It can be shown that the set of graphs with $N$ vertices $G_N$ has cardinality:
\begin{equation} \lvert G_N \rvert = 2^{N \choose 2} \tag{1} \end{equation}
Recently, I wondered how much bigger $\lvert G_N \rvert$ is compared to the number of graphs with $N$ vertices that are path-connected, $PC_N$.
If we denote the set of Hamiltonian paths by $H_N$ we can easily show that:
\begin{equation} H_N \subset PC_N \tag{2} \end{equation}
and by Stirling's approximation:
\begin{equation} \lvert H_N \rvert = N! \sim \sqrt{2\pi N} \big(\frac{N}{e}\big)^N \tag{3} \end{equation}
and therefore $\lvert PC_N \rvert$ must grow exponentially fast.
Now, I'm curious about asymptotic formulas for $\lvert PC_N \rvert$ and I'm fairly confident that for large $N$:
\begin{equation} \frac{\lvert PC_N \rvert}{\lvert G_N \rvert} \leq e^{-N} \tag{4} \end{equation}
but I suspect that a proof for this statement would be fairly subtle.
Note: I didn't make use of a computer before formulating this hypothesis but in retrospect I should have analysed the connectivity of random graphs, as suggested by Olivier Fouquet. This would have given me more insight into the problem.