Spectral gap bounds for the simplicial Laplacian and an application to random complexes

Article Type

Research Article

Publication Title

Journal of Combinatorial Theory. Series A

Abstract

In this article, we derive two spectral gap bounds for the reduced Laplacian of a general simplicial complex. Our two bounds are proven by comparing a simplicial complex in two different ways with a larger complex and with the corresponding clique complex respectively. Both of these bounds generalize the result of Aharoni et al. (2005) [1] which is valid only for clique complexes. As an application, we investigate the thresholds for vanishing of cohomology of the neighborhood complex of the Erdös-Rényi random graph. We improve the upper bound derived in Kahle (2007) [12] by a logarithmic factor using our spectral gap bounds and we also improve the lower bound via finer probabilistic estimates than those in Kahle (2007) [12].

DOI

10.1016/j.jcta.2019.105134

Publication Date

1-1-2020

Comments

Open Access, Green

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