The spectrum of the (r-1)-up-Laplacian is majorized by the conjugate degree sequence of (r-1)-faces, and the Duval-Reiner conjecture fails for all r≥2.
Eckmann, Harmonische Funktionen und Randwertaufgaben in einem Komplex,Comment
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Degree Majorization and Laplacian Eigenvalue Sums for Simplicial Complexes
The spectrum of the (r-1)-up-Laplacian is majorized by the conjugate degree sequence of (r-1)-faces, and the Duval-Reiner conjecture fails for all r≥2.