Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the spectra of k-uniform threshold hypergraphs

Miriam Abdón, Lucas Portugal, Renata Del-Vecchio, Renata de Freitas

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.

On the spectra of k-uniform threshold 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.
Paper Structure (9 sections, 8 theorems, 30 equations)

This paper contains 9 sections, 8 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Hypergraphs and Thresholds hypergraphs
  3. Hypergraphs
  4. Thresholds hypergraphs
  5. Spectral properties of threshold hypergraphs
  6. Infinite family of $k$-uniform threshold hypergraphs with at most $3$ distinct eigenvalues
  7. Infinite family of $k$-uniform threshold hypergraphs with at most 4 distinct eigenvalues
  8. Infinite family of k-uniform threshold hypergraphs with at most 5 distinct eigenvalues
  9. Conclusions

Key Result

Proposition 1

Let $\mathcal{H}=(V,E)$ be a $k$-uniform threshold hypergraph defined by a binary sequence. Then for all $x,y\in V$, either $y\ll x$ or $x\ll y$.

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Example 1
  • Remark 2
  • Proposition 1
  • proof
  • Example 2
  • Proposition 2
  • ...and 17 more