On hypergraph Lagrangians

It is conjectured by Frankl and F\"uredi that the $r$-uniform hypergraph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-uniform hypergraphs with $m$ edges in \cite{FF}. Motzkin and Straus' theorem confirms this conjecture when $r=2$. For $r=3$, it is shown by Talbot in \cite{T} that this conjecture is true when $m$ is in certain ranges. In this paper, we explore the connection between the clique number and Lagrangians for $r$-uniform hypergraphs. As an implication of this connection, we prove that the $r$-uniform hypergraph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-uniform graphs with $t$ vertices and $m$ edges satisfying ${t-1\choose r}\leq m \leq {t-1\choose r}+ {t-2\choose r-1}-[(2r-6)\times2^{r-1}+2^{r-3}+(r-4)(2r-7)-1]({t-2\choose r-2}-1)$ for $r\geq 4.$Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:1312.7529, arXiv:1211.7057, arXiv:1211.6508, arXiv:1311.140

On Frankl and Furedi's conjecture for 3-uniform hypergraphs

The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. In most applications, we need an upper bound for the Lagrangian of a hypergraph. Frankl and Furedi in \cite{FF} conjectured that the $r$-graph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-graphs with $m$ edges. In this paper, we give some partial results for this conjecture.Comment: 19 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1211.650

On Motzkin-Straus Type of Results and Frankl-F\"uredi Conjecture for Hypergraphs

A remarkable connection between the order of a maximum clique and the Graph-Lagrangian of a graph was established by Motzkin and Straus in 1965. This connection and its extension were useful in both combinatorics and optimization. Since then, Graph-Lagrangian has been a useful tool in extremal combinatorics. In this paper, we give a parametrized Graph-Lagrangian for non-uniform hypergraphs and provide several Motzkin-Straus type results for nonuniform hypergraphs which generalize results from [1] and [2]. Another part of the paper concerns a long-standing conjecture of Frankl-F\"uredi on Graph-Lagrangians of hypergraphs. We show the connection between the Graph-Lagrangian of $\{1, r_1, r_2, \cdots, r_l\}$-hypergraphs and $\{ r_1, r_2, \cdots, r_l\}$-hypergraphs. Some of our results provide solutions to the maximum value of a class of polynomial functions over the standard simplex of the Euclidean space.Comment: 24 page

Extremal problems for the p-spectral radius of graphs

The $p$-spectral radius of a graph $G\$of order $n$ is defined for any real number $p\geq1$ as $$\lambda^{\left( p\right) }\left( G\right) =\max\left\{ 2\sum_{\{i,j\}\in E\left( G\right) \ }x_{i}x_{j}:x_{1},\ldots,x_{n}\in\mathbb{R}\text{ and }\left\vert x_{1}\right\vert ^{p}+\cdots+\left\vert x_{n}\right\vert ^{p}=1\right\} .$$ The most remarkable feature of $\lambda^{\left( p\right) }$ is that it seamlessly joins several other graph parameters, e.g., $\lambda^{\left( 1\right) }$ is the Lagrangian, $\lambda^{\left( 2\right) }$ is the spectral radius and $\lambda^{\left( \infty\right) }/2$ is the number of edges. This paper presents solutions to some extremal problems about $\lambda^{\left( p\right) }$, which are common generalizations of corresponding edge and spectral extremal problems. Let $T_{r}\left( n\right)$ be the $r$-partite Tur\'{a}n graph of order $n.$ Two of the main results in the paper are: (I) Let $r\geq2$ and $p>1.$ If $G$ is a $K_{r+1}$-free graph of order $n,$ then $$\lambda^{\left( p\right) }\left( G\right) <\lambda^{\left( p\right) }\left( T_{r}\left( n\right) \right) ,$$ unless $G=T_{r}\left( n\right) .$ (II) Let $r\geq2$ and $p>1.$ If $G\$is a graph of order $n,$ with $$\lambda^{\left( p\right) }\left( G\right) >\lambda^{\left( p\right) }\left( T_{r}\left( n\right) \right) ,$$ then $G$ has an edge contained in at least $cn^{r-1}$ cliques of order $r+1,$ where $c$ is a positive number depending only on $p$ and $r.$Comment: 21 pages. Some minor corrections in v