L = D − A eigenvalues, Fiedler vector, spectral gap — partition paths, cycles, trees, complete
The graph Laplacian L = D − A has eigenvalues 0 = λ₁ ≤ λ₂ ≤ … ≤ λ_n. The number of zero eigenvalues equals the number of connected components (Kirchhoff). The Fiedler value λ₂ measures algebraic connectivity — it is zero iff the graph is disconnected. The Fiedler vector (eigenvector of λ₂) provides the optimal spectral graph bisection. For paths λ_k = 2−2cos(πk/n), for complete graphs λ_k = n for k≥2.