# Number and different kinds of spnning trees in a weighted graph

We know that for a unweighted graph the number $$\tau(\mathcal{G})$$ of unique spanning trees of $$\mathcal{G}$$ is $$\tau(\mathcal{G})=\det L_\mathcal{G}^{\{n-1\}},$$ where $$L_\mathcal{G}^{\{n-1\}}$$ is constructed by removing an arbitrary $$i$$th row and column of graph Laplacian $$L_\mathcal{G}$$.

Now, I am concerned the weighted graph.

For example, consider complete graph $$n=3$$ with weights $$e_1 = w_1$$, $$e_2 = w_2$$, $$e_3 = w_3$$. We know $$\det(L_g^{n-1} ) = w_1w_2+w_1w_3+w_2w_3.$$ And actually, we have these three kinds of spanning tree. (and it also tells us what kind of spanning tree.)

For example, consider tree graph $$n=3$$, i.e., $$1-2-3$$ with weight $$e_1 = w_1$$, $$e_2 = w_2$$. We know $$\det(L_g^{n-1} ) = w_1w_2.$$ And of course, we only have one spanning tree.

So conclusion from these examples:

1. number of terms $$=$$ number of spanning trees
2. each term tells the type of spanning tree

My question is that are both conclusions general for any connected, weighted, undirected and no self-loop graph?

Any related papers are also welcome. I know $$\tau(\mathcal{G})$$ is related to loop entropy of $$\mathcal{G}$$ from a few papers; however, I have not seen the version for weighted graphs. So also hope for related papers if they are closely related to my problems.

Thanks!