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

Fluorite Topology in Lanthanoid Coordination Polymers with Di- and Trimetallic Building Blocks

Reaction of the pseudotetrahedral tetracarboxylic acid proligand tetrakis­(4-carboxyphenyl)­silane (H₄L) with salts of various lanthanoid metals has afforded three new lanthanoid coordination polymers: {[H₃O]₂[Ce₂(L)₂(H₂O)₂]·2DMF·2H₂O}_∞ (<b>1</b>), {[Eu₃L₂(NO₃)­(DMF)₄(H₂O)]·0.5DMF·6H₂O}_∞ (<b>2</b>), and {Eu­(HL)­(DMF)₂(H₂O)]·DMF·2.5H₂O}_∞ (<b>3</b>) (DMF = <i>N</i>,<i>N</i>′-dimethylformamide), which have been structurally characterized by single-crystal X-ray diffraction. Compounds <b>1</b> and <b>2</b> are both noninterpenetrated three-dimensional networks that display the rare fluorite (CaF₂) topology. In compound <b>1</b>, the fully deprotonated L^4– ligands act as pseudotetrahedral 4-connecting nodes and dinuclear cerium-carboxylate building blocks act as 8-connecting nodes, while in compound <b>2</b> the L^4– ligands and trinuclear europium-carboxylate units are 4- and 8-connecting, respectively. In contrast, compound <b>3</b> exhibits a two-dimensional layered structure with triply deprotonated HL^3– ligands acting as 3-connecting units, linking single europium centers within each layer. Gas sorption studies of <b>2</b> show a high affinity of the pretreated microcrystalline solid for carbon dioxide gas

Visible-Light-Driven “On”/“Off” Photochromism of a Polyoxometalate Diarylethene Coordination Complex

Herein we report the first photochromic polyoxometalate (POM)-based diarylethene (DAE) coordination complex, prepared by ligation of two cobalt­(III)-incorporated borotungstates [B^IIIW^VI₁₁O₃₉Co^III]^6– with the ditopic pyridyl-containing diarylethene (C₂₅H₁₆N₂F₆S₂). The solution-state composition, structure, and stability of the assembly were probed using ¹H and ¹⁹F nuclear magnetic resonance spectroscopy (NMR), electrospray ionization quadrupolar time-of-flight mass spectrometry (ESI-QTOF-MS), ultraviolet–visible spectroscopy (UV–vis), and small-angle X-ray scattering (SAXS), revealing that the complex self-organizes to adopt a molecular dumbbell structure due to electrostatic and steric considerations. This conformation is a prerequisite for the photocyclization reaction. The assembly was found to be switchable between two states using visible light due to the perturbation of the DAE electronic structure on coordination to the POM. We present photophysical data, including the reaction quantum efficiency of the molecular switch in both directions measured using a custom-built quantum yield determination setup in addition to fatigue resistance on prolonged irradiation

Redox Activity and Two-Step Valence Tautomerism in a Family of Dinuclear Cobalt Complexes with a Spiroconjugated Bis(dioxolene) Ligand

A family of dinuclear cobalt complexes with bridging bis­(dioxolene) ligands derived from 3,3,3′,3′-tetramethyl-1,1′-spirobis­(indane-5,5′,6,6′-tetrol) (spiroH₄) and ancillary ligands based on tris­(2-pyridylmethyl)­amine (tpa) has been synthesized and characterized. The bis­(dioxolene) bridging ligand is redox-active and accessible in the (spiro^cat–cat)^4–, (spiro^SQ–cat)^3–, and (spiro^SQ–SQ)^2– forms, (cat = catecholate, SQ = semiquinonate). Variation of the ancillary ligand (Me_<i>n</i>tpa; <i>n</i> = 0–3) by successive methylation of the 6-position of the pyridine rings influences the redox state of the complex, governing the distribution of electrons between the cobalt centers and the bridging ligands. Pure samples of salts of the complexes [Co₂(spiro)­(tpa)₂]²⁺ (<b>1</b>), [Co₂(spiro)­(Metpa)₂]²⁺ (<b>2</b>), [Co₂(spiro)­(Me₂tpa)₂]²⁺ (<b>3</b>), [Co₂(spiro)­(Me₃tpa)₂]²⁺ (<b>4</b>), [Co₂(spiro)­(tpa)₂]³⁺ (<b>5</b>), and [Co₂(spiro)­(tpa)₂]⁴⁺ (<b>6</b>) have been isolated, and <b>1</b>, <b>4</b>, and <b>6</b> have been characterized by single crystal X-ray diffraction. Studies in the solid and solution states using multiple techniques reveal temperature invariant redox states for <b>1</b>, <b>2</b>, and <b>4</b>–<b>6</b> and provide clear evidence for four different charge distributions: <b>1</b> and <b>2</b> are Co^III-(spiro^cat–cat)-Co^III, <b>4</b> is Co^II-(spiro^SQ–SQ)-Co^II, <b>5</b> is Co^III-(spiro^SQ–cat)-Co^III, and <b>6</b> is Co^III-(spiro^SQ–SQ)-Co^III. Of the six complexes, only <b>3</b> shows evidence of temperature dependence of the charge distribution, displaying a rare thermally induced two-step valence tautomeric transition from the Co^III-(spiro^cat–cat)-Co^III form to Co^II-(spiro^SQ–cat)-Co^III and then to Co^II-(spiro^SQ–SQ)-Co^II in both solid and solution states. This is the first time a two-step valence tautomeric (VT) transition has been observed in solution. Partial photoinduction of the VT transition is also possible in the solid. Magnetic and spectroscopic studies of <b>5</b> and <b>6</b> reveal that spiroconjugation of the bis­(dioxolene) ligand allows electronic interaction across the spiro bridge, suggesting that thermally activated vibronic coupling between the two cobalt-dioxolene moieties plays a key role in the two-step transition evident for <b>3</b>