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

Pressure Dependence of the Magnetic Anisotropy in the "Single-Molecule Magnet" [Mn4O3Br(OAc)3(dbm)3]

The anisotropy splitting in the ground state of the single-molecule magnet [Mn4O3Br(OAc)3(dbm)3] is studied by inelastic neutron scattering as a function of hydrostatic pressure. This allows a tuning of the anisotropy and thus the energy barrier for slow magnetisation relaxation at low temperatures. The value of the negative axial anisotropy parameter $D_{\rm cluster}$ changes from -0.0627(1) meV at ambient to -0.0603(3) meV at 12 kbar pressure, and in the same pressure range the height of the energy barrier between up and down spins is reduced from 1.260(5) meV to 1.213(9) meV. Since the $\rm Mn-Br$ bond is significantly softer and thus more compressible than the $\rm Mn-O$ bonds, pressure induces a tilt of the single ion Mn$^{3+}$ anisotropy axes, resulting in the net reduction of the axial cluster anisotropy.Comment: 4 pages, 3 figure

Slow magnetisation relaxation in tetraoxolene-bridged rare earth complexes

Two tetraoxolene-bridged dinuclear Dy(iii) complexes exhibit slow relaxation in ac magnetic susceptibility studies with zero-field quantum tunnelling of the magnetisation that is suppressed by the application of a dc magnetic field

Valence tautomerism versus spin crossover in pyridinophane-cobalt-dioxolene complexes: an experimental and computational study

Compounds [Co(L-N4R2)(dbdiox)](BPh4) (L-N4R2 = N,N′-di-alkyl-2,11-diaza[3.3]-(2,6)pyridinophane, R = iPr (1a), Et (2a); dbdiox = 3,5-di-tert-butyldioxolene) and [M(L-N4iPr2)(dbdiox)](BPh4) (M = Mn (3a), Fe (4a)) have been synthesized and investigated with a view to possible valence tautomeric (VT) or spin crossover (SCO) interconversions. Single crystal X-ray diffraction data for all compounds at 100 or 130 K indicate trivalent metal cations and di-tert-butylcatecholate (dbcat2−) dioxolene ligands. Variable temperature magnetic susceptibility data for all species between 2 and 340 K are consistent with these redox states, with low spin configurations for the cobalt(III) ions and high spin for the manganese(III) and iron(III) ions. Above 340 K, compound 1a exhibits an increase in magnetic susceptibility, suggesting the onset of a VT interconversion from low spin Co(III)-dbcat to high spin Co(II)-dbsq (dbsq− = di-tert-butylsemiquinonate) that is incomplete up to 400 K. In solution, variable temperature electronic absorption spectra and Evans NMR method magnetic susceptibility data indicate reversible VT interconversions for 1a in several solvents, with the transition temperature varying with solvent. Variable temperature electronic absorption spectra are temperature-invariant for 3a and 4a, while spectra for 2a in 1,2-dichloroethane suggest the onset of a VT transition at the highest temperatures measured. Density functional theory calculations on all four compounds and literature analogues provide key insights into the relative energies of the different electromeric forms and the possibilities for VT versus SCO interconversions in this family of compounds

Polyoxometalate-supported lanthanoid single-molecule magnets

Polyoxometalates are robust and versatile multidentate oxygen-donor ligands, eminently suitable for coordination to trivalent lanthanoid ions. To date, 10 very different structural families of such complexes have been found to exhibit slow magnetic relaxation due to single-molecule magnet (SMM) behaviour associated with the lanthanoid ions. These families encompass complexes with between one and four of the later lanthanoid ions: Tb, Dy, Ho, Er, and Yb. The lanthanoid coordination numbers vary between six and eleven and a range of coordination geometries are evident. The highest energy barrier to magnetisation reversal measured to date for a lanthanoid–polyoxometalate SMM is Ueff/kB = 73 K for the heterodinuclear Dy–Eu compound (Bu4N)8H4[DyEu(OH)2(γ-SiW10O36)2]. </jats:p

Synthesis, structure and magnetic properties of a novel Tb4 spin cluster and synthesis of a Tb chain

The syntheses of two new polynuclear TbIII compounds are reported. A tetranuclear complex [Tb4(H2L)2(H4L)2(OAc)8] (1), with H4L=2-((2-hydroxy-benzylidene)-amino)-2-hydroxy-methyl-propane-1,3-diol, has a butterfly-type structure, while [Tb(OAc)3MeOH]∞ (2) is a linear chain. A novel binding mode for H2L2− is observed in 1, with two alkoxides each bridging three TbIII ions. The magnetic properties of 1 are determined by the superposition of the ligand field split 7F6 states of TbIII single-ions located on two crystallographically distinct sites. Weak antiferromagnetic exchange interactions possibly contribute to the lowering of the magnetic moment at the lowest temperatures. © 2007, Elsevier Ltd

Mixed-Valent Polynuclear Cobalt Complexes Incorporating Tetradentate Phenoxyamine Ligands

The new potentially tetradentate ligand precursor 2-[(bis(2-hydroxyethyl)amino)methyl]-4,6-bis-tert-butylphenol (H3L) has been synthesized and characterized. The reaction of H3L with cobalt(ii) acetate has afforded the novel mixed-valent tetra- and pentanuclear cobalt complexes [CoII 2 CoIII 2 (OAc)2(L)2(HL)] (1) and [CoIICoIII4 (OAc)2(L)4] (2). Single-crystal X-ray diffraction studies of these complexes indicate different coordination geometries for the divalent cobalt centres in each complex, with distorted trigonal bipyramidal and distorted tetrahedral coordination evident in 1 and 2, respectively. The variable temperature magnetic susceptibility data for complex 1 reveal behaviour dominated by antiferromagnetic coupling between the two cobalt(ii) centres. Their approximate trigonal bipyramidal coordination geometries give rise to a 4A2 ground term, allowing a spin-only treatment assuming local spin quantum numbers of Si =3/2. Fitting the data to the Heisenberg exchange Hamiltonian (ˆHex=−2JS1·S2) results in J =−6.9(1) cm−1 and g =2.30(5)