The object is configuration graphs with N vertices, numbered from 1 to N, whose vertex degrees are independent identically distributed random variables. The second degree η⁽²⁾ of an arbitrary vertex A of a configuration graph is equal to the sum of the degrees of the vertices adjacent to the vertex A excluding the edges going to A. As N →∞, the form of the generating function and the distribution of the random variable η⁽²⁾ are found for graphs whose vertex degrees have the Poisson distribution. Also, as N →∞, the form of the generating function of a random variable η⁽²⁾ is obtained for graphs with the vertex degree distribution p_k > 0, k= 1, 2, ..., such that p_k ∼ d/k^g( lnk)^h, h 0, g>7/3, d>0, as k→∞.