Posts Tagged ‘국웅’

Woong Kook, Combinatorial Laplacians and high dimensional tree numbers

Saturday, March 1st, 2014
Combinatorial Laplacians and high dimensional tree numbers
Woong Kook
Seoul National University
2014/05/08 Thursday 4PM-5PM
Room 1409
Combinatorial Laplacians provide important enumeration methods in topological combinatorics. For a finite chain complex \(\{C_{i},\partial_{i}\}\), combinatorial Laplacians \(\Delta_{i}\) on \(C_{i}\) are defined by

\(\Delta_{i}=\partial_{i+1}\partial_{i+1}^{t}+\partial_{i}^{t}\partial_{i}\, .\)

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.

Woong Kook (국웅), A Combinatorial Formula for Information Flow in a Network

Thursday, March 25th, 2010
A Combinatorial Formula for Information Flow in a Network
Woong Kook (국웅)
Department of Mathematics, University of Rhode Island, Kingston, Rhode Island, U.S.A.
2010/04/09 Fri 4PM-5PM

In 1989, Stephenson and Zelen derived an elegant formula for the information Iab contained in all possible paths between two nodes a and b in a network, which is described as follows. Given a finite weighted graph G and its Laplacian matrix L, the combinatorial Green’s function \(\mathcal{G}\), of G is the inverse of L+J, where J is the all 1’s matrix. Then, it was shown that Iab=(gaa+gbb-2gab)-1, where gij is the (i,j)-th entry of \(\mathcal{G}\). In this talk, we prove an intriguing combinatorial formula for Iab:

\(I_{ab}=\kappa(G)/\kappa(G_{a\ast b})\),

where \(\kappa(G)\) is the complexity, or tree-number, of G, and Ga*b is obtained from G by identifying two vertices a and b. We will discuss several implications of this new formula, including the equivalence of Iab and the effective conductance between two nodes in electrical networks.