Nearly planar graphs and graph minor obstructions for embedding on the spindle surface
Kwang Ju Choi
NIMS
NIMS
2014/05/19 Monday 4PM-5PM
Room 1409
Room 1409
In this talk, we define nearly planar graphs, that is, graphs that are edgeless or have an edge whose deletion results in a planar graph. We show that all but finitely many graphs that are not nearly planar and do not contain one particular graph have a well-understood structure based on large Mobius ladders. D. Archdeacon and C. Bonnington proved that a cubic obstruction for near-planarity is the same as an obstruction for embedding on the spindle surface and they gave the topological obstruction set for cubic nearly planar graphs. Now, we are searching graph minor obstructions for embedding on the spindle surface. This is a joint work with Bogdan Oporowski and Guoli Ding.
, combinatorial Laplacians 
on 
are defined by
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.