Applications of Geometric and Spectral Methods in Graph Theory

Networks, or graphs, are useful for studying many things in today’s world. Graphs can be used to represent connections on social media, transportation networks, or even the internet. Because of this, it’s helpful to study graphs and learn what we can say about the structure of a given graph or what properties it might have. This dissertation focuses on the use of the probabilistic method and spectral graph theory to understand the geometric structure of graphs and find structures in graphs. We will also discuss graph curvature and how curvature lower bounds can be used to give us information about properties of graphs. A rainbow spanning tree in an edge-colored graph is a spanning tree in which each edge is a different color. Carraher, Hartke, and Horn showed that for n and C large enough, if G is an edge-colored copy of Kn in which each color class has size at most n/2, then G has at least [n/(C log n)] edge-disjoint rainbow spanning trees. Here we show that spectral graph theory can be used to prove that if G is any edge-colored graph with n vertices in which each color appears on at most δλ1/2 edges, where δ ≥ C log n for n and C sufficiently large and λ1 is the second-smallest eigenvalue of the normalized Laplacian matrix of G, then G contains at least [δλ1/ C log n] edge-disjoint rainbow spanning trees. We show how curvature lower bounds can be used in the context of understanding (personalized) PageRank, which was developed by Brin and Page. PageRank ranks the importance of webpages near a seed webpage, and we are interested in how this importance diffuses. We do this by using a notion of graph curvature introduced by Bauer, Horn, Lin, Lippner, Mangoubi, and Yau

A Spacial Gradient Estimate for Solutions to the Heat Equation on Graphs

The study of positive solutions of the heat equation $\frac{\partial}{\partial \alpha} u = \Delta u$, on both manifolds and graphs, gives an analytic way of extracting geometric information about the object. In the manifold case, one of the most effective ways of studying how solutions to the heat equation evolve is to derive a local “gradient estimate” of heat change, using curvature lower bounds. Recently, notions of curvature for graphs have been developed which enable proving similar estimates for graphs. In this article, we derive a gradient estimate for positive heat solutions that considers only how heat varies in space and the time derivative. This result, due in the manifold case to Hamilton, applies to both finite graphs and infinite graphs of bounded degree. As a corollary, a heat comparison theorem is also developed. This in turn yields results about the mixing of the continuous time random walk on graphs