We consider configuration graphs with N vertices. The degrees of the vertices are independent identically distributed random variables according to power-law distribution. Node degrees form semiedges that are numbered in an arbitrary order. The graph is constructed by joining all the stubs 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 equal to n. The properties of the graph depend on the value of the parameter τ of the vertex degree distribution. Let µr be the number of vertices with degree r. We obtained the limit distributions of µr as N, n → ∞ with all possible values of r and τ . Also in our model the parameter τ can be changed together with N, n.