Randomly twisted hypercubes -- between structure and randomness

TL;DR

This paper investigates twisted hypercubes, a generalization of hypercubes formed by connecting graph instances with random matchings, revealing their structural properties and potential for efficient routing.

Contribution

It extends the study of twisted hypercubes to dependent and identical instances, analyzing their spectral, expansion, and automorphism properties, and highlighting their routing advantages.

Findings

Share properties with random regular graphs, such as small diameter and large expansion

Exhibit a semicircle law for eigenvalues

Allow for short routing schemes

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.