TITLE: The matrix tree theorem for weighted graphs
QUESTION [21 upvotes]: I am interested in the general form of the Kirchoff Matrix Tree Theorem for weighted graphs, and in particular what interesting weightings one can choose.
Let $G = (V,E, \omega)$ be a weighted graph where $\omega: E \rightarrow K$, for a given field $K$; I assume that the graph is without loops.
For any spanning tree $T \subseteq G$ the weight of the tree is given to be,
$$m(T) = \prod_{e \in T}\omega(e)$$
and the tree polynomial (or statistical sum) of the graph is given to be the sum over all spanning trees in G,
$$P(G) = \sum_{T \subseteq G}m(T)$$
The combinatorial laplacian of the graph G is given by $L_G$, where:
$$L_G = \begin{pmatrix} \sum_{k = 1}^n\omega(e_{1k}) & -\omega(e_{12}) & \cdots & -\omega(e_{1n}) \\\ -\omega(e_{12}) & \sum_{k = 1}^n\omega(e_{2k}) & \cdots & -\omega(e_{2n})\\\
\vdots & \vdots & \ddots & \vdots \\\
-\omega(e_{1n}) & -\omega(e_{2n}) & \cdots & \sum_{k = 1}^n\omega(e_{nk}) \end{pmatrix} $$
where $e_{ik}$ is the edge between vertices $i$ and $k$, if no edge exists then the entry is 0 (this is the same as considering the complete graph on n vertices with an extended weighting function that gives weight 0 to any edge not in G). The matrix tree theorem then says that the tree polynomial is equal to the absolute value of any cofactor of the matrix. That is,
$$P(G) = \det(L_G(s|t))$$
where $A(s|t)$ denotes the matrix obtained by deleting row $s$ and column $t$ from a matrix $A$.
By choosing different weightings one would expect to find interesting properties of a graph G. Two simple applications are to give the weighting of all 1's. Then the theorem allows us to count the number of spanning trees with ease (this yields the standard statement of the Matrix Tree Theorem for graphs). Alternatively, by giving every edge a distinct formal symbol as its label then by computing the relevant determinant, the sum obtained can be read as a list of all the spanning trees.
My question is whether there are other interesting weightings that can be used to derive other interesting properties of graphs, or for applied problems.

REPLY [2 votes]: Another interesting weighting comes from stochastic matrices, and was used by Tuncel and me to find a completely new invariant for Markov shifts (Douglas Lind and Selim Tuncel, A Spanning Tree Invariant for Markov Shifts, Codes, Systems, and Graphical Models, IMA Volumes in Mathematics and its Applications 123 (2001), 487--497).
Let $P=[p_{ij}]$ be an $r \times r$ stochastic matrix, so that $p_{ij}\ge0$ and row sums are all 1. Let $G_P$ denote the directed graph with $r$ vertices and with one edge from $i$ to $j$ if and only if $p_{ij}>0$ and no other edges. Assume that $G_P$ is irreducible. The shift of finite type $X_P$ determined by $G_P$ is the space of all bi-infinite trips on $G_P$, which is compact. Also, $P$ determines a unique probability measure $\mu_P$ (called the Parry measure) that is invariant under the left shift $\sigma_P $ on $X_P$. These measure-preserving transformations are key examples in ergodic theory and have been extensively studied.
Given $P$, assign an edge in $G_P$ from $i$ to $j$ the weight $p_{ij}>0$. If $T$ is a spanning tree in $G_P$, define the weight of $T$ to be the product of the weights of its edges. Finally, define $\tau(P)$ to be the sum of the weights of all spanning trees in $G_P$.
If $Q$ is another such matrix, of possibly a different size, we say that the dynamical systems $(X_P,\sigma_P,\mu_P)$ and $(X_Q,\sigma_Q,\mu_Q)$ are block isomorphic if there is a shift commuting measure-preserving homeomorphism between them. The main result is that $\tau$ is an invariant of block isomorphism.
We initially observed this experimentally using Mathematica. The $\tau$ invariant is defined using a simple finite sum, and differs from more standard dynamical invariants like entropy that use asymptotic limiting behavior.