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Discrete & Computational Geometry

, Volume 59, Issue 2, pp 293–330 | Cite as

Local Spectral Expansion Approach to High Dimensional Expanders Part I: Descent of Spectral Gaps

  • Izhar Oppenheim
Article

Abstract

We introduce the notion of local spectral expansion of a simplicial complex as a possible analogue of spectral expansion defined for graphs. We then show that the condition of local spectral expansion for a complex yields various spectral gaps in both the links of the complex and the global Laplacians of the complex.

Keywords

High dimensional expanders Graph Laplacian Simplicial complexes Spectral gap 

Mathematics Subject Classification

Primary 05E45 Secondary 05A20 05C81 

Notes

Acknowledgements

The author would like to thank Matthew Kahle for many useful discussions and Alex Lubotzky for the inspiration to pursue this subject.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsBen-Gurion University of the NegevBeershebaIsrael

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