We consider configuration graphs with N vertices. The degrees of the vertices are independent identically distributed random variables following the power-law distribution with positive parameter τ. We study the random graphs under the conditions that the sum of vertex degrees does not exceed n and the parameter τ is a random variable uniformly distributed on the interval [a,b], 1≤a<b<∞. We obtain the limit distributions of maximum vertex degree for for different relations between the parameters N and n tending to infinity.