The configuration graph where vertex degrees are independent identically distributed random variables is often used for modeling of complex networks such as the Internet. We consider a random graph consisting of N vertices. The random variables η1,….,ηN are equal to the degrees of vertices with the numbers 1,… ,N The probability P{ηi=k}, i=1,…, N, k=1,2,…., is equivalent to h(k)/kτ as k→∞, where h(x) is a slowly varying function integrable in any finite interval, τ>1. We obtain the limit distributions of the maximum vertex degree and the number of vertices with a given degree under the condition that the sum of degrees is equal to n and N,n→∞.