On one characteristic of a conditional distribution of configuration graph

We consider configuration graphs with $N$ vertices. The vertex degrees are independent identically distributed random variables and for any vertex of the graph the distribution of its degree $\eta$ satisfies the following condition:
$$ \mathbf{P}\{\eta=k\}\sim \frac{d}{k^{g}\ln^h k},\quad k\to\infty, $$
where $d>0$, $h\geqslant 0$, $2< g<3$. We obtain the limit distributions of the maximal degree of vertices in the configuration graph as $N,n\to\infty$ and $n/N^{(3g-4)/(2g-2)}\to\infty$ under the conditions that the sum of vertex degrees is $n$.