arXiv 1803.063852018-03-16math.CO

计算一致超图的$p$ -谱半径的$α$ -正规标号方法

The $α$-normal labeling method for computing the $p$-spectral radii of uniform hypergraphs

Lele Liu, Linyuan Lu

备注：: 24 pages

摘要：

设$G$是阶为$n$的$r$ -一致超图.对于每个$p\geq 1$，$p$ -谱半径$λ^{(p)}(G)$定义为\[ λ^{(p)}(G):=\max_{|x_1|^p+\cdots+|x_n|^p=1} r\sum_{\{i_1,\ldots,i_r\}\in E(G)}x_{i_1}\cdots x_{i_r}. \] $p$ -谱半径由Keevash-Lenz-Mubayi引入，随后由Nikiforov在2014年进行研究。最广泛研究的情况是当$p=r$，和$λ^{(r)}(G)$被称为光谱半径的$G$。由Lu和Man在2014年提出的$α$ -正规标号方法是计算一致超图谱半径的有效方法。它用一个正数来标记边的每个角，使得任何顶点处的角标签之和是$1$，而任何边处的所有角标签的乘积是$α$。从那时起，这种方法已被许多研究人员用于研究$λ^{(r)}(G)$。本文将Lu和Man的$α$ -正规标号方法推广到$p\ne r$的一致超图的$p$ -谱半径，并得到了一些应用.

英文摘要：

Let $G$ be an $r$-uniform hypergraph of order $n$. For each $p\geq 1$, the $p$-spectral radius $λ^{(p)}(G)$ is defined as \[ λ^{(p)}(G):=\max_{|x_1|^p+\cdots+|x_n|^p=1} r\sum_{\{i_1,\ldots,i_r\}\in E(G)}x_{i_1}\cdots x_{i_r}. \] 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 $λ^{(r)}(G)$ is called the spectral radius of $G$. The $α$-normal labeling 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 $λ^{(r)}(G)$. In this paper, we extend Lu and Man's $α$-normal labeling method to the $p$-spectral radii of uniform hypergraphs for $p\ne r$; and find some applications.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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