826 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
On the spectrum of hypergraphs
Here we study the spectral properties of an underlying weighted graph of a
non-uniform hypergraph by introducing different connectivity matrices, such as
adjacency, Laplacian and normalized Laplacian matrices. We show that different
structural properties of a hypergrpah, can be well studied using spectral
properties of these matrices. Connectivity of a hypergraph is also investigated
by the eigenvalues of these operators. Spectral radii of the same are bounded
by the degrees of a hypergraph. The diameter of a hypergraph is also bounded by
the eigenvalues of its connectivity matrices. We characterize different
properties of a regular hypergraph characterized by the spectrum. Strong
(vertex) chromatic number of a hypergraph is bounded by the eigenvalues.
Cheeger constant on a hypergraph is defined and we show that it can be bounded
by the smallest nontrivial eigenvalues of Laplacian matrix and normalized
Laplacian matrix, respectively, of a connected hypergraph. We also show an
approach to study random walk on a (non-uniform) hypergraph that can be
performed by analyzing the spectrum of transition probability operator which is
defined on that hypergraph. Ricci curvature on hypergraphs is introduced in two
different ways. We show that if the Laplace operator, , on a hypergraph
satisfies a curvature-dimension type inequality
with and then any non-zero eigenvalue of can be bounded below by . Eigenvalues of a normalized Laplacian operator defined on a connected
hypergraph can be bounded by the Ollivier's Ricci curvature of the hypergraph
The proof of a conjecture on largest Laplacian and signless Laplacian H-eigenvalues of uniform hypergraphs
Let and be the adjacency tensor, Laplacian tensor and signless Laplacian
tensor of uniform hypergraph , respectively. Denote by the largest H-eigenvalue of tensor . Let be a
uniform hypergraph, and be obtained from by inserting a new
vertex with degree one in each edge. We prove that Denote by
the th power hypergraph of an ordinary graph with maximum degree
. We will prove that
is a strictly decreasing sequence, which imply Conjectrue 4.1 of Hu, Qi and
Shao in \cite{HuQiShao2013}. We also prove that
converges to when goes to
infinity. The definiton of th power hypergraph has been generalized
as We also prove some eigenvalues properties about which generalize some known results. Some related results
about are also mentioned
H-Eigenvalues of Laplacian and Signless Laplacian Tensors
We propose a simple and natural definition for the Laplacian and the signless
Laplacian tensors of a uniform hypergraph. We study their H-eigenvalues,
i.e., H-eigenvalues with nonnegative H-eigenvectors, and H-eigenvalues,
i.e., H-eigenvalues with positive H-eigenvectors. We show that each of the
Laplacian tensor, the signless Laplacian tensor and the adjacency tensor has at
most one H-eigenvalue, but has several other H-eigenvalues. We
identify their largest and smallest H-eigenvalues, and establish some
maximum and minimum properties of these H-eigenvalues. We then define
analytic connectivity of a uniform hypergraph and discuss its application in
edge connectivity
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