MIT Combinatorics Seminar

A simplicial matrix tree theorem

Caroline Klivans  (University of Chicago)

Friday, April 06, 2007    4:15 pm    Room 2-136


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.