Mixing time, graph expansion, the first non-rigid vibration mode, and iterative-solver convergence are taught as separate results. Each panel below measures its quantity by simulation, then compares it against λ₂ of the same Laplacian — computed once, at the top.
Node color is the Fiedler vector — the eigenvector belonging to λ₂. Watch it do all four jobs at once: it is the slow direction of the walk, the shape of the cut, the mode that vibrates, and the error the solver cannot kill.
A lazy random walk is released on one node. Total-variation distance to the stationary distribution, per step.
Every one of the 2ⁿ⁻¹ vertex subsets is tested by brute force to find the true sparsest cut. Cheeger brackets it from λ₂ alone.
Unit masses, unit springs, released from a random shape. The frequency spectrum is read off the motion — the lowest peak is the first non-rigid mode.
Richardson iteration with optimal damping, solving Lx = b. Error norm after each pass.
Every graph here is regular — same degree at every node. That is not decoration. On an irregular graph the four quantities stop being one number: mixing and expansion follow the normalized Laplacian I − D^(−1/2) A D^(−1/2), while the vibration of unit masses follows the combinatorial one, D − A. They differ by exactly the degree weighting, so on a regular graph the two coincide up to the factor d and the correspondence is exact.
There is a way to glue them back together: give each mass a weight equal to its node's degree. The generalized problem Lx = λDx is the normalized spectrum, and all four line up again on any graph. The unifying object was never the graph — it was the quadratic form xᵀLx over a choice of inner product.