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 ${cal A}=(a_{i_1cdots i_r}^{j_1cdots j_s}) in ({mathbb R}^n)^rtimes ({mathbb R}^m)^s$ 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.