Sharp lower bounds on the spectral radius of uniform hypergraphs concerning degrees

TL;DR

This paper establishes sharp lower bounds on the spectral radii of adjacency and signless Laplacian tensors of uniform hypergraphs based on vertex degrees, solving a problem posed by Nikiforov and characterizing extremal hypergraphs.

Contribution

It provides new lower bounds for spectral radii of hypergraph tensors and characterizes the extremal hypergraphs that attain these bounds.

Findings

Lower bound on spectral radius $ ho(H)$ in terms of degrees

Lower bound on spectral radius $ ho( ext{Q}(H))$ in terms of degrees

Characterization of extremal hypergraphs attaining the bounds

Abstract

Let $\mathcal{A}(H)$ and $\mathcal{Q}(H)$ be the adjacency tensor and signless Laplacian tensor of an $r$-uniform hypergraph $H$. Denote by $\rho(H)$ and $\rho(\mathcal{Q}(H))$ the spectral radii of $\mathcal{A}(H)$ and $\mathcal{Q}(H)$, respectively. In this paper we present a lower bound on $\rho(H)$ in terms of vertex degrees and we characterize the extremal hypergraphs attaining the bound, which solves a problem posed by Nikiforov [V. Nikiforov, Analytic methods for uniform hypergraphs, Linear Algebra Appl. 457 (2014) 455-535]. Also, we prove a lower bound on $\rho(\mathcal{Q}(H))$ concerning degrees and give a characterization of the extremal hypergraphs attaining the bound.