Eigenvalue bounds for combinatorial Laplacians and an application to random complexes

TL;DR

The paper addresses eigenvalue bounds for the $k$-th combinatorial Laplacian $L_k(X)$ of a simplicial complex $X$ and links these bounds to cohomology vanishing via the simplicial Hodge theorem. It develops an elementary matrix-method framework, expressing $L_k(X)$ as $Q-P$ and employing additive-compound matrix techniques to relate $\lambda_i(L_k(X))$ to the spectrum of $L(G_X)+J$ with a discrepancy term involving $\sigma[j]$, thereby extending bounds from flag complexes to general complexes. The main contributions include explicit lower bounds for $\lambda_i(L_k(X))$, a subcomplex eigenvalue comparison, and a bound on $\dim \tilde{H}^k(X;\mathbb{R})$, along with refined cohomology-vanishing results for random neighborhood complexes of Erdős–Rényi graphs. As an application, the authors derive strengthened connectivity and cohomology-vanishing thresholds for $\mathcal{N}[G(n,p)]$, showing that under the condition $\binom{n}{k+2}(1-p^{k+2})^{n-k-2}=o(n^s)$ the complex is a.a. $(k-s)$-connected and has $\tilde{H}^{k-s+1}=0$, improving prior work by Kahle and by Shukla–Yogeshwaran. The work offers new tools for probabilistic topology and clarifies how spectral information of the underlying graph controls high-dimensional topology in random settings.

Abstract

This paper establishes new eigenvalue bounds for combinatorial Laplacians of simplicial complexes, extending previous results for flag complexes by Lew (2024) and general complexes by Shukla and Yogeshwaran (2020). Using elementary matrix-theoretic methods, we derive lower bounds for the eigenvalues of the combinatorial Laplacian in terms of the graph Laplacian spectrum and combinatorial parameters that measure the deviation from a flag complex. As a consequence, we obtain upper bounds on the dimension of cohomology groups. We also generalize an eigenvalue comparison inequality between a simplicial complex and its subcomplexes to arbitrary eigenvalues. As an application of the dimension bounds, we refine a result by Kahle (2007) on the vanishing of cohomology and connectivity in the neighborhood complex of the Erdős--Rényi random graph.

Theorems & Definitions (21)

• Theorem 1.1: Simplicial Hodge theorem, Eck44
• Theorem 1.2: Lew24
• Theorem 1.3: SY20
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6: SY20
• Theorem 1.7
• Theorem 1.8: Kah07
• Theorem 1.9
• Lemma 2.1: DR02