1 if **lim**_{n ––> +oo} {N(n) \over A(n)} = 0 or = +oo. A(n) is a "regular threshold function" of A if there exists a probability distribution function F(x) such that **lim**_{n ––> +oo} P_{n,N(n)}(A) = F(x) if **lim**_{n ––> +oo} {N(n) \over A(n)} = x, where 0 < x <+oo and x is a point of continuity of F(x). The investigated properties are as follows: the presence of certain subgraphs (e. g. trees, complete subgraphs, cycles, etc.) or connectedness, number of components etc. The results are of the following type: Theorem 3a. Suppose that N(n) ~ cn, where c > 0. Let \gamma_{k} denote the number of cycles of order k contained in \Gamma_{n,N} (k = 3,4,...). Then we have **lim**_{n ––> +oo} P_{n,N(n)} (\gamma_{k} = j) = \lambda^{j} e^{-\lambda}/j!, where j = 0,1,... and \lambda = (2c)^{k}/2k. Thus the threshold distribution corresponding to the threshold function A(n) = n for the property that the graph contains a cycle of order k is 1-e^{-(2c)k/2k}.

**Reviewer: ** K.Culik

**Classif.: ** * 05C80 Random graphs

**Index Words: ** topology

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