Hypergraphs with Spectral Radius at most $(r-1)!\sqrt[r]{2+\sqrt{5}}$

TL;DR

This paper classifies r-uniform hypergraphs with spectral radius at most (r-1)! times the r-th root of (2+√5), showing they must have a quipus-structure, extending previous spectral radius classifications.

Contribution

It extends spectral radius classifications from graphs to hypergraphs, identifying the structure of hypergraphs with spectral radius slightly above a known threshold.

Findings

Hypergraphs with spectral radius ≤ (r-1)!√[r]{2+√5} have a quipus-structure.

Generalizes previous spectral radius classifications from graphs to hypergraphs.

Provides structural characterization for hypergraphs near spectral radius bounds.

Abstract

In our previous paper, we classified all $r$-uniform hypergraphs with spectral radius at most $(r-1)!\sqrt[r]{4}$, which directly generalizes Smith's theorem for the graph case $r=2$. It is nature to ask the structures of the hypergraphs with spectral radius slightly beyond $(r-1)!\sqrt[r]{4}$. For $r=2$, the graphs with spectral radius at most $\sqrt{2+\sqrt{5}}$ are classified by [{\em Brouwer-Neumaier, Linear Algebra Appl., 1989}]. Here we consider the $r$-uniform hypergraphs $H$ with spectral radius at most $(r-1)!\sqrt[r]{2+\sqrt{5}}$. We show that $H$ must have a quipus-structure, which is similar to the graphs with spectral radius at most $\frac{3}{2}\sqrt{2}$ [{\em Woo-Neumaier, Graphs Combin., 2007}].