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Expander Graphs — Both Local and Global

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Abstract

Let G = (V, E) be a finite graph. For v ∈ V we denote by Gv the subgraph of G that is induced by v’s neighbor set. We say that G is (a,b)-regular for a>b> 0 integers, if G is a-regular and Gv is b-regular for every v ∈ V. Recent advances in PCP theory call for the construction of infinitely many (a,b)-regular expander graphs G that are expanders also locally. Namely, all the graphs {Gv ∣ v ∈ V} should be expanders as well. While random regular graphs are expanders with high probability, they almost surely fail to expand locally. Here we construct two families of (a,b)-regular graphs that expand both locally and globally. We also analyze the possible local and global spectral gaps of (a,b)-regular graphs. In addition, we examine our constructions vis-a-vis properties which are considered characteristic of high-dimensional expanders.

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Acknowledgements

We thank Irit Dinur, whose questions prompted us to work on these problems. Our conversations with Irit took place during a special year on high dimensional combinatorics held in the Israel Institute of Advanced Studies. We thank the IIAS and the organizers of this beautiful year.

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Authors and Affiliations

  1. Einstein institute of Mathematics, The Hebrew University, Jerusalem, 91904, Israel

    Michael Chapman

  2. Department of Computer Science, The Hebrew University, Jerusalem, 91904, Israel

    Nati Linial

  3. Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY, 10012, USA

    Yuval Peled

Authors
  1. Michael Chapman
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  2. Nati Linial
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  3. Yuval Peled
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Corresponding author

Correspondence to Michael Chapman.

Additional information

The second author is supported by ERC grant 339096 “High dimensional combinatorics”.

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Chapman, M., Linial, N. & Peled, Y. Expander Graphs — Both Local and Global. Combinatorica 40, 473–509 (2020). https://doi.org/10.1007/s00493-019-4127-8

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  • DOI: https://doi.org/10.1007/s00493-019-4127-8

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