We consider configuration graphs with N vertices, whose degrees are independent and identically distributed according to an unknown distribution law that depends on an arbitrary slowly varying function. The degree of each vertex is equal to the number of incident numbered semi-edges. The graph is constructed by joining all of the semi-edges pairwise equiprobably to form edges. Such models can be used to adequately describe various communication networks and Internet topologies. 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. The paper finds the limit distributions of the maximum vertex degree and the number vertices with a given degree as N, n → ∞ in such a way that 0 < C₁ ≤ n/N ≤ C₂ < ∞.