# Graphs with Eigenvalues of High Multiplicity

Summer 8-15-2015

## Author's School

Graduate School of Arts and Sciences

Mathematics

## Degree Name

Doctor of Philosophy (PhD)

Dissertation

## Abstract

Given a graph G we can form a matrix A_G indexed by the vertices of G and which encodes the edges of G. A_G is called the adjacency matrix of G. From the adjacency matrix we may find the eigenvalues. We would now like to know what information we may garner from the eigenvalues. It turns out quite a bit may be determined from the eigenvalues, collectively called the spectrum.

One big question is to ask whether or not a graph can be uniquely determined by its spectrum. Much research has been done in this area, and it is conjectured that almost all graphs may in fact be determined by their spectra. This is however a difficult task. In this dissertation we look at a subset of all graphs, namely those with either -1 or 0 in their spectrum with a given multiplicity. We first show that any such graph must either be primitive in a sense, or that it is obtained from a primitive graph by an elementary operation of blowing up or splitting vertices. We then show that the set of primitive graphs is finite, for a fixed multiplicity. Lastly, we analyze graphs with -1 or 0 in their spectra with multiplicities up to 4, and show many which are uniquely determined by their spectra.

English (en)

John Shareshian

## Committee Members

Renato Feres, Robert Pless, Ari Stern, David Wright

## Comments

Permanent URL: https://doi.org/10.7936/K7ZK5DXR

Available for download on Thursday, August 15, 2115

## Share

COinS

#### DOI

https://doi.org/10.7936/K7ZK5DXR