The matrix tree theorem for weighted graphs
A very interesting weighting is obtained by just working with directed multigraphs (dimgraphs). 7 or 8 years ago, I applied the matrix-tree theorem applied to dimgraphs in conjunction with the BEST theorem to provide a structure theory for the equilibrium thermodynamics of hybridization for oligomeric (short) DNA strands.
Briefly, the SantaLucia model of DNA hybridization takes a finite word $w$ on four letters (A, C, G, T) and associates to it various thermodynamical characteristics (e.g., free energy $\Delta G$ of hybridization) based on the sequence. Assuming the words are cyclic (which is not fair, but also not a very bad approximation practically), one has $\Delta G (w) = \sum_k \Delta G (w_kw_{k+1})$ where the index $k$ is cyclic and the 16 parameters $\Delta G(AA), \dots, \Delta G(TT)$ are given.
Assuming for convenience that the $\Delta G(\cdot,\cdot)$ are independent over $\mathbb{Q}$, it is not hard to see that $\Delta G$ projects from the space of all words of length $N$ to the space of dimgraphs on 4 vertices (again, A, C, G, T) with $N$ edges, where (e.g.) an edge from A to G corresponds to the subsequence AG. Using matrix-tree and BEST provides a functional expression for the number of words of length $N$ with a given number of AA's, AC's, ... and TT's, and thus for the desired $\Delta G$.
Going a step further, one can use generalized Euler-Maclaurin identities for evaluating sums of functions over lattice polytopes to characterize the space of all (cyclic) words of length $N$ with $\Delta G$ lying in a narrow range. This effectively completes the structure theory and shows how one can construct DNA sequences having desired thermodynamical and combinatorial properties. Among other things, I used this to design a protocol for (as David Harlan Wood put it) "simulating simulated annealing by annealing DNA".
For signed graphs we have an interesting matrix-tree theorem. A signed graph is a graph with the additional structure of edge signs (weights) being either +1 or -1. We say that a cycle in a signed graph is balanced if the product of the edges in the cycle is +1. A signed graph is balanced if all of its cycles are balanced. Otherwise, we say a signed graph is unbalanced.
We say a subgraph $H$ of a connected signed graph $G$ is an essential spanning tree of $\Gamma$ if either
1) $\Gamma$ is balanced and $H$ is a spanning tree of $G$, or
2) $\Gamma$ is unbalanced, $H$ is a spanning subgraph and every component of $H$ is a unicyclic graph with the unique cycle having sign -1.
The matrix-tree theorem for signed graphs can be stated as follows:
Let $G$ be a connected signed graph with $N$ vertices and let $b_k$ be the number of essential spanning subgraphs which contain $k$ negative cycles. Then
$$ \det(L(G))=\sum_{k=0}^n 4^k b_k.$$
For some references see:
T. Zaslavsky, Signed Graphs, Discrete Appl. Math, 4 (1982), 47-74.
S. Chaiken, A combinatorial proof of the all minors matrix tree theorem. SIAM J. Algebraic Discrete Methods, 3 (1982), 319-329.
Signed graphs are used in spin glass theory and network applications.
Considering mixed graphs, which are directed graphs that have some undirected edges, we actually get the same theory. This is immediate since the matrix definitions are identical for signed graphs and mixed graphs. This seems to escape many of the people studying matrix properties of mixed graphs however. See:
R. Bapat, J. Grossman and D. Kulkarni, Generalized Matrix Tree Theorem for Mixed Graphs, Lin. and Mult. Lin. Algebra, 46 (1999), 299-312.
One can use the matrix-tree theorem with a suitable weight in order to compute the resultant of two polynomials in one variable. This is done in this 1908 article by Arthur Lee Dixon. There is also an older 1860 article by Borchardt in a similar vein.