Perron-Frobenius Theorem for Rectangular Tensors and Directed Hypergraphs

For any positive integers $r$, $s$, $m$, $n$, an $(r,s)$-order $(n,m)$-dimensional rectangular tensor ${\cal A}=(a_{i_1\cdots i_r}^{j_1\cdots j_s}) \in ({\mathbb R}^n)^r\times ({\mathbb R}^m)^s$ 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 $p,q \geq r+s$. We improved their results by extending to all $(p,q)$ satisfying $\frac{r}{p} +\frac{s}{q}\leq 1$. 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 $\cal A$. Finally, we applied these results to study the $(p,q)$-spectral radius of $(r,s)$-uniform directed hypergraphs.