Conformal Rigidity and Topological Obstructions: A Spectral Perspective on Graph Genus

Exploring the relationship between conformal rigidity, spectral properties, and genus of graphs.

Setup

Choose one of Pyenv or Conda to manage your Python environment.

With Pyenv

1. Install Python 3.10.12
   • Install pyenv
   • Run pyenv install 3.10.12
   • Pyenv should automatically use this version in this directory. If not, run pyenv local 3.10.12
2. Create a virtual environment
   • Run python -m venv ./venv to create it
   • Run . venv/bin/activate when you want to activate it
     • Run deactivate when you want to deactivate it
   • Pro-tip: select the virtual environment in your IDE, e.g. in VSCode, click the Python version in the bottom left corner and select the virtual environment
3. Install dependencies with pip install -r requirements.txt

With Conda

1. Install miniconda or anaconda
   • Install miniconda
   • Or install anaconda
2. Create a virtual environment
   • Run conda create --prefix ./venv python=3.10.12 to create it
   • Run conda activate ./venv when you want to activate it
     • Run conda deactivate when you want to deactivate it
   • Pro-tip: select the virtual environment in your IDE, e.g. in VSCode, click the Python version in the bottom left corner and select the virtual environment
3. Install dependencies with pip install -r requirements.txt

Usage

Open any of the .ipynb notebooks in VS Code or Jupyter Notebook to run the code.

• conformal_rigidity.ipynb contains the code for checking whether a graph is conformally rigid.
• spectral_bound.ipynb contains delta_g plots

References

• Conformally Rigid Graphs by Steinerberger and Thomas
• Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem by Asma Hassannezhad