On the principal eigenvectors of uniform hypergraphs

Lele Liu, Liying Kang, Xiying Yuan

Published 2016-05-30Version 1

Let $\mathcal{A}(H)$ be the adjacency tensor of $k$-uniform hypergraph $H$. If $H$ is connected, the unique positive eigenvector $x=(x_1,x_2,\cdots,x_n)^T$ with $||x||_k=1$ corresponding to spectral radius $\rho(H)$ is called the principal eigenvector of $H$. In this paper, we investigate the bounds of the maximal entry in the principal eigenvector of $H$. Meanwhile, we also obtain some bounds of the ratio $\frac{x_i}{x_j}$ for any $i$, $j\in [n]$. Based on these previous results, we finally give an estimate of the gap of spectral radii between $H$ and proper sub-hypergraph $H'$ of $H$.