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{η = k} ∼ d / (k^g ln^h k), d > 0, h 0, g +h > 1, 2 < g < 3, and the random value η is equal to the degree of any vertex of the graph. We study the random graphs under the condition that the sum of vertex degrees is n. The limit distributions of the number of vertices with the given degree in the conditional graph are derived as N, n → ∞ and n/N^{(3g−4)/(2g−2)} →∞.