KAIST Discrete Math Seminar

Combinatorial Laplacians provide important enumeration methods in topological combinatorics. For a finite chain complex , combinatorial Laplacians on are defined by

We will review applications of in computing the tree numbers for graphs and in solving discrete Laplace equations for networks. In general, the boundary operators are used to define high-dimensional trees as a generalization of spanning trees for graphs. We will demonstrate an intriguing relation between high-dimensional tree numbers and for acyclic complexes, based on combinatorial Hodge theory. As an application, a formula for the top-dimensional tree-number of matroid complexes will be derived. If time permits, an important role of combinatorial Laplacians in topological data analysis (TDA) will be briefly discussed.