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We consider configuration graphs with  N  vertices. The degrees of vertices are independent identically distributed random variables having the power-law distribution with parameter  τ  &gt; 0. There are two critical values of this parameter:  τ  = 1 and  τ  = 2. The properties of a graph change significantly when  τ  =  τ ( N ) passes these points as  N  → ∞. Let  G   N , n   be the subset of random graphs under the condition that sum of degrees of its vertices is equal to  n . The limit theorem for the maximum vertex degree in  G   N , n   as  N , n  → ∞ and  τ  → 1 or  τ  → 2 is proved.

On limit behavior of maximum vertex degree in a conditional configuration graph near critical points