3 Planarity via Generalized Laplacians

The main topic of this course has been the unexpected connection between graph theory and spectral theory: we’ve seen the relationships between graph spectra and connectivity, bipartiteness, independent sets, colorings, trees, cuts, flows, and others. Today we see another connection with a standard concept in graph theory: planarity. In many branches of mathematics, it is frequently helpful to visualize the information and find properties and relationships by seeing them rather than by deriving them. In graph theory, it is particularly nice if the arcs and vertices of interest are able to be neatly laid out for viewing. This is captured in the concept of a planar graph. However, a graph may not necessarily have such a nice visual form. In this lecture, we introduce that notion of planarity, ask how to tell if a graph is planar, and investigate the connections between this property and spectrum of matrices associated with the graph.