The maximum $p$-Spectral Radius of Hypergraphs with $m$ Edges

Abstract

For $r\ge 2$ and $p\ge 1$, the $p$-spectral radius of an $r$-uniform hypergraph $H=(V,E)$ on $n$ vertices is defined to be $${\rho }_p(H)=\underset{\mathbf{x}\in {\mathbb{R}}^n:\Vert \mathbf{x}{\Vert }_p=1}{\max }r\cdot \sum _{\{i_1,i_2,\dots ,i_r\}\in E(H)}{x}_{i_1}{x}_{i_2}\cdots x_{i_r},$$ where the maximum is taken over all $\mathbf{x}\in {\mathbb{R}}^{\mathbf{n}}$ with the $p$-norm equals 1. In this paper, we proved for any integer $r\ge 2$, and any real $p\ge 1$, and any $r$-uniform hypergraph $H$ with $m=\left(\genfrac{}{}{0ex}{}{s}{r}\right)$ edges (for some real $s\ge r-1$), we have $${\lambda }_p(H)\le \frac{rm}{s^{r/p}}.$$ The equality holds if and only if $s$ is an integer and $H$ is the complete $r$-uniform hypergraph $K_s^r$ with some possible isolated vertices added. Thus, we completely settled a conjecture of Nikiforov. In particular, we settled all the principal cases of the Frankl-Füredi's Conjecture on the Lagrangians of $r$-uniform hypergraphs for all $r\ge 2$.

16 pages

References in corpus (2)

• Lagrangians of hypergraphs: The Frankl-Füredi conjecture holds almost everywhere
• Symmetric functions and the principal case of the Frankl-Füredi conjecture

Cited by in corpus (3)

• On Lagrangians of $3$-uniform hypergraphs
• The extremal $p$-spectral radius of Berge-hypergraphs
• On Hypergraph Lagrangians and Frankl-Füredi's Conjecture