1,392 research outputs found
The minimum number of nonnegative edges in hypergraphs
An r-unform n-vertex hypergraph H is said to have the
Manickam-Mikl\'os-Singhi (MMS) property if for every assignment of weights to
its vertices with nonnegative sum, the number of edges whose total weight is
nonnegative is at least the minimum degree of H. In this paper we show that for
n>10r^3, every r-uniform n-vertex hypergraph with equal codegrees has the MMS
property, and the bound on n is essentially tight up to a constant factor. This
result has two immediate corollaries. First it shows that every set of n>10k^3
real numbers with nonnegative sum has at least nonnegative
k-sums, verifying the Manickam-Mikl\'os-Singhi conjecture for this range. More
importantly, it implies the vector space Manickam-Mikl\'os-Singhi conjecture
which states that for n >= 4k and any weighting on the 1-dimensional subspaces
of F_q^n with nonnegative sum, the number of nonnegative k-dimensional
subspaces is at least . We also discuss two additional
generalizations, which can be regarded as analogues of the Erd\H{o}s-Ko-Rado
theorem on k-intersecting families
The extremal spectral radii of -uniform supertrees
In this paper, we study some extremal problems of three kinds of spectral
radii of -uniform hypergraphs (the adjacency spectral radius, the signless
Laplacian spectral radius and the incidence -spectral radius).
We call a connected and acyclic -uniform hypergraph a supertree. We
introduce the operation of "moving edges" for hypergraphs, together with the
two special cases of this operation: the edge-releasing operation and the total
grafting operation. By studying the perturbation of these kinds of spectral
radii of hypergraphs under these operations, we prove that for all these three
kinds of spectral radii, the hyperstar attains uniquely the
maximum spectral radius among all -uniform supertrees on vertices. We
also determine the unique -uniform supertree on vertices with the second
largest spectral radius (for these three kinds of spectral radii). We also
prove that for all these three kinds of spectral radii, the loose path
attains uniquely the minimum spectral radius among all
-th power hypertrees of vertices. Some bounds on the incidence
-spectral radius are given. The relation between the incidence -spectral
radius and the spectral radius of the matrix product of the incidence matrix
and its transpose is discussed
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