We consider configuration graphs with N vertices. The degrees of the verticesare independent identically distributed random variables according to power-lawdistribution with positive parameter . They are equal to the number of vertex’ssemiedges that are numbered in an arbitrary order. The graph is constructed byjoining all of the semiedges pairwise equiprobably to form edges. Such models canbe used for describing different communication networks and Internet topology. Westudy the subset of random graphs under the condition that the sum of vertexdegrees is known and it is equal to n. The properties of the graph depend on thebehaviour of the parameter . We assume that is a random variable followinguniform distribution on the interval [a; b]; 0 < a < b < 1. Let (N) and r be themaximum vertex degree and the number of vertices with a given degree r. Limitdistributions of these random variables as N; n ! 1 in such a way that n=N ! 1were known only if a 6 1. In the paper we proved limit theorems for (N) and rin the case a > 1