Expander Graphs — Both Local and Global

Abstract

Let G = (V, E) be a finite graph. For vV 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 vV. 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 {GvvV} 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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Correspondence to Michael Chapman.

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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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Mathematics Subject Classification (2010)

  • 05C48
  • 05C81
  • 05E45