Uniform Hypergraphs with $\alpha$-Spectral Radius at Most That of a Loose Cycle

Abstract

Let $k\geq 3$ and $\alpha\in[0,1)$, and let $\lambda_k(\alpha)$ denote the $\alpha$-spectral radius of a $k$-uniform loose cycle. Lu and Man classified all connected $k$-uniform hypergraphs with adjacency spectral radius at most $\lambda_k(0)$. In this paper, we extend their classification to the $\alpha$-spectral setting. In particular, for every $k\geq 3$ and $0<\alpha<1$, we prove that $\lambda_k(\alpha)$ is the smallest limit point of the $\alpha$-spectral radii of connected $k$-uniform hypergraphs. Moreover, we give a complete comparison between the $\alpha$-spectral radius of an arbitrary finite connected $k$-uniform hypergraph $H$ and that of a loose cycle by determining the sign of $\rho_\alpha(H)-\lambda_k(\alpha)$. As a consequence, for each fixed $0<\alpha<1$, we classify all finite connected $k$-uniform hypergraphs with $\alpha$-spectral radius at most $\lambda_k(\alpha)$.