## The Smallest Eigenvalues of a Graph Laplacian

Given a graph $ G = (V, E) $, its adjacency matrix $ A $ contains an entry at $ A_{ij} $ if vertices $ i $ and $ j $ have an edge between them. The degree matrix $ D $ contains the degree of each vertex along its diagonal.

The graph laplacian of $ G $ is given by $ D - A $.

Several popular techniques leverage the information contained in this matrix. This blog post focuses on the two smallest eigenvalues.

First, we look at the eigenvalue 0 and its eigenvectors. A very elegant result about its multiplicity forms the foundation of spectral clustering.

Then we look at the second smallest eigenvalue and the corresponding eigenvector. A slightly more involved result (YMMV) allows us to partition the graph in question. A recent publication by John Urschel (who apparently moonlights as a sportsperson) focused on this quantity.

The insights provided here sacrifice some rigor for the sake of brevity. I find such descriptions help me study without getting bogged down too much with details. A bibliography provided at the end contains links to actual proofs.