Hello,

I have two questions that have been bugging me recently. The first is about the number of spanning forests in a graph and the second is about enumerating these with edge labels.

Q1: I am aware of Kirchhoff's Matrix-Tree theorem regarding the number of spanning trees in a graph. I was wondering if there is a generalization to this theorem that counts the number of spanning k-forests in a graph. What I am mostly interested in is this: is there a method of finding the number of k-forests in a graph by taking a determinant of some matrix?

Q2: Suppose you label each edge as $e_{i,j}$ meaning that you are taking the undirected edge from $v_i$ to $v_j$ in the graph. Then in the Laplacian matrix if you plug in the sum of $e_{i,j}$'s instead of $\deg(v_i)$ and $-e_{i,j}$ instead of -1 when that edge connects vertices $i$ and $j$, you get the combinatorial Laplacian. Taking the determinant of a minor of this matrix gives the Kirchhoff polynomial which is an enumeration of the spanning trees of the graph, where each monomial contains the variables for all the edges in the given tree. My question is whether we can generalize this to spanning forests.