## Abstract

Let G = (V, E) be a finite graph. For d _{0} > 0 we say that G is d _{0}-regular, if every v ∈ V has degree d _{0}. We say that G is (d _{0}, d _{1})-regular, for 0 < d _{1} < d _{0}, if G is d _{0} regular and for every v ∈ V, the subgraph induced on v’s neighbors is d _{1}-regular. Similarly, G is (d _{0}, d _{1},⋯,d_{n−1})-regular for 0 < d_{n−1} < ⋯ < d _{1} < d _{0}, if G is d _{0} regular and for every v ∈ V, the subgraph induced on v’s neighbors is (d _{1},⋯,d_{n−1})-regular (i.e., for every 1 ≤ i ≤ n − 1, the joint neighborhood of every clique of size i is d_{i}-regular); in that case, we say that G is an n-dimensional hyper-regular graph (HRG). Here we define a new kind of graph product, through which we build examples of infinite families of n-dimensional HRG such that the joint neighborhood of every clique of size at most n − 1 is connected. In particular, relying on the work of Kaufman and Oppenheim, our product yields an infinite family of n-dimensional HRG for arbitrarily large n with good expansion properties. This answers a question of Dinur regarding the existence of such objects.

Original language | English |
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Pages (from-to) | 233-267 |

Number of pages | 35 |

Journal | Israel Journal of Mathematics |

Volume | 256 |

Issue number | 1 |

DOIs | |

State | Published - Sep 2023 |

## All Science Journal Classification (ASJC) codes

- General Mathematics