On Some Geometry of Graphs

Abstract

In this thesis we study the intrinsic geometry of graphs via the constants that appear in discretized partial differential equations associated to those graphs. By studying the behavior of a discretized version of Bochner\u27s inequality for smooth manifolds at the cone point for a cone over the set of vertices of a graph, a lower bound for the internal energy of the underlying graph is obtained. This gives a new lower bound for the size of the first non-trivial eigenvalue of the graph Laplacian in terms of the curvature constant that appears at the cone point and the size of the vertex set for the underlying graph. For the sake of completeness, the main analysis for cones is actually done for cones over subsets of the vertex set. We follow this analysis up by studying which types of functions can achieve equality in the discrete Bochner inequality, in particular functions which yield the largest possible curvature bound at the cone point come with a dynamical definition. We are then able to classify the space of all such functions via spectral graph theory and recast the regularity of a graph in terms of the dimension of this space of functions