Is there a lower bound on the number of perfect matchings in a $k$-regular bipartite graph?


One can use Hall's marriage theorem and induction on $k$ to derive the lower bound of $k$. I can't come up with an example where this bound is actually tight. Is there a better lower bound than this?



Minimum number of perfect matchings in a regular bipartite graph
as already mentioned in the comments,
that is answered by Alexander Schrijver in his publication "**Counting 1-factors in regular bipartite graphs**":



> 
> any $k$-regular bipartite graph with $2n$ vertices has at least $$\left(\frac{(k-1)^{k-1}}{k^{k-2}}\right)^n$$ perfect matchings.
> 
> 
> 


