Graph Laplacian L = D − A

The Laplacian governs diffusion, random walks, synchronization, and spectral clustering. Its eigenvectors reveal community structure.

Network — click to inject heat at a node
Diffusion / Laplacian spectrum
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λ₂ Fiedler value
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components
0
diffusion steps
The graph Laplacian L = D − A (degree matrix minus adjacency) has eigenvalues 0 = λ₁ ≤ λ₂ ≤ ... ≤ λₙ. The Fiedler value λ₂ = algebraic connectivity: λ₂ = 0 iff graph is disconnected; larger λ₂ means more robustly connected. The Fiedler vector (eigenvector of λ₂) partitions the graph optimally for spectral bisection — click "Fiedler vector" to see the community structure.