I read that one of the current challenging problems in mathematics is [constructing a minimal graph with a specified number of spanning trees](https://mathoverflow.net/questions/93656/minimal-graphs-with-a-prescribed-number-of-spanning-trees) (say, $k$).


* However, is there a quick way to create some graph $G$ (*not* necessarily minimal) that has $k$ spanning trees ?


We can compute the number of spanning trees of a graph $G$ using the pseudo-determinant of the Laplacian matrix of $G$ and the number of labelled vertices ([Kirchoff's theorem](https://en.wikipedia.org/wiki/Kirchhoff%27s_theorem)).


Perhaps answering the following question helps in answering the original question:


* is there a way to generate a Laplacian matrix given its pseudo-determinant value?


Thanks in advance!



Create a graph with a specified number of spanning trees
A $k$-cycle works if $k>2$. For $k=1$ any tree works. I don't think $k=2$ is possible unless you allow double edges: if the graph is not a tree then it has an $m$-cycle $C$ for some $m>2$; remove edges off $C$ until the graph is connected but has no cycle other than $C$, and then removing an arbitrary edge of $C$ yields at least $m$ spanning trees.


