On the spectrum and linear programming bound for hypergraphs

TL;DR

This paper develops a linear programming approach to bound the size of regular uniform hypergraphs based on their spectral properties, extending spectral graph theory to hypergraphs and identifying extremal structures.

Contribution

It introduces a linear programming method for hypergraph bounds based on eigenvalues, extending spectral graph theory results to hypergraphs and identifying extremal configurations.

Findings

Derived upper bounds on hypergraph order using spectral parameters

Identified largest hypergraphs with given second eigenvalue constraints

Established extremal structures like orthogonal arrays and Moore geometries

Abstract

The spectrum of a graph is closely related to many graph parameters. In particular, the spectral gap of a regular graph which is the difference between its valency and second eigenvalue, is widely seen an algebraic measure of connectivity and plays a key role in the theory of expander graphs. In this paper, we extend previous work done for graphs and bipartite graphs and present a linear programming method for obtaining an upper bound on the order of a regular uniform hypergraph with prescribed distinct eigenvalues. Furthermore, we obtain a general upper bound on the order of a regular uniform hypergraph whose second eigenvalue is bounded by a given value. Our results improve and extend previous work done by Feng-Li (1996) on Alon-Boppana theorems for regular hypergraphs and by Dinitz-Schapira-Shahaf (2020) on the Moore or degree-diameter problem. We also determine the largest order of an $r$-regular $u$-uniform hypergraph with second eigenvalue at most $\theta$ for several parameters $(r,u,\theta)$. In particular, orthogonal arrays give the structure of the largest hypergraphs with second eigenvalue at most $1$ for every sufficiently large $r$. Moreover, we show that a generalized Moore geometry has the largest spectral gap among all hypergraphs of that order and degree.