Based on Kirchoff's theorem, I'm wondering what can be said about the determinant of the Laplacian, after removing the columns and rows corresponding to two vertices. The motivation would be to see if iterating this process would give insight on the factorization or growth of the number of spanning trees as a function of the vertices.
I would appreciate any references to the literature! Thank you.
Induction over the number of vertices whose corresponding rows and columns you are removing from the Laplacian matrix (hence removing those vertices from the graph): whatever is left from the original Laplacian matrix is guaranteed to be positive definite, if and only if at least one of the remaining vertices was connected to any of the vertices you removed.