The forcing number of toroidal polyhexes

The forcing number, denoted by f(G), of a graph G with a perfect matching is the minimum number of independent edges that completely determine the perfect matching of G. In this paper, we consider the forcing number of a toroidal polyhex H(p,q,t) with a torsion t, a cubic graph embedded on torus with every face being a hexagon. We obtain that f(H(p,q,t)) ≥ min{p,q}, and equality holds for p ≤ q or p > q and t∈{ 0,p−q,p−q + 1,..., p−1}. In general, we show that f(H(p,q,t)) is equal to the side length of a maximum triangle on H(p,q,t). Based on this result, we design a linear algorithm to compute the forcing number of H(p,q,t).     

On unicyclic conjugated molecules with minimal energies

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let U(k) be the set of all unicyclic graphs with a perfect matching. Let C _g(G) be the unique cycle of G with length g(G), and M(G) be a perfect matching of G. Let U ⁰(k) be the subset of U(k) such that g(G)≡ 0 (mod 4), there are just g/2 independence edges of M(G) in C _g(G) and there are some edges of E(G)\ M(G) in G\ C _g(G) for any G∈U ⁰(k). In this paper, we discuss the graphs with minimal and second minimal energies in U ^*(k) = U(k)\ U ⁰(k), the graph with minimal energy in U ⁰(k), and propose a conjecture on the graph with minimal energy in U(k).      

A complete characterization for <Emphasis Type="Italic">k</Emphasis>-resonant Klein-bottle polyhexes

A hexagonal tessellation K(p, q, t) on Klein bottle, a non-orientable surface with cross-cap number 2, is a finite-sized elemental benzenoid which can be produced from a p × q-parallelogram of hexagonal lattice with usual identifications of sides and with torsion t. Unlike torus, Klein bottle polyhex K(p, q, t) is not transitive except for some degenerated cases. We shall show, however, that K(p, q, t) does not depend on t. Accordingly, criteria for K(p, q, t) to be k-resonant for every positive integer k will be given. Moreover, we shall show that K(3, q, t) of 3-resonance are fully-benzenoid.      

The stability condition of phase equilibrium for various concentration variables

The stability conditions of phase equilibrium for various concentration variables are educed according to thermodynamic principle. When a system with k components arrives at stable equilibrium, if the mole number n_i or the mole fraction y_i(=n_i/n_k) or molality m_i[= n_i/(n_kM_k)] of component i(i = 1,2,...,k − 1) are elected as concentration variables, thermodynamic theory is able to confirm that the sign of every order determinant composed of the second-order partial differential of chemical potential with respect to these concentration variables is positive; if the mole fraction x_i(= n_i/n) or mass fraction w_i(= n_iM_i/W are elected as the concentration variables, thermodynamic theory is only able to confirm that the sign of (k−1) order determinant is positive; if molarity c_i(= n_i/V are elected as the concentration variables, thermodynamic theory is not able to confirm the sign of every order determinant.     

The spread of unicyclic graphs with given size of maximum matchings

The spread s(G) of a graph G is defined as s(G) = max _i,j |λ _i − λ _j |, where the maximum is taken over all pairs of eigenvalues of G. Let U(n,k) denote the set of all unicyclic graphs on n vertices with a maximum matching of cardinality k, and U ^*(n,k) the set of triangle-free graphs in U(n,k). In this paper, we determine the graphs with the largest and second largest spectral radius in U ^*(n,k), and the graph with the largest spread in U(n,k).      

Two New C-21 Steroidal Glycosides from the Stems of Stephanotis mucronata

Two new C-21 steroidal glycosides, mucronatosides M（1） and N（2）, were isolated from the stems of Stephanotis mucronata, together with one known compound stephanoside M（3）. On the basis of chemical evidence and extensive spectroscopic methods, including one-dimensional and two-dimensional NMR, the structures of the two new compounds were identified as 12-O-tigloyl-20-O-N-methylanthraniloyl sarcoslin 3-O-β-D-glucopyranosyl-（1→4）-6- deoxy-3-O-methyl-β-D-allopyranosyl-（→4）-β-D-cymaropyranosyl-（1→4）-β-D-cymaropyranoside （1）, and 12-O- cinnamoyl-20-O-nicotinoyl（ 2OS）-pregn-6-ene-3 β,5α,8βm12 β,14β,17β,20-heptanol 3-O-β-D-glucopyranosyl-（1→4）-6- deoxy-3-O-methyl-β-D-allopyranosyl-（1→4）-β-D-cymaropyranosyl-（1→4）-β-D-cymaropyranoside （2）.     

