We consider the model of an $N$-vertex configuration graph where the number of edges is at most $n$ and the degrees of vertices are independent and identically distributed (i.i.d.) random variables (r.v.'s). The distribution of the r.v. $xi$, which is defined as the degree of any given vertex, is assumed to satisfy the condition $p_k=mathbf{P}{xi=k}simfrac{L}{k^gln^h k}$ as $k oinfty$, where $L>0$, $g>1$, $hge0$. Limit theorems for the number of vertices of a given degree as $N, n oinfty$ are proved.