I read that one of the current challenging problems in mathematics is [constructing a minimal graph with a specified number of spanning trees][1] (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][2]).

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!


  [1]: http://mathoverflow.net/questions/93656/minimal-graphs-with-a-prescribed-number-of-spanning-trees
  [2]: https://en.wikipedia.org/wiki/Kirchhoff%27s_theorem