Perron-Frobenius Theorem for Rectangular Tensors and Directed Hypergraphs

Abstract

For any positive integers $r$, $s$, $m$, $n$, an $(r,s)$-order $(n,m)$-dimensional rectangular tensor $\mathcal{A}=(a_{i_1\cdots i_r}^{j_1\cdots j_s})\in ({\mathbb{R}}^n{)}^r\times ({\mathbb{R}}^m{)}^s$ is called partially symmetric if it is invariant under any permutation on the lower $r$ indexes and any permutation on the upper $s$ indexes. Such partially symmetric rectangular tensor arises naturally in studying directed hypergraphs. Ling and Qi [Front. Math. China, 2013] first studied the $(p,q)$-spectral radius (or singular values) and proved a Perron-Fronbenius theorem for such tensors when both $p,q\ge r+s$. We improved their results by extending to all $(p,q)$ satisfying $\frac{r}{p}+\frac{s}{q}\le 1$. We also proved the Perron-Fronbenius theorem for general nonnegative $(r,s)$-order $(n,m)$-dimensional rectangular tensors when $\frac{r}{p}+\frac{s}{q}>1$. We essentially showed that this is best possible without additional conditions on $\mathcal{A}$. Finally, we applied these results to study the $(p,q)$-spectral radius of $(r,s)$-uniform directed hypergraphs.

1. One of the main results "Theorem 3.2" has already been proved under a more general setting by Antoine Gautier and Francesco Tudisco in the paper arXiv:1801.04215. 2. Example 2.1 contains an error; the strong eigenvalue-eigenvectors triple actually exists