Eigenvalues and Linear Quasirandom Hypergraphs

TL;DR

This paper establishes a spectral characterization of quasirandom properties in k-uniform hypergraphs, linking combinatorial quasirandomness with eigenvalues, and extends classical graph spectral results to hypergraphs.

Contribution

It introduces a spectral framework for hypergraph quasirandomness, connecting previous notions and answering an open question about spectral gaps.

Findings

Hypergraph quasirandom properties are characterized by spectral gaps.

Defines largest and second largest eigenvalues for hypergraph properties.

Extends spectral results from graphs to hypergraphs.

Abstract

Let p(k) denote the partition function of k. For each k >= 2, we describe a list of p(k)-1 quasirandom properties that a k-uniform hypergraph can have. Our work connects previous notions on linear hypergraph quasirandomness of Kohayakawa-R\"odl-Skokan and Conlon-H\`{a}n-Person-Schacht and the spectral approach of Friedman-Wigderson. For each of the quasirandom properties that are described, we define a largest and second largest eigenvalue. We show that a hypergraph satisfies these quasirandom properties if and only if it has a large spectral gap. This answers a question of Conlon-H\`{a}n-Person-Schacht. Our work can be viewed as a partial extension to hypergraphs of the seminal spectral results of Chung-Graham-Wilson for graphs.