Efficient method to find Cofactor of Laplacian Matrix

Question

I want to find number of Spanning tree of "Simple connected graph". After some research i found it is equal to co-factor of any element of its "Laplacian Matrix".

I remove first row and first column of Laplacian matrix, and then convert it into upper triangular matrix to find its Determinant. But this approach takes $O(n^3)$ time for graph with $n$ nodes.

Since it is Laplacian Matrix i think there must exist some better approach with less complexity.

What if Laplacian matrix is sparse or only few elements are $0$ in it. Can these cases will solve much more efficiently.

Please Suggest some link, theorem or solution to efficiently solve this problem.