2 research outputs found
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
Luminescent Tetrahedral Molecular Cages Containing Ruthenium(II) Chromophores
We have designed
linear metalloligands which contain a central photoactive [RuĀ(N<sup>ā§</sup>N)<sub>3</sub>]<sup>2+</sup> unit bordered by peripheral
metal binding sites. The combination of these metalloligands with
ZnĀ(II) and FeĀ(II) ions leads to heterometallic tetrahedral cages,
which were studied by NMR spectroscopy, mass spectrometry, and photophysical
methods. Like the parent metalloligands, the cages remain emissive
in solution. This approach allows direct incorporation of the favorable
properties of rutheniumĀ(II) polypyridyl complexes into larger self-assembled
structures