spectral-graph-theory #

Shows a small graph whose adjacency (A) and degree (D) matrices combine into the Laplacian L = D − A. A pulsing “bridge” edge smoothly turns a disconnected graph into a connected one; the Laplacian eigenvalue bars update in real time to demonstrate that the number of near-zero eigenvalues matches the number of connected components, and that the second-smallest eigenvalue (algebraic connectivity λ1) becomes positive exactly when the graph is connected.

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practical uses #

01.Detecting connected components and connectivity from matrix spectra
02.Spectral clustering / graph partitioning via small Laplacian eigenvalues and eigenvectors
03.Analyzing diffusion, random walks, and network robustness using Laplacian properties

technical notes #

Uses a small (6×6) Laplacian and a lightweight Jacobi iteration to approximate eigenvalues each frame. The bridge edge is treated as a weighted adjacency entry so λ1 lifts smoothly during the animation. All drawing is grid-snapped for a retro blocky look with green-on-black styling.

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