University of North Carolina at Chapel Hill Graduate School
Doi
Abstract
This thesis proposes and develops a graph spectral flow for computing nodal counts and nodal deficiencies of graph Laplacian eigenvectors. Background on Laplace eigenfunctions and their nodal domains, as well as the corresponding results in the spectral graph theory literature, is given. We also review some effective tools that adapt spectral methods for the analysis of data, in particular the use of ratio cuts for partitioning and diffusion maps for dimensionality reduction. We then define two versions of the graph spectral flow and develop properties of each, after which examples of the spectral flow on a number of graphs are provided. Finally we mention ongoing lines of research related to both theoretical and applied aspects of graphs' nodal counts.Doctor of Philosoph
