Every vertex and its neighbors cancel exactly. Step off that null space and the cancellation leaks into sound.
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.
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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