Cospectral constructions and spectral properties of variations of the distance matrix

Abstract

A graph is a collection of objects (vertices) and connections between the objects (edges). Graphs can be associated with matrices by assigning matrix entries corresponding to the graph structure. As the graph grows large so does the matrix making it difficult to understand the graph's properties. The spectrum (multi-set of eigenvalues) of a matrix for a graph gives a snapshot of the graph structure independent of labeling. We know not all structural properties are captured by the spectrum by the existence of pairs of graphs that share a spectrum (cospectral graphs). In this dissertation, we investigate cospectrality for several graph matrices as well as discuss spectral properties of two recent matrix variants.