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There is a remarkable connection between the clique number and the Lagrangian of a 2-graph proved by Motzkin and Straus in 1965. It is useful in practice if similar results hold for hypergraphs. However the obvious generalization of Motzkin and Straus result to hypergraphs is false. Frankl and F{u}redi conjectured 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. For $r=2$, Motzkin and Straus theorem confirms this conjecture. For $r=3$, it is shown by Talbot 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 $3$-uniform hypergraphs. As an application of this connection, we confirm that Frankl and F{u}redis conjecture holds for bigger ranges of $m$ when $r$=3. We also obtain two weak
Determine the size of $r$-graphs with given graph parameters is an interesting problem. Chvatal and Hanson (JCTB, 1976) gave a tight upper bound of the size of 2-graphs with restricted maximum degree and matching number; Khare (DM, 2014) studied the
We bound the number of minimal hypergraph transversals that arise in tri-partite 3-uniform hypergraphs, a class commonly found in applications dealing with data. Let H be such a hypergraph on a set of vertices V. We give a lower bound of 1.4977 |V | and an upper bound of 1.5012 |V | .
The spectral radius (or the signless Laplacian spectral radius) of a general hypergraph is the maximum modulus of the eigenvalues of its adjacency (or its signless Laplacian) tensor. In this paper, we firstly obtain a lower bound of the spectral radi
There is a remarkable connection between the maximum clique number and the Lagrangian of a graph given by T. S. Motzkin and E.G. Straus in 1965. This connection and its extensions were successfully employed in optimization to provide heuristics for t
Let K_4^3-2e denote the hypergraph consisting of two triples on four points. For an integer n, let t(n, K_4^3-2e) denote the smallest integer d so that every 3-uniform hypergraph G of order n with minimum pair-degree delta_2(G) geq d contains floor{n