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