Ordering of the $k$-uniform unicyclic hypergraphs by $\alpha$-spectral radii

Abstract

Let $\mathbb{U}^k_n$ be the set of the $k$-uniform unicyclic hypergraphs with $n$ vertices and $m$ edges, where $k\geq 3$ and $m=\frac{n}{k-1}\geq 2$.
A new transformation and a new $\rho_{\alpha}$-normal labeling method for comparing the $\alpha$-spectral radii of the $k$-uniform hypergraphs are developed. By the methods proposed here, we obtain the hypergraphs with the first to the fourth largest $\alpha$-spectral radii among $\mathbb{U}^k_n$, where $k\geq 3$ and $m=\frac{n}{k-1}\geq 13$.