We consider configuration graphs with N vertices. The degrees of the vertices are independent random variables identically distributed according to the power law, with a positive parameter τ . They are equal to the number of vertex semiedges that are numbered in an arbitrary order. The graph is constructed by joining all of the semiedges pairwise equiprobably to form edges. We study the subset of such random graphs under the condition that the sum of vertex degrees is known and it is equal to n. Let τ be a random variable following a truncated normal distribution on an arbitrary fixed finite interval. We obtained the limit distributions of the maximum vertex degree and the number of vertices with a given degree for various zones of N and n tendency to infinity.