ABSTRACT

Let G be an r-uniform hypergraph of order n. For each $p\ge 1$, the p-spectral radius ${\lambda }^{\left(p\right)}\left(G\right)$ is defined as ${\lambda }^{\left(p\right)}\left(G\right):=\underset{|x_1{|}^p+\cdots +|x_n{|}^p=1}{max}r\sum _{\{i_1,\dots ,i_r\}\in E\left(G\right)}{x}_{i_1}\cdots x_{i_r}.$ The p-spectral radius was introduced by Keevash-Lenz-Mubayi, and subsequently studied by Nikiforov in 2014. The most extensively studied case is when p = r, and ${\lambda }^{\left(r\right)}\left(G\right)$ is called the spectral radius of G. The α-normal labelling method, which was introduced by Lu and Man in 2014, is effective method for computing the spectral radii of uniform hypergraphs. It labels each corner of an edge by a positive number so that the sum of the corner labels at any vertex is 1 while the product of all corner labels at any edge is α. Since then, this method has been used by many researchers in studying ${\lambda }^{\left(r\right)}\left(G\right)$. In this paper, we extend Lu and Man's α-normal labelling method to the p-spectral radii of uniform hypergraphs for $p\ne r$; and find some applications.

摘要

令G为n阶的r均匀超图。对于每个$p\ge 1$, p谱半径${\lambda }^{\left(p\right)}\left(G\right)$定义为${\lambda }^{\left(p\right)}\left(G\right):=\underset{|X_1{|}^p+\cdots +|X_n{|}^p=1}{最大限度}r\sum _{\{{\mathrm{一世}}_1,\dots ,{\mathrm{一世}}_r\}\in 乙\left(G\right)}{X}_{{\mathrm{一世}}_1}\cdots X_{{\mathrm{一世}}_r}.$p谱半径由 Keevash-Lenz-Mubayi 引入，随后由 Nikiforov 在 2014 年进行了研究。研究最广泛的情况是当p =  r 时，并且${\lambda }^{\left(r\right)}\left(G\right)$称为G的光谱半径。Lu和Man在2014年提出的α-法线标注方法是计算均匀超图谱半径的有效方法。它用正数标记边的每个角，使得任何顶点的角标签的总和为 1，而任何边的所有角标签的乘积为α。从那时起，这种方法被许多研究人员用于研究${\lambda }^{\left(r\right)}\left(G\right)$. 在本文中，我们将 Lu 和 Man 的α法线标记方法扩展到均匀超图的p谱半径$p\ne r$; 并找到一些应用程序。