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Two adventures in spectral graph theory

Resource type
Thesis type
(Thesis) M.Sc.
Date created
2023-03-03
Authors/Contributors
Abstract
In his landmark paper, How to Draw a Graph, Tutte explores the spring energy model of straight line graph drawings. Using semidefinite programming, Goemans and Williamson further explore the utility of energy in graph embeddings in their investigation of maxcut of a graph. For our first problem we take ideas from both Tutte and Goemans and Williamson. We use semidefinite programming to find minimal energy graph representations, and investigate a natural parameter we define which arises from this, ρ. For G a random regular graph, or regular graph of high girth, we show that ρ is bounded by the second largest eigenvalue of the adjacency matrix of G. For our second problem, we again find inspiration from Tutte. A fundamental family of totally unimodular matrices introduced by Tutte are network matrices. A network matrix is defined based on a graph G and a tree T on a common vertex set. The positive semidefinite matrix NNT obeys the Matrix Tree Theorem and generalizes the Laplacian of G. We investigate the spectrum of this matrix, in particular the highest eigenvalue λmax. When G = Kn we find the trees maximizing and minimizing λmax. When T is restricted to be a subgraph of G we give a rough structure theorem determining when λmax is bounded. Additionally we find a natural edge connectivity property that implies λmax is minimized by taking T to be a star. Lastly we find an alternate proof of correctness of the famous Gomory-Hu tree theorem.
Document
Extent
77 pages.
Identifier
etd22378
Copyright statement
Copyright is held by the author(s).
Permissions
This thesis may be printed or downloaded for non-commercial research and scholarly purposes.
Supervisor or Senior Supervisor
Thesis advisor: DeVos, Matt
Language
English
Member of collection
Download file Size
etd22378.pdf 12.58 MB

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