Degree Majorization and Laplacian Eigenvalue Sums for Simplicial Complexes

2026/07/23 by Huan-Zhi Zhang, Yi-Min Song, Yi-Zheng Fan
#math.CO

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Abstract

Let K be an r-dimensional simplicial complex. We prove that the spectrum of its (r - 1)-dimensional up-Laplacian is majorized by the conjugate degree sequence of its (r - 1)-dimensional faces: \mathbfλ_r-1(K) \preccurlyeq \mathbf d_r-1^\top(K). We also establish a Brouwer-type inequality: for every integer ℓ ≥ 1, ∑_{i = 1}^ℓλ_r-1,i(K) ≤ (r + 1)/(2)f_r(K) + \fracf_{r - 2}(K)r \binomℓ + 12, where λ_r-1,i(K) denotes the i-th largest eigenvalue in the spectrum \mathbfλ_r-1(K), and f_t(K) denotes the number of t-dimensional faces of K. These results provide higher-dimensional analogs of the Grone-Merris-Bai theorem and the Brouwer-Kothari-Tudose theorem and recover the corresponding graph results when r=1. We show that the Duval-Reiner conjecture on the majorization by the conjugate degree sequence of vertices fails in every dimension r ≥ 2. More precisely, for every n ≥ r + 5, we construct a pure r-dimensional complex on n vertices that violates the conjectured inequality at the fifth partial sum.

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