TITLE: Number of spanning forests in a graph
QUESTION [9 upvotes]: 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.

REPLY [3 votes]: I am surprised no one mentioned the work of Alan Sokal and coworkers on precisely this issue
of weighted enumeration of spanning forests which is related to the $q\rightarrow 0$ limit of the
Potts model as well as the multivariate Tutte polynomial.
A determinant expression corresponds to a Fermionic (Grassmann/Berezin) integral with a quadratic
`action' in the exponential. There is an analogue of the matrix-tree theorem for spanning forests
with a Fermionic integral with quartic action see:
http://arxiv.org/abs/cond-mat/0403271
which appeared in PRL.
Follow ups such as
http://arxiv.org/abs/0706.1509
can be found by looking at Sokal's papers on arxiv:
http://arxiv.org/find/grp_math/1/au:+sokal/0/1/0/all/0/1
Note that there was a whole semester at the Isaac Newton Institute revolving around this
topic:
http://www.newton.ac.uk/programmes/CSM/
One can even watch the videos of the talks.
The one perhaps most relevant to this question is the talk by Andrea Sportiello in the fourth
workshop.