Abstract:
Let $\mathfrak{G}(n)$ be the set of all nonoriented graphs with $n$ enumerated points without loops or multiple lines, and let $\nu_k(G)$ be the number of mutually nonisomorphic $k$-point subgraphs of $G\in\mathfrak{G}(n)$. It is proved that at least $|\mathfrak{G}(n)|(1-1/n)$ graphs $G\in\mathfrak{G}(n)$ possess the following properties: a) for any $k\in[6\log_2n, cn+5\log_2n]$, where $c=-c\log_2c-(1-c)\times\log_2(1-c)$ and $c>1/2$, we have $\nu_k(G)>C_n^k(1-1/n^2)$; b) for any $k\in[cn+5\log_2n, n]$ we have $\nu_k(G)=C_n^k$. Hence “almost all” graphs $G\in\mathfrak{G}(n)$ contain $\nu(G)\sim2^n$ pairwise nonisomorphic subgraphs.
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