Redetermination of poly[aquadi-μ3-oxy­diacetato-dicopper(II)]

The title complex, [Cu2(C4H4O5)2(H2O)]n, has a two-dimensional layer structure. The Cu atom has a distorted octa­hedral (CuO6) environment and is coordinated by four carboxyl­ate group O atoms from three different oxydiacetate ligands in a planar arrangement and one half-occupancy water mol­ecule and an ether O atom in the axial positions. In the crystal structure, weak intra- and inter­molecular O—H⋯O hydrogen bonds help to stabilize the crystal packing. The structure has already been published [Whitlow & Davey (1975 ▶). J. Chem. Soc. Dalton. Trans. pp. 1228–1232]; this redetermination reports the structure with higher precision

Set Representations of Linegraphs

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A family $\mathcal{S}$ of nonempty sets $\{S_1,\ldots,S_n\}$ is a set representation of $G$ if there exists a one-to-one correspondence between the vertices $v_1, \ldots, v_n$ in $V(G)$ and the sets in $\mathcal{S}$ such that $v_iv_j \in E(G)$ if and only if S_i\cap S_j\neq \es. A set representation $\mathcal{S}$ is a distinct (respectively, antichain, uniform and simple) set representation if any two sets $S_i$ and $S_j$ in $\mathcal{S}$ have the property $S_i\neq S_j$ (respectively, $S_i\nsubseteq S_j$, $|S_i|=|S_j|$ and $|S_i\cap S_j|\leqslant 1$). Let $U(\mathcal{S})=\bigcup_{i=1}^n S_i$. Two set representations $\mathcal{S}$ and $\mathcal{S}'$ are isomorphic if $\mathcal{S}'$ can be obtained from $\mathcal{S}$ by a bijection from $U(\mathcal{S})$ to $U(\mathcal{S}')$. Let $F$ denote a class of set representations of a graph $G$. The type of $F$ is the number of equivalence classes under the isomorphism relation. In this paper, we investigate types of set representations for linegraphs. We determine the types for the following categories of set representations: simple-distinct, simple-antichain, simple-uniform and simple-distinct-uniform

On the multiplicity of Laplacian eigenvalues of graphs

summary:In this paper we investigate the effect on the multiplicity of Laplacian eigenvalues of two disjoint connected graphs when adding an edge between them. As an application of the result, the multiplicity of 1 as a Laplacian eigenvalue of trees is also considered

Bis(μ-3-nitro­phthalato-κ2 O 1:O 2)bis­[(thio­urea-κS)zinc] dihydrate

In the title complex, [Zn2(C8H3NO6)2(CH4N2S)4]·2H2O, the carboxyl­ate groups of the 3-nitro­phthalate ligands coordinate in a bis-monodentate mode to the ZnII cations. This results in the formation of a centrosymmetric dimer containing two ZnII cations with distorted tetra­hedral geometries provided by the O atoms of two different 3-nitro­phthalate dianions and the S atoms of two non-equivalent coordinated thio­urea mol­ecules. The crystal structure exhibits N—H⋯O and O—H⋯O hydrogen bonds which link the dimers into a three-dimensional network

catena-Poly[[bis­(1-methyl-1H-imidazole-κN 3)zinc]-μ-3-nitro­phthalato-κ2 O 1:O2]

In the title complex, [Zn(C8H3NO6)(C4H6N2)2]n, the carboxyl­ate groups of the 3-nitro­phthalate dianion ligand coordinate the ZnII ion in a bis-monodentate mode. The ZnII ion shows distorted tetra­hedral coordination as it is bonded to two O atoms from the carboxyl­ate groups of symmetry-related 3-nitro­phthalate anions and two N atoms of two independent 1-methyl­imidazole mol­ecules. The bridging 3-nitro­phthalate ligand allows the formation of one-dimensional chains in the c direction. The crystal structure is further stabilized by weak inter­molecular C—H⋯O hydrogen bonds

A scheme for tunable quantum phase gate and effective preparation of graph-state entanglement

A scheme is presented for realizing a quantum phase gate with three-level atoms, solid-state qubits--often called artificial atoms, or ions that share a quantum data bus such as a single mode field in cavity QED system or a collective vibrational state of trapped ions. In this scheme, the conditional phase shift is tunable and controllable via the total effective interaction time. Furthermore, we show that the method can be used for effective preparation of graph-state entanglement, which are important resources for quantum computation, quantum error correction, studies of multiparticle entanglement, fundamental tests of non-locality and decoherence.Comment: 7 pages, 5 figure

Theory of mobility edge and non-ergodic extended phase in coupled random matrices

The mobility edge, as a central concept in disordered models for localization-delocalization transitions, has rarely been discussed in the context of random matrix theory (RMT). Here we report a new class of random matrix model by direct coupling between two random matrices, showing that their overlapped spectra and un-overlapped spectra exhibit totally different scaling behaviors, which can be used to construct tunable mobility edges. This model is a direct generalization of the Rosenzweig-Porter model, which hosts ergodic, localized, and non-ergodic extended (NEE) phases. A generic theory for these phase transitions is presented, which applies equally well to dense, sparse, and even corrected random matrices in different ensembles. We show that the phase diagram is fully characterized by two scaling exponents, and they are mapped out in various conditions. Our model provides a general framework to realize the mobility edges and non-ergodic phases in a controllable way in RMT, which pave avenue for many intriguing applications both from the pure mathematics of RMT and the possible implementations of ME in many-body models, chiral symmetry breaking in QCD and the stability of the large ecosystems.Comment: 7+10 pages, 5+7 figure