# Seminar: February 16, 2009

## Combinatorial Laplacians

The graph Laplacian is a well-known and well-studied matrix associated to a graph. The eigenvalues, for example, have been used for embeddings, clustering and coloring.

The combinatorial Laplacian is a higher dimensional analogue for more general cell complexes. The combinatorial Laplacian first appeared as a discrete version of the usual Laplacian on differential forms for a Riemannian manifold and was later utilized for efficient computations of Betti numbers. These results gave rise to a number of questions concerning the Laplacian spectra and its combinatorial significance.

I will give a brief survey of this operator and discuss recent work on cellular matrix tree theorems which, analogous to the graphical case, enumerate spanning trees of a complex using the combinatorial Laplacian.