The $(p,q)$-spectral radii of $(r,s)$-directed hypergraphs

An $(r,s)$-directed hypergraph is a directed hypergraph with $r$ vertices in tail and $s$ vertices in head of each arc. Let $G$ be an $(r,s)$-directed hypergraph. For any real numbers $p$, $q\geq 1$, we define the $(p,q)$-spectral radius $\lambda_{p,q}(G)$ as $$\lambda_{p,q}(G):=\max_{||{\bf x}||_p=||{\bf y}||_q=1} \sum_{e\in E(G)}\Bigg(\prod_{u\in T(e)}x_u\Bigg)\Bigg(\prod_{v\in H(e)}y_v\Bigg),$$ where ${\bf x}=(x_1, \ldots, x_m)^{{\rm T}}$, ${\bf y}=(y_1,\ldots, y_n)^{{\rm T}}$ are real vectors; and $T(e)$, $H(e)$ are the tail and head of arc $e$, respectively. We study some properties about $\lambda_{p,q}(G)$ including the bounds and the spectral relation between $G$ and its components. The $\alpha$-normal labeling method for uniform hypergraphs was introduced by Lu and Man in 2014. It is an effective method in studying the spectral radii of uniform hypergraphs. In this paper, we develop the $\alpha$-normal labeling method for calculating the $(p,q)$-spectral radii of $(r,s)$-directed hypergraphs. Finally, some applications of $\alpha$-normal labeling method are given.