Ionothermal Synthesis,Structure and Luminescent Properties of a New 2-D Bismuth(Ⅲ) Coordination Polymer with(6,5)-Connected Topological Sheet

A new 2-D Bi（3＋） coordination polymer {Bi4（μ3-O）2（μ4-O）2（C7H3NO4）2}n（1,C7H3NO4 = pyridine-2,6-dicarboxylate） containing（6,5）-connected topological sheet with the Schl？fli symbol of（4~4·6~3·8~4·12~4）（4~5·6~4·8） has been synthesized by an ionothermal method using the ionic liquid 1-ethyl-3-methylimidazolium bromide（[Emim]Br） as solvent,and characterized by C,H and N elemental analysis,energy-dispersive X-ray spectroscopy（EDS）,infrared spectrum（IR spectrum） and single-crystal X-ray diffraction.The results indicate that 1 belongs to the monoclinic system,space group P21/n with a = 13.766（7）,b = 4.142（2）,c = 16.439（8） ？,β = 99.099（9）°,Z = 2,V = 925.6（8） ？~3,Dc = 4.414 g·cm^（-3）,F（000） = 1064,μ = 37.985 mm-1,the final R = 0.0326,wR = 0.0727 and S = 1.022.The molecular structural units [Bi4（μ3-O）2（μ4-O）2（C7H3NO4）2] are connected to each other via bridging O atoms,extending into an interesting 2-D layered structure.These resulting layers are further assembled by weak C-H…O hydrogen bonds into a 3-D supramolecular architecture.Moreover,its thermal stability and the luminescent properties have been also studied.

An 8-Connected Chiral Lanthanide Metal-organic Framework Constructed from Naturally Camphoric Acid:Crystal Structure,Vibrational Circular Dichroism Spectroscopy and Second-order Nonlinear Optical Effect

A chiral lanthanide metal-organic framework based on enantiopure camphoric acid （D-H2cam）, [Nd3（D-cam）8（H2O）4Cl]n （1）, has been synthesized and characterized by single-crystal X-ray structural analysis, elemental analysis, IR, thermal gravimetric, and X-ray powder diffraction. Crystal data for the title compound are as follows： orthorhombic system, space group P212121 with a = 13.8287（7）, b = 14.0715（7）, c = 25.7403（12） A^°, V = 5008.8（4） A^°3, Mr = 1333.08, Z = 4, F（000） = 2644, Dc = 1.768 g/cm^3, μ（MoKα） = 3.189 mm^-1, the final R = 0.0351 and wR = 0.0814 （I 〉 2σ（I））. Compound 1 displays an 8-connected bcu topology 3D framework and hydrogen-bonding interactions stabilize the solid-state structure. The vibrational circular dichroism （VCD） spectrum and second-order nonlinear optical effect of compound 1 have been studied in the solid state.

A Novel 3D Pillar-layered Co(Ⅱ)-MOF with(4,8)-Connected Topology:Synthesis,Structure and Magnetic Properties

By using ＂pillaring＂ strategy, a new Co（Ⅱ）-MOF, [Co_2（abtc）（bimb）_2]·2H_2O（1, H_4abtc = 3,3？,5,5？-azobenzenetetracarboxylic acid, bimb = 4,4？-bis（imidazole-1-ylmethyl）biphenyl）, has been solvothermally synthesized and structurally characterized. The structural determination revealed that 1 features a 3D pillar-layered framework with（4,8）-connected {4^4.6^2}{4^8.6^（20）} topology based on dinuclear Co（Ⅱ）-SBUs. The magnetic investigation shows that the dominant antiferromagnetic coupling is observed in compound 1.

