Eigenvalue bounds for combinatorial Laplacians and an application to random complexes

TL;DR

This paper derives new eigenvalue bounds for combinatorial Laplacians of simplicial complexes, linking spectral properties to topological features and applying these results to random graph complexes.

Contribution

It introduces elementary matrix-theoretic bounds for Laplacian eigenvalues, extending previous results to general complexes and subcomplex comparisons.

Findings

Lower bounds for eigenvalues in terms of graph spectra

Upper bounds on cohomology group dimensions

Refined conditions for cohomology vanishing in random complexes

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 Erd\H{o}s--R\'{e}nyi random graph.