# Tag Info

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• 109k
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### Are these three different notions of a graph Laplacian?

These are usually known as the Laplacian, the normalized Laplacian and the unsigned Laplaian. All three are positive semidefinite. If the graph is regular, they all provide the same information. If ...
• 11.8k

### Why decompose a function with eigenvectors of Laplace operator?

The exponentials used in Fourier series are eigenvalues of shifts, and thus of any operator commuting with shifts, not just Laplacian. Similarly, spherical harmonics carry irreducible representations ...
• 6,044
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### is it possible to have two non-isomorphic non-regular graphs with the same adjacent spectrum and the same laplacian spectrum?

Yes, Brendan McKay showed that almost all trees have mates that are simultaneously cospectral in both adjacency and Laplacian spectra. And more. http://users.cecs.anu.edu.au/~bdm/papers/SpectralTrees....
• 11.3k

### Explicit eigenvalues of the Laplacian

Jeffrey Weeks has computed the spectra of homogeneous elliptic manifolds. For arithmetic hyperbolic manifolds, the spectrum is in principle computable in the sense that one may define a Selberg zeta ...
• 62.2k
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• 31.9k

### Graph Laplacian Operator

Your operator is a rank one perturbation of the multiplication operator $(Mf)(x) = (x/2)f(x)$, which has (purely) absolutely continuous spectrum equal to $[0,1/2]$. Since the ac spectrum is invariant ...

### On equation $\Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$ on a Riemannian manifold

This is an expansion on Igor's comment and fixes my previous mistake (see also Willie's answer). A direct computation gives  D(f) = 2f^{ab}\nabla_{(a}X_{b)} + X_a(\nabla^b\nabla_b\nabla^a-\nabla^a\...
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### On equation $\Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$ on a Riemannian manifold
The following formula is known among the experts but hard to find in the literature, so I figure I will document it here. Throughout $(M,g)$ denote an arbitrary pseudo-Riemannian manifold, and $\nabla$...