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Isoperimetric inequalities in simplicial complexes

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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 Gundert and Wagner, we give an estimate for the combinatorial expansion and geometric overlap of random Linial-Meshulam complexes.

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Author notes

  1. Ori Parzanchevski

    Present address: School of Mathematics, Institute for Advanced Study, Princeton, NJ, 08540, USA

  2. Ron Rosenthal

    Present address: Departement Mathematik, ETH Zürich, 8092, Zürich, Switzerland

Authors and Affiliations

  1. Einstein Institute of Mathematics, Hebrew University, Jerusalem, 91904, Israel

    Ori Parzanchevski, Ron Rosenthal & Ran J. Tessler

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  1. Ori Parzanchevski
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  2. Ron Rosenthal
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  3. Ran J. Tessler
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Correspondence to Ori Parzanchevski.

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Parzanchevski, O., Rosenthal, R. & Tessler, R.J. Isoperimetric inequalities in simplicial complexes. Combinatorica 36, 195–227 (2016). https://doi.org/10.1007/s00493-014-3002-x

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  • DOI: https://doi.org/10.1007/s00493-014-3002-x

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