Connection Laplacian, Hodge Laplacian, and Tensor Laplacian of a Graph
summary
Argues that the Cheeger constant equals the second eigenvalue of the 1‑Laplacian and defines that 1‑Laplacian as B^T sign(Bf) so the Rayleigh/Lagrange multiplier view produces nonlinear eigenvalues; shows cohomology equals the kernel of the Hodge Laplacian and gives an orthogonal Hodge decomposition for cochains; treats hypergraphs/tensors via a cubic Rayleigh quotient so the permanent becomes an adjacency tensor, proving Perron–Frobenius–type facts; claims near‑linear solvers apply when combinatorial Ricci curvature is nonnegative, but implementations are mostly unbuilt.
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chapters (16)
- 0:00Introduction
- 3:40Examples
- 13:58Eigenvalues
- 16:17Part 0 Motivation
- 17:11Part 1 Colmology
- 18:13Part 2 Hodge Theory
- 20:07Algebraic Topology
- 21:09Spectral Graph Theory
- 28:31topology language
- 30:42case of greatest interest
- 31:28relevance
- 34:17Hypergraphs
- 36:00Tensors
- 38:26Hypermatrix
- 44:20Manifold Laplacian
- 45:46Finding Eigenvalues
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