What graphs on $\mathbb{N}$ can be encoded as regular languages?
$\beta(E_1)$ is the language $s^nx,s^{n+1}x$. This language is straightforwardly not regular, by the pumping lemma.
If we assume that the language is regular, the pumping lemma tells us that there must exist some $p, q$ such that for all $n \ge p$, $s^{n+q}x,s^{n+1}x$ is also in the language. This is false, meaning that the language is not regular.
A similar argument also demonstrates that $\beta(E_2)$ is not regular, and more generally demonstrates that for a graph to be regular, for any $(n, m)$ in the graph with $n$ greater than some finite cutoff $p$, there must be infinitely many $n'$ such that $(n', m)$ is in the graph.
Since both $E_1$ and $E_2$ fail this condition, they are not regular.