7,114 research outputs found
The Largest Laplacian and Signless Laplacian H-Eigenvalues of a Uniform Hypergraph
In this paper, we show that the largest Laplacian H-eigenvalue of a
-uniform nontrivial hypergraph is strictly larger than the maximum degree
when is even. A tight lower bound for this eigenvalue is given. For a
connected even-uniform hypergraph, this lower bound is achieved if and only if
it is a hyperstar. However, when is odd, it happens that the largest
Laplacian H-eigenvalue is equal to the maximum degree, which is a tight lower
bound. On the other hand, tight upper and lower bounds for the largest signless
Laplacian H-eigenvalue of a -uniform connected hypergraph are given. For a
connected -uniform hypergraph, the upper (respectively lower) bound of the
largest signless Laplacian H-eigenvalue is achieved if and only if it is a
complete hypergraph (respectively a hyperstar). The largest Laplacian
H-eigenvalue is always less than or equal to the largest signless Laplacian
H-eigenvalue. When the hypergraph is connected, the equality holds here if and
only if is even and the hypergraph is odd-bipartite.Comment: 26 pages, 3 figure
Hankel Tensors: Associated Hankel Matrices and Vandermonde Decomposition
Hankel tensors arise from applications such as signal processing. In this
paper, we make an initial study on Hankel tensors. For each Hankel tensor, we
associate it with a Hankel matrix and a higher order two-dimensional symmetric
tensor, which we call the associated plane tensor. If the associated Hankel
matrix is positive semi-definite, we call such a Hankel tensor a strong Hankel
tensor. We show that an order -dimensional tensor is a Hankel tensor if
and only if it has a Vandermonde decomposition. We call a Hankel tensor a
complete Hankel tensor if it has a Vandermonde decomposition with positive
coefficients. We prove that if a Hankel tensor is copositive or an even order
Hankel tensor is positive semi-definite, then the associated plane tensor is
copositive or positive semi-definite, respectively. We show that even order
strong and complete Hankel tensors are positive semi-definite, the Hadamard
product of two strong Hankel tensors is a strong Hankel tensor, and the
Hadamard product of two complete Hankel tensors is a complete Hankel tensor. We
show that all the H-eigenvalue of a complete Hankel tensors (maybe of odd
order) are nonnegative. We give some upper bounds and lower bounds for the
smallest and the largest Z-eigenvalues of a Hankel tensor, respectively.
Further questions on Hankel tensors are raised
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