For any simplicial complex, the sum of the k largest upper-Laplacian eigenvalues is at most the sum of the (r+1)k largest r-degrees of (r-1)-faces, with applications to graphs and partite complexes.
The Grone Merris Conjecture and a quadratic eigenvalue problem
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We verify the Grone Merris conjecture for a class of graphs. We do this by curve sketching in the sense of first year calculus. That is, we do it by homotopy methods.
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Sums of Laplacian eigenvalues and sums of degrees
For any simplicial complex, the sum of the k largest upper-Laplacian eigenvalues is at most the sum of the (r+1)k largest r-degrees of (r-1)-faces, with applications to graphs and partite complexes.