Graph Editor
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Eigenvalue Spectrum (click bar to select eigenvector)
Eigenvector Selector
Laplacian L = D - A
Add nodes and edges to see L
Graph Properties
Spectral Graph Theory
The graph Laplacian L = D - A encodes connectivity. Its eigenvalues are always non-negative real numbers.
lambda_1 = 0 always. The multiplicity of zero eigenvalues equals the number of connected components.
The Fiedler vector (eigenvector for lambda_2) gives the optimal graph bisection. Nodes colored red vs blue are on opposite sides of the minimum cut.
Cheeger inequality: h/2 ≤ lambda_2/2 ≤ h, where h is the edge expansion. The barbell graph has small lambda_2 (bottleneck); the complete graph has large lambda_2 (highly connected).