English: The number ${\displaystyle {\widehat {\operatorname {G} }}_{n}}$ of connected labeled simple graphs with n vertices can be determined from the recursive formula

${\displaystyle {\widehat {\operatorname {G} }}_{n}={\begin{cases}1&&n=1\\\displaystyle 2^{\binom {n}{2}}-{\frac {1}{n}}\sum _{k=1}^{n-1}\ k{\binom {n}{k}}\ 2^{\binom {n-k}{2}}\ {\widehat {\operatorname {G} }}_{k}&&n>1\end{cases}}}$

This is sequence A001187 in OEIS.

Having indicated the generic graph with the notation ${\displaystyle \operatorname {G} =(V,E,\phi _{a})}$ with ${\displaystyle |V(G)|=n,\ |E(G)|=m}$ and

${\displaystyle \phi _{a}:V\to {\mathcal {P}}(V)}$

which is called adjacency function from V to its power set.

The number of edges is in the range ${\displaystyle m=0,1,\ldots ,{\binom {n}{2}}}$.