Lower bounds for the $\mathcal{A}_{\alpha}$-spectral radius of uniform hypergraphs

TL;DR

This paper establishes new lower bounds for the $\

Contribution

It introduces novel lower bounds for the $\

Findings

Lower bounds for the spectral radius difference from average degree.

Lower bounds based on maximum and minimum degrees.

Results applicable to connected $k$-uniform hypergraphs.

Abstract

For $0\leq \alpha < 1$, the $\mathcal{A}_{\alpha}$-spectral radius of a $k$-uniform hypergraph $G$ is defined to be the spectral radius of the tensor $\mathcal{A}_{\alpha}(G):=\alpha \mathcal{D}(G)+(1-\alpha) \mathcal{A}(G)$, where $\mathcal{D}(G)$ and $A(G)$ are diagonal and the adjacency tensors of $G$ respectively. This paper presents several lower bounds for the difference between the $\mathcal{A}_{\alpha}$-spectral radius and an average degree $\frac{km}{n}$ for a connected $k$-uniform hypergraph with $n$ vertices and $m$ edges, which may be considered as the measures of irregularity of $G$. Moreover, two lower bounds on the $\mathcal{A}_{\alpha}$-spectral radius are obtained in terms of the maximum and minimum degrees of a hypergraph.