Multiplicity bounds for nonzero upper-Laplacian eigenvalues of simplicial complexes
Abstract
Let K be a finite pure ridge-connected d-dimensional simplicial complex with at least two facets, and let mK(λ) denote the multiplicity of a positive eigenvalue λ of its upper Laplacian Ld−1up(K). We obtain upper bounds for mK(λ) in terms of the basic combinatorial parameters fd(K) and fd−1(K), the numbers of facets and ridges, pd(K), the number of pendant facets, and qd(K), the number of quasi-pendant ridges. The quantity θd(K)=d fd(K)−fd−1(K)+1 arises as the cyclomatic number of a natural reduced facet--ridge incidence graph. We prove mK(λ)≤ 2θd(K)+pd(K), mK(λ)≤ 2θd(K)+qd(K) (λ≠ d). We also characterize the complexes for which equality is attained in these bounds. Finally, for every finite pure d-dimensional simplicial complex, we prove the sharp inequality mK(d)≥ pd(K)−qd(K), which is a higher-dimensional analogue of Faria's inequality. For d=1, our results reduce to the corresponding multiplicity bounds for graph Laplacians.
Related
- Harmonic-Einstein maps and critical metrics for eigenvalues of the conformal Laplacian
- Complementary vectors of simplicial complexes
- Existence, multiplicity and asymptotic behavior of positive solutions to (p,q)-Laplacian elliptic systems
- Arithmetic classification of the Dirichlet eigenvalues for a one-dimensional nonlocal p-Laplacian problem
- Adaptive Uncertainty Quantification on Simplicial Complexes with Conformalized Simplicial Neural Networks
BibTeX
@misc{gupta2026multiplicity,
title = {{Multiplicity bounds for nonzero upper-Laplacian eigenvalues of simplicial complexes}},
author = {Vinayak Gupta},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29677},
eprint = {2609.29677},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29677},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Multiplicity bounds for nonzero upper-Laplacian eigenvalues of simplicial complexes
AU - Vinayak Gupta
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.29677
UR - https://arxiv.org/abs/2609.29677
AB - Let K be a finite pure ridge-connected d-dimensional simplicial complex with at least two facets, and let m_K(λ) denote the multiplicity of a positive eigenvalue λ of its upper Laplacian L_d−1^up(K). We obtain upper bounds for m_K(λ) in terms of the basic combinatorial parameters f_d(K) and f_d−1(K), the numbers of facets and ridges, p_d(K), the number of pendant facets, and q_d(K), the number of quasi-pendant ridges. The quantity θ_d(K)=d f_d(K)−f_d−1(K)+1 arises as the cyclomatic number of a natural reduced facet--ridge incidence graph. We prove m_K(λ)≤ 2θ_d(K)+p_d(K), m_K(λ)≤ 2θ_d(K)+q_d(K) (λ≠ d). We also characterize the complexes for which equality is attained in these bounds. Finally, for every finite pure d-dimensional simplicial complex, we prove the sharp inequality m_K(d)≥ p_d(K)−q_d(K), which is a higher-dimensional analogue of Faria's inequality. For d=1, our results reduce to the corresponding multiplicity bounds for graph Laplacians.
ER -