# Isoperimetric inequalities in simplicial complexes

@article{Parzanchevski2016IsoperimetricII, title={Isoperimetric inequalities in simplicial complexes}, author={Ori Parzanchevski and Ron Rosenthal and Ran J. Tessler}, journal={Combinatorica}, year={2016}, volume={36}, pages={195-227} }

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…

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