On the Maximal Degree of a Vertex of a Conditional Configuration Graph

We consider the model of an $N$-vertex configuration graph, where the degrees of vertices are independent and identically distributed random variables, and the distribution of the random variable $\eta$, which is the degree of each vertex, satisfies the condition $p_k=\mathbf{P}\{\eta=k\}\sim \frac{h(k)}{k^g}$, $2<g<3$, $k\to\infty$, where $h(x)$ is a function which slowly varies at infinity. We prove the limit theorem for the maximal degree of vertices in this configuration graph provided that the number $n$ of edges in the graph is known, $N, n\to\infty$, and $h(N)n^2N^{(4-3g)/(g-1)}\geqslant C>0$.