## Abstract

Let G = (V, E) be a finite graph. For v ∈ V we denote by G_{v} 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 G_{v} 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 {G_{v} ∣ 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.

Original language | American English |
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Pages (from-to) | 473-509 |

Number of pages | 37 |

Journal | Combinatorica |

Volume | 40 |

Issue number | 4 |

DOIs | |

State | Published - 1 Aug 2020 |

### Bibliographical note

Publisher Copyright:© 2020, János Bolyai Mathematical Society and Springer-Verlag.