Seoul National University

Room 1409

We will review applications of \(\Delta_{0}\) in computing the tree numbers for graphs and in solving discrete Laplace equations for networks. In general, the boundary operators \(\partial_{i}\) 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 \(\det\Delta_{i}\) 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.