On the spectra of k-uniform threshold hypergraphs

TL;DR

This paper introduces k-uniform threshold hypergraphs, characterizes their eigenvalues using combinatorial numbers, and provides methods to construct such hypergraphs with few distinct eigenvalues.

Contribution

It extends the concept of threshold graphs to hypergraphs and offers a spectral characterization enabling the construction of hypergraphs with few eigenvalues.

Findings

Eigenvalues characterized via combinatorial numbers

Construction methods for hypergraphs with few eigenvalues

Extension of threshold graph concepts to hypergraphs

Abstract

In this article we introduce a definition of k-uniform thresholds hypergraphs through a binary sequence, a natural extension of the classical definition for thresholds graphs. We characterize some of its eigenvalues and multiplicities by means of combinatorial numbers, derived from edge counts. An important problem addressed in Spectral Graph Theory is to find graphs with few distinct eigenvalues. Our characterization allows us to construct k-uniform threshold hypergraphs having an arbitrary number of vertices with few distinct eigenvalues.