Graph partitioning via the Fiedler vector — the eigenvector of the 2nd smallest eigenvalue of the graph Laplacian L = D − A. Sign of Fiedler vector components partitions the graph.
The Fiedler value (λ₂) measures algebraic connectivity. A barbell graph has near-zero λ₂ (weak connection). A complete graph has large λ₂ = n (strongly connected). Spectral bisection: partition by sign(v₂). Used in VLSI placement, image segmentation (Shi-Malik normalized cuts, 2000).