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

TL;DR

This paper introduces a new spectral radius concept for directed hypergraphs, explores its properties, and extends the $ ext{alpha}$-normal labeling method to compute these spectral radii, with applications.

Contribution

It develops the $ ext{alpha}$-normal labeling method for $(p,q)$-spectral radii of directed hypergraphs, expanding spectral analysis tools.

Findings

Derived bounds for $ ext{lambda}_{p,q}(G)$

Established spectral relations between hypergraph components

Applied the $ ext{alpha}$-normal labeling method to compute spectral radii

Abstract

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.