Two New Complexes with the Same(3,8)-Connected tfz-d Topology Based on [Co4(μ3-OH)2] Clusters and Different Aromatic Multicarboxylic Acids

Two new metal-organic frameworks（MOFs）, namely, [Co_2（L_1）（bix）（μ_3-OH）]·2H_2O（1） and [Co_2（L_2）（bix）（μ_3-OH）]·2.5H_2O（2）（H_3L_1 = 5-oxyacetate isophthalic acid, H_3L_2 = 3,5-bis-oxyacetate-benzoic acid, bix = 1,4-bis（imidazol-1-ylmethyl）benzene）, have been synthesized under hydrothermal conditions. Their structures were determined by single-crystal X-ray diffraction analysis and further characterized by elemental analysis, IR spectra, and powder X-ray diffraction（PXRD） analysis. Both complexes 1 and 2 demonstrate identical three-dimensional（3D）（3,8）-connected tfz-d nets with（4~3）_2（4~6·6^（18）·8~4） topologies, where the tetranuclear [Co_4（μ_3-OH）_2] clusters act as 8-connected nodes and aromatic multicarboxylic ligands as 3-connected nodes. The results show that the ligands with different geometrical conformations can form products with the same topological structures. Their thermal and magnetic properties were also investigated.

A Three-dimensional Cd(Ⅱ) Coordination Polymer with Unique 8-Connected LOMFOI Topology Based on Mixed Flexible N/O Ligands

A new metal-organic framework, {[Cd（DPA）（OBA）]＇（HzO）}n （1, DPA = 4,4＂-dipyri- dylamine, H2OBA = 4,4＂-oxybisbenzoic acid）, has been synthesized and characterized by elemental analysis, IR and single-crystal X-ray diffraction. Single-crystal X-ray analysis shows that it is a 3-D framework. Complex 1 crystallizes in monoclinic, space group C2/c with a = 21.699（7）, b = 12.192（4）, c = 17.566（5） A, β = 105.254（5）°, V = 4483.43 A3, Z = 8, Dc = 1.653 g/cm3, p = 1.021 mm^-1], F（000） = 2240, the final R = 0.0307, wR = 0.0867 （I 〉 2σ（I））. Complex I exhibits a rare three-dimensional highly 8-connected LOMFOI topology with a point symbol of {424＂64}. Thermogravimetric analysis shows that the decomposition temperature of the host framework of the complex is above 400 ℃.

On Graphs with Equal Connected Domination and 2-connected Domination Numbers

A subset S of V is called a k-connected dominating set if S is a dominating set and the induced subgraph S has at most k components.The k-connected domination number γck（G） of G is the minimum cardinality taken over all minimal k-connected dominating sets of G.In this paper,we characterize trees and unicyclic graphs with equal connected domination and 2-connected domination numbers.

k-Factors and Spanning Subgraph in Graphs

In this paper, we discussed k-factors and spanning subgraph, and propose a conjecture which will lead to a series of important conclusion.

Synthesis and Structure of a 3D Manganese Coordination Polymer with 1,3-Bis(benzimidazol-1-ylmethyl)-2,4,6-trimethylbenzene and 1,5-Naphthalenedisulfonate

The organic-inorganic hybrid polymer [Mn（mbbimb）2（1,5-nds）]n 1 has been synthesized using 1,5-naphthalenedisulfonate （1,5-nds） and 1,3-bis（benzimidazol-1-ylmethyl）- 2,4,structure analysis indicates that 1 crystallizes in monoclinic, space group C2/c with a=21.3546（17）, b=11.8841（7）, c=22.1588（16）A, β=108.058（3）°, C60H54MnN8O6S2, Mr=1102.17, Z=4, V=5346.5（7）A^3, Dc=1.369 g/cm^3, μ=0.386 mm^-1, F（000）=2300, S=1.091, R=0.0634 and wR=0.1559 for 4641 observed reflections （I 〉 2σ（I））. The Mn center is six-coordinated to furnish a distorted octahedral geometry, and the overall framework is a 6-connected 3D net.

