Graphs, Principal Minors, and Eigenvalue Problems

Abstract

This thesis considers four independent topics within linear algebra: determinantal point processes, extremal problems in spectral graph theory, force-directed layouts, and eigenvalue algorithms. For determinantal point processes (DPPs), we consider the classes of symmetric and signed DPPs, respectively, and in both cases connect the problem of learning the parameters of a DPP to a related matrix recovery problem. Next, we consider two conjectures in spectral graph theory regarding the spread of a graph, and resolve both. For force-directed layouts of graphs, we connect the layout of the boundary of a Tutte spring embedding to trace theorems from the theory of elliptic PDEs, and we provide a rigorous theoretical analysis of the popular Kamada-Kawai objective, proving hardness of approximation and structural results regarding optimal layouts, and providing a polynomial time randomized approximation scheme for low diameter graphs. Finally, we consider the Lanczos method for computing extremal eigenvalues of a symmetric matrix and produce new error estimates for this algorithm.