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

Abstract

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