On Single Valued Neutrosophic Regularity Spaces

This article aims to present new terms of single-valued neutrosophic notions in theˇSostak sense,known as singlevalued neutrosophic regularity spaces.Concepts such as r-single-valued neutrosophic semi￡-open,r-single-valued neutrosophic pre-￡-open,r-single valued neutrosophic regular-￡-open and r-single valued neutrosophicα￡-open are defined and their properties are studied as well as the relationship between them.Moreover,we introduce the concept of r-single valued neutrosophicθ￡-cluster point and r-single-valued neutrosophicγ￡-cluster point,r-θ￡-closed,andθ￡-closure operators and study some of their properties.Also,we present and investigate the notions of r-single-valued neutrosophicθ￡-connectedness and r-single valued neutrosophicδ￡-connectedness and investigate relationship with single-valued neutrosophic almost￡-regular.We compare all these forms of connectedness and investigate their properties in single-valued neutrosophic semiregular and single-valued neutrosophic almost regular in neutrosophic ideal topological spaces inˇSostak sense.The usefulness of these concepts are incorporated to multiple attribute groups of comparison within the connectedness and separateness ofθ￡andδ￡.

Synthesis,Structure,Luminescence and Thermal Stability Properties of a New(3,4)-Connected 2D Zn Coordination Polymer

A new zinc coordination polymer{[Zn2(ttmb)2(dca)4(H2O)].H2O}n(1.H2O)was synthesized by the reaction of 1,3,5-tris(1,2,4-triazol-1-ylmethyl)benzene(ttmb),Zn(Ⅱ)salt and sodium dicyanamide(Na(dca)).The structure of the title compound was determined by single-crystal X-ray diffraction,elemental analyses and IR spectra.1 exhibits a(3,4)-connected 2D double-layer network with the point symbol of{63}{(66}.The solid-state luminescence shows a strong emission band at 485 nm upon excitation at 384 nm.The thermal stability properties were also studied.

Minor and Minimum Cycle Bases of a 3-connected Planar Graph

In this paper, we prove that if any set of ｜E（G）｜- ｜V（G）｜ ＋ 1 facial cycles of a 3-connected planar graph G embedded in the plane doesn＇t form a minimum cycle base of G, then any minimum cycle base of G contains a separating cycle, and G has a minor isomorphic to T6, where T6 is the graph obtained from the complete graph K6 by deleting a path with four edges.

The Characterization of Graphs with No 2-connected Spanning Subgraph of V_(8)as a Minor

It is difficult to characterize graphs which contain no a 2-connected graph as a minor in graph theory.Let V_(8)be a graph constructed from an 8-cycle by connecting the antipodal vertices.There are thirteen 2-connected spanning subgraphs of V_(8).In particular,one of them is obtained from the Petersen graph by deleting two vertices and it is also a hard problem to characterize Petersen-minor-free graphs.In this paper,we characterize internally 4-connected graphs which contain a 2-connected spanning subgraph of V_(8)as a forbidden minor.

LONGEST CYCLES IN 2-CONNECTEDCLAW-FREE GRAPHS

M. Matthews and D. Sumner proved that if G is a 2-connected claw-free graph of order n, then c(G) min{2δb + 4, n}. In this paper, we prove that if G is a,2-connected claw-free graph on n venices, then c(G) min{3δ + 2, n} or G belongs to one exceptional class of graphs.

2-Connected Factor-critical Graphs G with Exactly ｜E(G)｜ ＋ 1 Maximum Matchings

A connected graph G is said to be a factor-critical graph if G - v has a perfect matching for every vertex v of G. In this paper, the 2-connected factor-critical graph G which has exactly ｜E（G）｜＋ 1 maximum matchings is characterized.

Solution to an Extremal Problem on Bigraphic Pairs with a Z3-connected Realization

Let S = （a1,...,am;b1,...,bn）, where a1,...,am and b1,...,bn are two nonincreasing sequences of nonnegative integers. The pair S= （a1,..., am; b1,..., bn） is said to be a bigraphic pair if there is a simple bipartite graph G = （X ∪ Y, E） such that a1,…, am and b1,..., bn are the degrees of the vertices in X and Y, respectively. Let Z3 be the cyclic group of order 3. Define σ（Z3, m, n） to be the minimum integer k such that every bigraphic pair S = （a1,..., am; b1,..., bn） with am, b ≥ 2 and σ（S） = a1 ＋ ... ＋ am ≥ k has a Z3-connected realization. For n = m, Yin [Discrete Math., 339, 2018-2026 （2016）] recently determined the values of σ（Z3,m,m） for m ≥ 4. In this paper, we completely determine the values of σ（Z3, m, n） for m ≥ n ≥4.

