独立点数为3的图的Z_(3)-连通性

Jaeger猜想为“5-边连通图是Z_(3)-连通的”,此猜想对于独立点数为2的图是成立的。利用收缩、点分裂、反证等方法,证明了此猜想对于独立点数为3且点连通度不大于5的图也是成立的。

独立点数为2的图的Z_(3)-连通性

设G是独立点数为2的连通简单图.主要证明两个结论:(1)当边连通为4时,G是Z 3-连通的;(2)当点连通为3时,G是Z 3-连通的当且仅当G{K 4,W 5}且G不是一类特殊的图.

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.

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.

Degree sum of a pair of independent edges and Z3-connectivity

Let G be a 2-edge-connected simple graph on n vertices. For an edge e = uv ∈ E（G）, define d（e） = d（u） ＋ d（v）. Let F denote the set of all simple 2-edge-connected graphs on n ） 4 vertices such that G∈ F if and only if d（e） ＋ d（e′） ≥ 2n for every pair of independent edges e, e′ of G. We prove in this paper that for each G ∈ F, G is not Z3-connected if and only if G is one of K2,n-2, K3,n-3, K^＋2,n-2,K^＋ 3,n-3 or one of the 16 specified graphs, which generalizes the results of X. Zhang et al. [Discrete Math., 20]0, 310： 3390-3397] and G. Fan and X. Zhou [Discrete Math., 2008, 308： 6233-6240].

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.

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.

Synthesis and spectroscopic investigations of trivalent europium- doped Z_(2)Si_(3)O_(8)(Z = Mg, Ca and Sr) nanophosphors for display applications

To explore the silicate lattice-based nanophosphors, a series of Eu3+-doped Z_(2)Si_(3)O_(8)(Z = Mg, Ca and Sr) materials were prepared by sol–gel procedure. The metal nitrates and silica powder were used as precursor components for the synthesis of these luminescent materials. Photoluminescence(PL) emission spectra, powder X-ray diffraction(PXRD), energy-dispersive X-ray spectroscopy(EDS), Fourier transform infrared spectroscopy(FTIR) and transmission electron microscopy(TEM)spectroscopic techniques were applied for the characterization of the fabricated materials. Three peaks were observed at 589, 613 and 650–652 nm corresponding to ^(5)D_(0)→^(7)F_(1-3) transition in PL emission spectra. Upon395 nm excitation and at 0.03 mol Eu3+, these nanophosphors displayed optimum photoluminescence with the most intense peak analogous to ^(5)D_(0)→^(7)F_(2)transition of dopant ion. The as-prepared phosphor materials were re-heated at1050 and 1150 ℃ to observe the consequences of higher temperatures on the emission intensity and crystal lattice.XRD analysis confirmed that all the synthesized materials were of crystalline nature, and the crystallinity was observed to be improved by increasing the temperature. In the FTIR spectrum, peaks at 483 and 610 cm^(-1) proved the existence of SiO_(4)group in Ca_(2)Si_(3)O_(8), and the peak centered at 417 cm^(-1) confirmed the presence of MgO6 octahedral in Mg_(2)Si_(3)O_(8) materials. TEM images were used to determine the particle size(13–35 nm) and to study the threedimensional structure of nanophosphor materials. The experimental studies indicate that these materials may be promising as red-emitting nanophosphors for white lightemitting diodes.

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.

Nowhere-zero 3-flows in matroid base graph

The base graph of a simple matroid M = （E, A） is the graph G such that V（G） = A and E（G） = {BB＇： B, B＇ B, [B ／ B＇｜ = 1}, where the same notation is used for the vertices of G and the bases of M. It is proved that the base graph G of connected simple matroid M is Z3-connected if ｜V（G）｜ ≥5. We also proved that if M is not a connected simple matroid, then the base graph G of M does not admit a nowhere-zero 3-flow if and only if IV（G）[ =4. Furthermore, if for every connected component Ei （ i≥ 2） of M, the matroid base graph Gi of Mi=MIEi has IV（Gi）｜≥5, then G is Z3-connected which also implies that G admits nowhere-zero 3-flow immediately.

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.

Connectivity of Minimum Non-5-injectively Colorable Planar Cubic Graphs

Suppose that G is a planar cubic graph withχi(G)>5.We show that ifχi(H)<χi(G)for each planar cubic graph H of order less thanG,thenG is either a 3-connected simple planar cubic graph,or a planar graph obtained from a simple cubic 3-connected planar graph by adding some earrings.This shows that a minimum non-5-injectively colorable simple planar cubic graph must be 3-connected.