kernel choir 2. move the points to hear new tones--
Kernel Choir
(A + I) x = 0
Every vertex and its neighbours cancel exactly. Step off that null space and the cancellation leaks into sound.
Spectrum of A
One oscillator, read two ways
Signal and IMFs
Fourier partials
Hilbert frequency in time
How it plays
Each vertex is a speaker. The kernel of A + I is the set of weightings where a vertex and its neighbours sum to zero. Its dimension is the number of independent oscillators, so a graph with nullity 3 gets three voices. Every speaker plays a mix of those voices in proportion to its weights, and a listener on a vertex hears the sum of its closed neighbourhood. For a kernel weighting that sum is zero for every wave at every instant.
Each voice is a bent wave. cos(θ + ε sin pθ) is one oscillation whose frequency swells and relaxes inside every cycle. Fourier describes it as a stack of harmonics. EMD sifts it back into a single intrinsic mode, and the Hilbert transform then reads its frequency from moment to moment. Run the transform on the raw signal and the reading can be off, because the cycles are not centred on zero. EMD centres each cycle on its own local mean first.
Leak adds the wound weights back onto the kernel weights. Listeners in a wounded vertex's neighbourhood then hear the voices, scaled by how far their neighbourhood is from cancelling.
Higher dimensions
The figures turn in every plane of their own space. A 5-cube has ten rotation planes, and what you see is one flat shadow of it, with depth shown by size and brightness. The projected positions pan the speakers, so the turning moves the sound across the stereo field. Drag the figure to turn it yourself.
A hypercube carries a −1 kernel only in odd dimension. For Qₙ the eigenvalue n−2j occurs C(n, j) times, and the eigenvectors are the Walsh characters, one for each set of j axes. At −1 that means sets of (n+1)/2 axes: three voices for the cube, ten for the 5-cube, 35 for the 7-cube. In the 5-cube tuning each voice sounds at the product of its axes' ratios (3/2, 5/4, 7/4, 11/8, 13/8), folded into an octave.
Other eigenvalues have kernels too. The listener rule generalises: it hears the neighbours minus λ times its own vertex, and for λ = −1 that is the closed neighbourhood. Pick any integer eigenvalue under Kernel of. This is how the tesseract, the 24-cell and the 600-cell can be played, although none of them has eigenvalue −1.
| Graph | Kernel at −1 |
|---|---|
| Hypercube Qₙ, n odd | C(n, (n+1)/2): 1, 3, 10, 35 for n = 1, 3, 5, 7 |
| Hypercube Qₙ, n even | none |
| Halved 6-cube, 32 vertices | 15 |
| Halved 4-, 5- and 7-cube | none |
| Hamming graph H(4,3), 81 vertices | 32 |
| Hamming graphs H(2,3), H(3,3), H(2,4) | none |
| Johnson graph J(6,3), 20 vertices | 9 |
| Johnson graphs J(6,2), J(7,3) | none |
| 24-cell and 600-cell | none. Kernels at other integer eigenvalues, for example 36 voices at −2 on the 600-cell |
Checked against the paper
The null space is computed exactly, in rational arithmetic, and each family in Mohiaddin and Sharaf (2019) was compared with it.
| Claim in the paper | Exact result |
|---|---|
| Kₙ has nullity n−1 | confirmed |
| Pₙ has nullity 1 only when n ≡ 2 (mod 3) | confirmed |
| Kₙ with t pendants: n−1−t | confirmed |
| Fan of n triangles: n | confirmed |
| Pₜ∘Kₙ, and Kₐ,ᵦ only for K₁,₁ | confirmed |
| Cₙ∘3 pendants, and Kₙ∘n pendants, are nut graphs | confirmed for every n tried (up to 7) |
| Cₙ with n divisible by 3, n ≠ 3: nullity 1 | nullity 2 (C₆, C₉, C₁₂ …) |
| Star K₁,₂ₙ with every edge subdivided: nullity n | nullity 2n−1 |
| Pₜ∘Pₙ and Pₜ∘Cₙ counts | several differ, for example P₂∘C₆ has 5 where the paper gives 2 |
| K₁,₃∘P₃ is a nut tree | order 16, nullity 2 |
Sources: G. H. Mohiaddin and K. R. Sharaf, Null spaces dimension of the eigenvalue −1 in a graph, Science Journal of University of Zakho 7(4), 2019. N. E. Huang et al., The empirical mode decomposition and the Hilbert spectrum, Proc. R. Soc. Lond. A 454, 1998.
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