Removable Edges in a 5-Connected Graph

An edge e of a k-connected graph G is said to be a removable edge if G O e is still k-connected, where G e denotes the graph obtained from G by deleting e to get G - e, and for any end vertex of e with degree k - 1 in G- e, say x, delete x, and then add edges between any pair of non-adjacent vertices in NG-e （x）. The existence of removable edges of k-connected graphs and some properties of 3-connected and 4-connected graphs have been investigated [1, 11, 14, 15]. In the present paper, we investigate some properties of 5-connected graphs and study the distribution of removable edges on a cycle and a spanning tree in a 5- connected graph. Based on the properties, we proved that for a 5-connected graph G of order at least 10, if the edge-vertex-atom of G contains at least three vertices, then G has at least （3│G│ ＋ 2）/2 removable edges.

Z3-CONNECTIVITY OF 4-EDGE-CONNECTED TRIANGULAR GRAPHS

A graph G is k-triangular if each of its edge is contained in at least k triangles. It is conjectured that every 4-edge-connected triangular graph admits a nowhere-zero 3-flow. A triangle-path in a graph G is a sequence of distinct triangles T1T2%…Tk in G such that for 1 〈 i 〈 k - 1, IE（Ti）∩E（Ti＋1）1= 1 and E（Ti） n E（Tj）=φ if j 〉 i＋1. Two edges e, e＇∈ E（G） are triangularly connected if there is a triangle-path T1, T2,... , Tk in G such that e ∈ E（T1） and er ∈ E（Tk）. Two edges e, e＇ ∈E（G） are equivalent if they are the same, parallel or triangularly connected. It is easy to see that this is an equivalent relation. Each equivalent class is called a triangularly connected component. In this paper, we prove that every 4-edge-connected triangular graph G is Z3-connected, unless it has a triangularly connected component which is not Z3-connected but admits a nowhere-zero 3-flow.

MINIMALLY (n, λ)-CONNECTED GRAPHS OF LOW ORDER AND MAXIMAL SIZE

A. Kaneko and K. Ota proved that for a minimally （n, λ）-connected graph G, if ｜G｜ = p ≥ 3n - 1, then e（G） ≤ nλ（｜G｜ - n）; and if e（G） = nλ（｜G｜ - n）, then G is isomorphic to the graph Kn^λ,p-n which is obtained from the complete bipartite graph Kn,p-n by replacing each edge with A multiple edges; if 3n - 1 ≥ ｜G}≥ n ＋ 1, then e（G） ≤λ（｜G｜ ＋ n）^2/8. In this paper, we determine all the minimally （n, λ）-connected graphs with order p and the maximum size λ（p＋n）^2/8 for 3n- 1 ≥ p ≥ n＋ 1 for 3n-1≥p≥n＋1.

Reticular exploration of uranium-based metal-organic frameworks with hexacarboxylate building units

The rational reticular design of metal-organic frameworks(MOFs)from building units of known geometries is essential for enriching the diversity of MOF structures.Unexpected and intriguing structures,however,can also arise from subtle changes in the rigidity/length of organic linkers and/or synthetic conditions.Herein,we report three uranium-based MOF structures—i.e.,NU-135X(X=0,1,2)—synthesized from trigonal planar uranyl nodes and triptycene-based hexacarboxylate ligands with variable arm lengths.A new chiral 3,6-connected nuc net was observed in NU-1350,while the extended versions of the ligand led to 3-fold catenated MOFs(NU-1351 and NU-1352)with rare 3,6-connected cml-c3 nets.The differences in the topology of NU-1350 and NU-1351/NU-1352 could be attributed to the slight distortions of the shorter linker in the former from the ideal trigonal prism geometry to an octahedral geometry when coordinated to the trigonal planar uranyl nodes.

K-Factors and Hamilton Cycles in Graphs

We discuss k-factors and Hamiltonian Graphs in graph theory. We prove a general version of the conjecture by R. Haggkvist; as a result, we prove two extended versions of two well-known theorems due to O. Ore and B. Jachson, respectively.