Hypergraph expanders from Cayley graphs

TL;DR

This paper introduces a simple, randomized method for constructing sparse 3-uniform hypergraphs with strong expansion properties using Cayley graphs over rac{Z}_2^t, leading to rapid mixing and uniform edge distribution.

Contribution

It presents a novel construction of hypergraph expanders based on Cayley graphs, with polylogarithmic vertex degree and multiple expansion properties.

Findings

Hypergraphs exhibit strong expansion properties derived from Cayley graphs.

Constructed hypergraphs have polylogarithmic vertex degree.

Hypergraphs demonstrate rapid mixing and uniform edge distribution.

Abstract

We present a simple mechanism, which can be randomised, for constructing sparse $3$-uniform hypergraphs with strong expansion properties. These hypergraphs are constructed using Cayley graphs over $\mathbb{Z}_2^t$ and have vertex degree which is polylogarithmic in the number of vertices. Their expansion properties, which are derived from the underlying Cayley graphs, include analogues of vertex and edge expansion in graphs, rapid mixing of the random walk on the edges of the skeleton graph, uniform distribution of edges on large vertex subsets and the geometric overlap property.