On the principal eigenvectors of uniform hypergraphs

TL;DR

This paper studies the properties of the principal eigenvector of the adjacency tensor of connected uniform hypergraphs, providing bounds on its entries and ratios, and estimating spectral radius gaps between hypergraphs and their sub-hypergraphs.

Contribution

It introduces bounds for the maximum and minimum entries of the principal eigenvector and the ratios between its entries, advancing understanding of spectral properties of hypergraphs.

Findings

Bounds for $x_{max}$ and $x_{min}$ in the principal eigenvector.

Bounds for the ratio $x_i/x_j$ and the principal ratio $ ho(H)$.

Estimate of spectral radius gaps between hypergraphs and sub-hypergraphs.

Abstract

Let $\mathcal{A}(H)$ be the adjacency tensor of $r$-uniform hypergraph $H$. If $H$ is connected, the unique positive eigenvector $x=(x_1,x_2,\ldots,x_n)^{\mathrm{T}}$ with $||x||_r=1$ corresponding to spectral radius $\rho(H)$ is called the principal eigenvector of $H$. The maximum and minimum entries of $x$ are denoted by $x_{\max}$ and $x_{\min}$, respectively. In this paper, we investigate the bounds of $x_{\max}$ and $x_{\min}$ in the principal eigenvector of $H$. Meanwhile, we also obtain some bounds of the ratio $x_i/x_j$ for $i$, $j\in [n]$ as well as the principal ratio $\gamma(H)=x_{\max}/x_{\min}$ of $H$. As an application of these results we finally give an estimate of the gap of spectral radii between $H$ and its proper sub-hypergraph $H'$.