A complete solution of the $k$-uniform supertrees with the eight largest $\alpha$-spectral radii

TL;DR

This paper fully proves a conjecture about the eight largest $ ho_ ext{spectral}$ radii of certain $k$-uniform supertrees using a new labeling method, advancing spectral hypergraph theory.

Contribution

The paper introduces a new $ ho_ ext{spectral}$-normal labeling method and applies it to completely prove a conjecture for $k$-uniform supertrees with specific parameters.

Findings

The conjecture holds for $0 \\leq \\alpha < 1$ and $m \\geq 13$.

The new labeling method effectively computes $ ho_ ext{spectral}$ for hypergraphs.

Complete proof of the conjecture for specified parameters.

Abstract

Let $\mathcal T (n, k)$ be the set of the $k$-uniform supertrees with $n$ vertices and $m$ edges, where $k\geq 3$, $n\geq 5$ and $m=\frac{n-1}{k-1}$. % Let $m$ be the number of the edges of the supertrees in $\mathcal T (n, k)$, where $m=\frac{n-1}{k-1}$. A conjecture concerning the supertrees with the fourth through the eighth largest $\alpha$-spectral radii in $\mathcal T (n, k)$ was proposed by You et al.\ (2020), where $0 \leq \alpha<1$, $k\geq 3$ and $m \geq 10$. This conjecture was partially solved for $1-\frac{1}{m-2}\leq \alpha <1$ and $m\geq 10$ by Wang et al.\ (2022). When $0\leq \alpha <1-\frac{1}{m-2}$ and $m \geq 10$, whether this conjecture is correct or not remains a problem to be further solved. By using a new $\rho_{\alpha}$-normal labeling method proposed in this article for computing the $\alpha$-spectral radius of the $k$-uniform hypergraphs, we completely prove that this conjecture is right for $0\leq\alpha<1$ and $m\geq 13$.