Hyper-regular graphs and high dimensional expanders

Ehud Friedgut, Yonatan Iluz

Research output: Contribution to journalArticlepeer-review

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,⋯,dn−1)-regular for 0 < dn−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,⋯,dn−1)-regular (i.e., for every 1 ≤ i ≤ n − 1, the joint neighborhood of every clique of size i is di-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 languageEnglish
Pages (from-to)233-267
Number of pages35
JournalIsrael Journal of Mathematics
Volume256
Issue number1
DOIs
StatePublished - Sep 2023

All Science Journal Classification (ASJC) codes

  • General Mathematics

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