On the principal eigenvectors of uniform hypergraphs

Abstract

Let $\mathcal{A}(H)$ be the adjacency tensor of $r$-uniform hypergraph $H$. If $H$ is connected, the unique positive eigenvector $x=(x_1,x_2,\dots ,x_n{)}^{\mathrm{T}}$ 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 $H^{\prime }$.

In this version, we corrected a reference for the fact Page 6 Line 1, which shoud be [15], not [5] as before

References in corpus (1)

• Principal eigenvector and spectral radius of uniform hypergraphs

Cited by in corpus (2)

• Principal eigenvectors of general hypergraphs
• A homogeneous polynomial associated with general hypergraphs and its applications