Minimum number of Hamiltonian Cycle(s)

Question

Given a graph of size n (n vertices). State the minimum number of Hamiltonian Cycle(s) it can have In a document I was looking for, it is

2^[(n−3)/2]

Is it true? How to prove it?

Answer 1

In The minimum number of Hamilton cycles in a hamiltonian threshold graph of a prescribed order, Theorem 6 stated:

Theorem 6. The minimum number of Hamilton cycles in a hamiltonian threshold graph of order $n$ is $2^{\lfloor(n-3) / 2\rfloor}$ and this minimum number is attained uniquely by the graph $G_n$.

You can take a look at the proof there.