IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

The maximum p-Spectral Radius of Hypergraphs with m Edges

  • Oct. 26, 2018
  • 2:30 p.m.
  • LeConte 317R

Abstract

For $r\geq 2$ and $p\geq 1$, the $p$-spectral radius of an $r$-uniform hypergraph $H=(V,E)$ on $n$ vertices is defined to be $$\rho _ p(H)=\max _ {{\bf x}\in \mathbb{R}^n: \|{\bf x}\| _ p=1}r \cdot \!\!\!\! \sum _ {\{i _ 1,i _ 2,\ldots, i _ r\}\in E(H)} x _ {i _ 1}x _ {i _ 2}\cdots x _ {i _ r},$$ where the maximum is taken over all ${\bf x\in \mathbb{R}^n}$ with the $p$-norm equals 1.

In this talk, we proved for any integer $r\geq 2$, and any real $p\geq 1$, and any $r$-uniform hypergraph $H$ with $m={s\choose r}$ edges (for some real $s\geq r-1$), we have $$\lambda _ p(H)\leq \frac{rm}{s^{r/p}}.$$ The equality holds if and only if $s$ is an integer and $H$ is the complete $r$-uniform hypergraph $K^r _ s$ 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-F├╝redi's Conjecture on the Lagrangians of $r$-uniform hypergraphs for all $r\geq 2$.

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