On the limit distributions of the degrees of vertices in configuration graphs with a bounded number of edges

A model of a configuration graph on $N$ vertices is considered in which the degrees of the vertices are distributed identically and independently according to the law $\mathbf P\{\xi=k\}=k^{-\tau}-(k+1)^{-\tau}$, $k=1,2,\dots$, $\tau>0$, and the number of edges is no greater than $n$. Limit theorems for the number of vertices of a particular degree and for the maximum vertex degree as $N,n\to\infty$ are established.
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