Graph Laplacian L = D - A encodes graph structure in a matrix. Eigenvalues: 0 = λ₁ ≤ λ₂ ≤ ... ≤ λₙ. λ₂ = algebraic connectivity (Fiedler, 1973). Product λ₂·...·λₙ / n = spanning tree count τ(G).
How to Use
Select graph:
- L: Laplacian matrix
- Spectrum: All eigenvalues
- λ₂: Algebraic connectivity
Fiedler Value
λ₂ = second smallest eigenvalue of L. λ₂=0 iff disconnected. For K_n: λ₂=n. For P_n: λ₂=2(1-cos(π/n))≈π²/n². Higher λ₂ = harder to disconnect. The Fiedler vector partitions vertices optimally.
Spectral Clustering
The k smallest eigenvectors of L embed vertices into R^k. Clusters in the embedding correspond to graph communities. This is the foundation of spectral clustering, widely used in machine learning and data science.
Step-by-Step Instructions
- 1Select graph.
- 2Compute L.
- 3Find spectrum.
- 4Read λ₂.
- 5Interpret connectivity.