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An $(r,s)$-directed hypergraph is a directed hypergraph with $r$ vertices intail and $s$ vertices in head of each arc. Let $G$ be an $(r,s)$-directedhypergraph. For any real numbers $p$, $q\geq 1$, we define the $(p,q)$-spectralradius $\lambda_{p,q}(G)$ as \[ \lambda_{p,q}(G):=\max_{||{\bf x}||_p=||{\bfy}||_q=1} \sum_{e\in E(G)}\Bigg(\prod_{u\in T(e)}x_u\Bigg)\Bigg(\prod_{v\inH(e)}y_v\Bigg), \] where ${\bf x}=(x_1, \ldots, x_m)^{{\rm T}}$, ${\bfy}=(y_1,\ldots, y_n)^{{\rm T}}$ are real vectors; and $T(e)$, $H(e)$ are thetail 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 byLu and Man in 2014. It is an effective method in studying the spectral radii ofuniform hypergraphs. In this paper, we develop the $\alpha$-normal labelingmethod for calculating the $(p,q)$-spectral radii of $(r,s)$-directedhypergraphs. Finally, some applications of $\alpha$-normal labeling method aregiven.

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