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arXiv · 1605.09281

On the principal eigenvectors of uniform hypergraphs

Lele Liu·Liying Kang·Xiying Yuan
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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 $H'$.

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Lele Liu, Liying Kang, Xiying Yuan. 2016-05-30. On the principal eigenvectors of uniform hypergraphs. https://doi.org/10.1016/j.laa.2016.09.015

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