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We consider configuration graphs with  N  vertices. The degrees of vertices are independent identically distributed random variables having the power-law distribution with positive parameterτ  .    $\tau .$   We study properties of random graphs such that the sum of vertex degrees does not exceed  n  and the parameter is a random variable uniformly distributed on the interval[   a  ,        b   ]   ,  0  &lt;  a  &lt;  b  &lt;  ∞  .    $\left[ a,\,\,b \right],0&lt;a&lt;b&lt;\infty .$   We find limit distributions of the numberμ  r${{\mu }_{r}}$   of vertices with degree  r  for various types of variation of  N, n  and r.

On the asymptotics of degree structure of configuration graphs with bounded number of edges