Isoperimetric inequalities in simplicial complexes

Ori Parzanchevski*, Ron Rosenthal, Ran J. Tessler

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

36 Scopus citations


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.

Original languageAmerican English
Pages (from-to)195-227
Number of pages33
Issue number2
StatePublished - 1 Apr 2016

Bibliographical note

Publisher Copyright:
© 2015, János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.


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