https://github.com/prof-faustus/m840
Spectral stability analysis of weighted complete graph Laplacians under structured perturbations, focusing on algebraic connectivity, Rayleigh quotient bounds, cut-capacity bottlenecks, vertex augmentation, Schur complements, and reproducible Python figures.
https://github.com/prof-faustus/m840
Last synced: about 2 months ago
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Spectral stability analysis of weighted complete graph Laplacians under structured perturbations, focusing on algebraic connectivity, Rayleigh quotient bounds, cut-capacity bottlenecks, vertex augmentation, Schur complements, and reproducible Python figures.
- Host: GitHub
- URL: https://github.com/prof-faustus/m840
- Owner: prof-faustus
- License: mit
- Created: 2026-05-27T00:02:37.000Z (about 2 months ago)
- Default Branch: main
- Last Pushed: 2026-05-27T00:08:54.000Z (about 2 months ago)
- Last Synced: 2026-05-27T02:12:08.063Z (about 2 months ago)
- Language: TeX
- Size: 943 KB
- Stars: 0
- Watchers: 0
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
- Citation: CITATION.cff
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README
# 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
```text
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:
```bash
python -m venv .venv
source .venv/bin/activate
pip install -r requirements.txt
```
Regenerate the figures and table:
```bash
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:
```bash
pdflatex main.tex
pdflatex main.tex
```
For a full BibTeX rebuild:
```bash
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.