Relabel to front, what does topological sort according to admissible network mean?
By definition an "admissible edge" $(u,v)$ is an edge that has a residual capacity $c_f(u,v) &gt; 0$ and the height of the vertex it enters is greater than the vertex it leaves by exactly one $h.u = h.v + 1$.

1- When the list is initialized, there exist no admissible-edges in the admissible-network because the height difference between all pairs of vertices in the network is either $|V|$ or $0$; only the source vertex has a height of $|V|$ while all other vertices has a height of $0$. Therefore, no matter which order of vertices we use, the list will be topologically sorted.

2- After relabeling a vertex $u$, there is at least on edge leaving $u$, but there are no edges entering it (according to lemma 26.28 in CLRS book p.750). Therefore, there can exist no other vertex $v$ such that $(v,u)$ is an edge and $v$ precedes $u$ in the topological order. So moving $u$ to the front of the list keeps it topologically sorted.