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The Laplacian matrix is the representation of a graph in matrix form.

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Explicit eigenvalues of the Laplacian

Edit: Also, for $M$ a compact semisimple Lie group it is well known (due apparently to Freudenthal (1954))1 that the Laplacian (= Casimir) acts on the $\lambda$-subspace in the Peter-Weyl decomposition …
Francois Ziegler's user avatar