I read that one of the current challenging problems in mathematics is constructing a minimal graph with a specified 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).

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!