On the $\alpha$-spectral radius of uniform hypergraphs

TL;DR

This paper investigates the $eta$-spectral radius of uniform hypergraphs, providing bounds, transformations to increase it, and characterizing hypergraphs with maximum spectral radius in certain classes.

Contribution

It introduces bounds and transformations for the $eta$-spectral radius and characterizes extremal hypergraphs, advancing spectral hypergraph theory.

Findings

Established upper bounds for the $eta$-spectral radius.

Proposed transformations that increase the spectral radius.

Identified hypergraphs with maximum spectral radius in specific classes.

Abstract

For $0\le\alpha<1$ and a uniform hypergraph $G$, the $\alpha$-spectral radius of $G$ is the largest $H$-eigenvalue of $\alpha \mathcal{D}(G) +(1-\alpha)\mathcal{A}(G)$, where $\mathcal{D}(G)$ and $\mathcal{A}(G)$ are the diagonal tensor of degrees and the adjacency tensor of $G$, respectively. We give upper bounds for the $\alpha$-spectral radius of a uniform hypergraph, propose some transformations that increase the $\alpha$-spectral radius, and determine the unique hypergraphs with maximum $\alpha$-spectral radius in some classes of uniform hypergraphs.