Spectral Stability of Weighted Complete Graph Laplacians under Structured Perturbations

This repository contains the LaTeX source, bibliography, reproducibility code, generated figures, and compiled PDF for the M840 dissertation:

Spectral Stability of Weighted Complete Graph Laplacians under Structured Perturbations

Student number: J0840966
Date: May 2026

Repository contents

main.tex                 LaTeX dissertation source
references.bib           BibTeX bibliography database
main.bbl                 Compiled bibliography file used for reproducible builds
main.pdf                 Compiled dissertation PDF
requirements.txt         Python dependencies for regenerating figures and tables
code/make_figures.py     Full Python script for numerical figures and table output
figures/                 Generated PDF figures used in Chapter 7
two_block_table.tex      Generated numerical table used in Chapter 7
CITATION.cff             Citation metadata for this repository
UPLOAD_CHECKLIST.md      Practical upload and verification checklist

Mathematical focus

The dissertation is a proof-led algebraic graph theory study of weighted complete graph Laplacians. It focuses on:

• algebraic connectivity;
• Rayleigh quotient methods;
• cut-capacity certificates;
• two-block weighted complete graph spectra;
• vertex augmentation;
• Schur complements;
• interlacing and perturbation estimates;
• numerical illustrations of structured versus diffuse perturbations.

The main conclusion is that complete support alone does not guarantee spectral robustness: algebraic connectivity is governed by the geometry of the edge weights, especially cross-cut capacity.

Reproducing the numerical material

Create a Python environment and install the dependencies:

python -m venv .venv
source .venv/bin/activate
pip install -r requirements.txt

Regenerate the figures and table:

python code/make_figures.py

This recreates the files in figures/ and regenerates the numerical table used in Chapter 7.

Compiling the dissertation

The repository includes main.bbl, so the document can be compiled even in environments without BibTeX:

pdflatex main.tex
pdflatex main.tex

For a full BibTeX rebuild:

pdflatex main.tex
bibtex main
pdflatex main.tex
pdflatex main.tex

Reproducibility note

The numerical examples are illustrative checks of the analytic formulae and inequalities. They are not used as substitutes for the proofs. Randomness appears only in the diffuse perturbation comparison, where the script fixes the random seed for reproducibility.

AI-use statement

AI was used for searching, LaTeX organisation, checking formatting, and reproducibility packaging.