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, $\Delta$, on a hypergraph satisfies a curvature-dimension type inequality $CD (\mathbf{m}, \mathbf{K})$ with $\mathbf{m}>1$ and $\mathbf{K}>0$ then any non-zero eigenvalue of $- \Delta$ can be bounded below by $\frac{ \mathbf{m} \mathbf{K}}{ \mathbf{m} -1 }$. Eigenvalues of a normalized Laplacian operator defined on a connected hypergraph can be bounded by the Ollivier's Ricci curvature of the hypergraph

Terbium(III) and Yttrium(III) Complexes with Pyridine-Substituted Nitronyl Nitroxide Radical and Different β‑Diketonate Ligands. Crystal Structures and Magnetic and Luminescence Properties

A terbium­(III) complex of nitronyl nitroxide free radical 2-(2-pyridyl)-4,4,5,5-tetramethyl-4,5-dihydro1<i>H</i>-imidazolyl-1-oxy-3-oxide (NIT2Py), [Tb­(acac)­3NIT2Py]·0.5H₂O (<b>3</b>) (acac = acetylacetonate), was synthesized for comparison with the previously reported [Tb­(hfac)₃NIT2Py]·0.5C₇H₁₆ (<b>1</b>) (hfac = hexafluoroacetylacetonate), together with their yttrium analogues [Y­(hfac)₃NIT2Py]·0.5C₇H₁₆ (<b>2</b>) and [Y­(acac)₃NIT2Py]·0.5H₂O (<b>4</b>). The crystal structures show that in all complexes the nitronyl nitroxide radical acts as a chelating ligand. Magnetic studies show that <b>3</b> like <b>1</b> exhibits slow relaxation of magnetization at low temperature, suggesting single-molecule magnet behavior. The luminescence spectra show resolved vibronic structure with the main interval decreasing from 1600 cm^–1 to 1400 cm^–1 between 80 and 300 K. This effect is analyzed quantitatively using experimental Raman frequencies

Interpreting Effects of Structure Variations Induced by Temperature and Pressure on Luminescence Spectra of Platinum(II) Bis(dithiocarbamate) Compounds

Luminescence spectra of two square-planar dithiocarbamate complexes of platinum­(II) with different steric bulk, platinum­(II) bis­(dimethyldithiocarbamate) (Pt­(MeDTC)₂) and platinum­(II) bis­(di­(<i>o</i>-pyridyl)­dithiocarbamate) (Pt­(dopDTC)₂), are presented at variable temperature and pressure. The spectra show broad d–d luminescence transitions with maxima at approximately 13500 cm^–1 (740 nm). Variations of the solid-state spectra with temperature and pressure reveal intrinsic differences due to subtle variations of molecular and crystal structures, reported at 100 and 296 K for Pt­(dopDTC)₂. Luminescence maxima of Pt­(MeDTC)₂ shift to higher energy as temperature increases by +320 cm^–1 for an increase by 200 K, mainly caused by a bandwidth increase from 3065 to 4000 cm^–1 on the high-energy side of the band over the same temperature range. Luminescence maxima of Pt­(dopDTC)₂ shift in the opposite direction by −460 cm^–1 for a temperature increase by 200 K. The bandwidth of approximately 2900 cm^–1 does not vary with temperature. Both ground and emitting-state properties and subtle structural differences between the two compounds lead to this different behavior. Luminescence maxima measured at variable pressure show shifts to higher energy by +47 ± 3 and +11 ± 1 cm^–1/kbar, for Pt­(MeDTC)₂ and Pt­(dopDTC)₂, respectively, a surprising difference by a factor of 4. The crystal structures indicate that decreasing intermolecular interactions with increasing pressure are likely to contribute to the exceptionally high shift for Pt­(MeDTC)₂