Multiplicity bounds for nonzero upper-Laplacian eigenvalues of simplicial complexes

Vinayak Gupta

paper

Multiplicity bounds for nonzero upper-Laplacian eigenvalues of simplicial complexes

arXiv:2609.29677

Abstract

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 \[ m_K(λ)\le 2θ_d(K)+p_d(K), \qquad m_K(λ)\le 2θ_d(K)+q_d(K)\quad(λ\ne d). \] 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 $m_{K} (d) \geq p_{d} (K) - q_{d} (K)$ , which is a higher-dimensional analogue of Faria's inequality. For $d = 1$ , our results reduce to the corresponding multiplicity bounds for graph Laplacians.