Randomly Twisted Hypercubes: Between Structure and Randomness

Itai Benjamini; Yotam Dikstein; Renan Gross; Maksim Zhukovskii

doi:https://doi.org/10.1002/rsa.21267

Randomly Twisted Hypercubes: Between Structure and Randomness

Itai Benjamini, Yotam Dikstein, Renan Gross, Maksim Zhukovskii

Research output: Contribution to journal › Article › peer-review

Abstract

Twisted hypercubes are generalizations of the Boolean hypercube, obtained by iteratively connecting two instances of a graph by a uniformly random perfect matching. Dudek et al. showed that when the two instances are independent, these graphs have optimal diameter. We study twisted hypercubes in the setting where the instances can have general dependence, and also in the particular case where they are identical. We show that the resultant graph shares properties with random regular graphs, including small diameter, large vertex expansion, a semicircle law for its eigenvalues and no non-trivial automorphisms. However, in contrast to random regular graphs, twisted hypercubes allow for short routing schemes.

Original language	English
Article number	e21267
Journal	Random Structures and Algorithms
Volume	66
Issue number	1
DOIs	https://doi.org/10.1002/rsa.21267
State	Published - Jan 2025
Externally published	Yes

Keywords

random graphs
semicircle law
shortest path
vertex expansion

All Science Journal Classification (ASJC) codes

Software
General Mathematics
Computer Graphics and Computer-Aided Design
Applied Mathematics

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title = "Randomly Twisted Hypercubes: Between Structure and Randomness",

abstract = "Twisted hypercubes are generalizations of the Boolean hypercube, obtained by iteratively connecting two instances of a graph by a uniformly random perfect matching. Dudek et al. showed that when the two instances are independent, these graphs have optimal diameter. We study twisted hypercubes in the setting where the instances can have general dependence, and also in the particular case where they are identical. We show that the resultant graph shares properties with random regular graphs, including small diameter, large vertex expansion, a semicircle law for its eigenvalues and no non-trivial automorphisms. However, in contrast to random regular graphs, twisted hypercubes allow for short routing schemes.",

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AU - Gross, Renan

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AB - Twisted hypercubes are generalizations of the Boolean hypercube, obtained by iteratively connecting two instances of a graph by a uniformly random perfect matching. Dudek et al. showed that when the two instances are independent, these graphs have optimal diameter. We study twisted hypercubes in the setting where the instances can have general dependence, and also in the particular case where they are identical. We show that the resultant graph shares properties with random regular graphs, including small diameter, large vertex expansion, a semicircle law for its eigenvalues and no non-trivial automorphisms. However, in contrast to random regular graphs, twisted hypercubes allow for short routing schemes.

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Randomly Twisted Hypercubes: Between Structure and Randomness

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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