On the Spectra of Threshold Hypergraphs

TL;DR

This paper investigates the Laplacian spectra of threshold hypergraphs, providing explicit spectral formulas from their string representations and degree sequences, and demonstrating their $r$-integrality property.

Contribution

It introduces a method to compute the Laplacian spectrum of threshold hypergraphs from their string representation and degree sequences, revealing their $r$-integrality.

Findings

Laplacian spectra of threshold hypergraphs are $r$-integral.

Spectra can be derived from string representations and Ferrer's diagrams.

A new class of hypergraphs with $r$-integral spectra is constructed.

Abstract

Starting with an isolated vertex, here we construct a threshold hypergraph by repeatedly adding an isolated vertex or a $k$-dominating vertex set. We represent a threshold hypergraph by a string of non-negative integers and find the Laplacian spectrum of threshold hypergraphs from their string representation. We also compute the complete Laplacian spectrum of certain threshold hypergraphs from the Ferrer's diagram of their degree sequences. We show that the Laplacian spectra of threshold hypergraphs are $r$-integral, i.e., integral multiple of $r$, for some $r\in \mathbb{Q}$. We also construct another class of hypergraphs whose Laplacian spectra are $r$-integral.