We consider configuration graphs with N vertices. The vertex degrees are independent equally distributed random variables whose distribution satisfies the following condition: as k→∞
p_k = P{η = k} ∼ h(k) / k^g , 2 < g < 3,
where h(k) is a slowly varying function. We study the random graphs under the conditions that the sum of vertex degrees is n. The limit distributions of the number of vertices with a given degree in the conditional graph are obtained as N, n → ∞ and h(N)n²N^{(4−3g)/(g−1)} ≥ C > 0 are obtained.