orbit $O$ of $G_{\alpha}$ containing $\gamma$. Suppose $|O|=k$ and let $\Delta$ be the orbital corresponding with $O$ by the above bijection. Then the number of pairs in $\Delta$ beginning with $\alpha$ is $k$.
Firstly, if we suppose the orbital graph $\Delta$ is self-paired (i.e. its orbital graph is undirected) it has a valency of $k$. It has indeed $k$ elements incident with $\alpha$, but how can I tell that it is the case for every other element of $\Delta$? In other words, how can I tell the orbital graphs are regular graphs.
Secondly, how can I prove that this graph has a diameter of at most $r-1$, where $r$ is the number of orbitals of $G$.