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

Abstract

Let $\mathcal{T}(n,k)$ be the set of the $k$-uniform supertrees with $n$ vertices and $m$ edges, where $k\ge 3$, $n\ge 5$ 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 $0\le \alpha <1$, $k\ge 3$ and $m\ge 10$. This conjecture was partially solved for $1-\frac{1}{m-2}\le \alpha <1$ and $m\ge 10$ by Wang et al.\ (2022). When $0\le \alpha <1-\frac{1}{m-2}$ and $m\ge 10$, 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 $0\le \alpha <1$ and $m\ge 13$.

21 pages,1 figure