Building upon the work of Kalai and Adin, we extend the
concept of a spanning tree from graphs to simplicial complexes.
For all complexes K satsifying a mild technical condition, we show
that the simplicial spanning trees of K can be enumerated using its
Laplacian matrices, generalizing the matrix-tree theorem. As in
the graphic case, replacing the Laplacian with a weighted analogue
yields homological information about the simplicial spanning trees
being counted. We find a nice expression for the resulting
weighted tree enumerator of shifted complexes, by generalizing a
formula for the Laplacian eigenvalues of a shifted complex to the
weighted case.

This is joint work with Art Duval and Jeremy Martin.