Isoperimetric Inequalities in Simplicial Complexes

Abstract

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In particular, we present a Cheeger-type inequality, and a high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach, we obtain a connection between spectral properties of complexes and Gromov's notion of geometric overlap. Using the work of Gunder and Wagner, we give an estimate for the combinatorial expansion and geometric overlap of random Linial-Meshulam complexes.

Cited by in corpus (13)

• What are higher-order networks?
• Topological Signal Processing over Simplicial Complexes
• Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian
• Toward a Spectral Theory of Cellular Sheaves
• Mixing in high-dimensional expanders
• On the chromatic number of a simplicial complex
• Complexity of Supersymmetric Systems and the Cohomology Problem
• A Notion of Harmonic Clustering in Simplicial Complexes
• Graphical Designs and Extremal Combinatorics
• Spectrum and combinatorics of two-dimensional Ramanujan complexes
• Free flags over local rings and powering of high dimensional expanders
• Hypergraph expanders of all uniformities from Cayley graphs
• Spectral Gap of The Discrete Laplacian On Triangulations