Preparation of C2‐Symmetric Bis[2‐(diphenylphosphino)ferrocen‐1‐yl]‐ methane and Its Use in Rhodium‐ and Ruthenium‐Catalyzed Hydrogenation

The two diphosphine ligands (R_p,R_p)‐ and (S_p,S_p)‐bis[2‐(diphenylphospino)ferrocenyl]methane, (R_p,R_p)‐ and (S_p,S_p)‐ 1 , resp., were prepared in six steps from (S)‐ and (R)‐ferrocenyl tolyl sulfoxide, respectively (Scheme). In the solid state, both the diborane complex (R_p,R_p)‐ 1 ? (BH₃)₂ and the palladium dichloride complex [PdCl₂((R_p,R_p)‐ 1 )] were found to adopt C₂‐pseudosymmetric structures according to X‐ray analyses (Figs. 2 and 3). In the Rh‐ and Ru‐catalyzed hydrogenation of selected alkenes and ketones in the presence of the new ligands, enantioselectivities of up to 55% ee were obtained.     

Triterpenoid Saponins from Vaccaria segetalis

Four novel triterpenoid saponins, Vaccariside B‐E (1–4), were isolated from the seeds of Vaccaria segetalis and their structures were elucidated as 3‐O‐β‐D‐galactopyranosyl‐(1–2)‐β‐D‐glucuronopyranosyl quillaic add 28‐O‐β‐D‐xylopyranosyl‐(1–3)‐α‐L rhamno‐pyranosyl‐(1–2)‐[α‐L‐arabinofura‐nosyl‐(1–3)]‐4‐O‐acetyl‐β‐D)‐fucopyranoside (1), 3‐O‐β‐D‐galactopyranosyl ‐ (1–2) ?3‐O‐acetyl‐β‐D ‐ glucuronopyranosyl quillaic acid 28‐O‐β‐D‐xylopyranosyl‐(1–3)‐α‐L‐rhamnopyra‐nosyl‐(1–2)‐[α‐L‐arabinofuranosyl‐(1–3)]‐4‐O‐acetyl‐β‐D‐fucopyranoside (2), 3‐O‐β‐D‐galactopyranosyl‐(1–2)‐β‐D‐glucuronopyranosyl quillaic add 28‐O‐α‐L‐arabinopyranosyl‐(1–3)‐α‐L‐rhamnopyranosyl‐(1–2)‐[α‐L‐arabinofuranosyl‐(1–3)]‐4‐O‐acetyl‐β‐D‐fucopyranoside (3), 3‐O‐β‐D‐galacto‐pyranosyl‐(1–2)‐[β‐D‐xytopyranosyl‐(1–3)]‐β‐D‐glucurono‐pyranosyl quillaic add 28‐O‐β‐D‐xylopyranosyl‐(1–3)‐α‐L‐rhamnopyranosyl‐(1–2)‐[α‐L‐arabinofuranosyl‐(1–3)]‐4‐O‐acetyl‐β‐D‐fucopyranoside (4), respectively.     

Ordering chemical unicyclic graphs by Wiener polarity index

Suppose G is a chemical graph with vertex set V(G). Define D(G) = {{u, v} ⊆ V (G) | d _G (u, v) = 3}, where d _G (u, v) denotes the length of the shortest path between u and v. The Wiener polarity index of G, W _p (G), is defined as the size of D(G). In this article, an ordering of chemical unicyclic graphs of order n with respect to the Wiener polarity index is given.     

A conjecture for some partial differential operators on L 2( R n)

OnX =L ²(R ⁿ), letQ = (Q ₁,Q ₂,…,Q _n) andP = (P ₁,P ₂, …,P _n) be the operators given by (Q _jf) (x) =x _jf(x),P _j = - i∂/∂x _j. For anyC ^∞ functionh:R ⁿ →R putH ₀ =h(P) andH =H ₀ + (1 +Q ²)^-δ, where δ > 1/2. By the method of scattering theory we prove thatH _ac, the absolutely continuous part ofH is unitarily equivalent toH ₀ when (a)n = 1 and (b) forn ≥ 2, whenh is in a large class of polynomials. It is conjectured that the results are true for any polynomialh. We use the techniques of Enss’ method and the idea of bound states for momentum.     

The PI index of phenylenes

The Padmakar–Ivan (PI) index is a graph invariant defined as the summation of the sums of n _eu (e|G) and n _ev (e|G) over all the edges e = uv of a connected graph G, i.e., PI(G) = ∑ _e∈E(G)[n _eu (e|G) + n _ev (e|G)], where n _eu (e|G) is the number of edges of G lying closer to u than to v and n _ev (e|G) is the number of edges of G lying closer to v than to u. An efficient formula for calculating the PI index of phenylenes is given, and a simple relation is established between the PI index of a phenylene and of the corresponding hexagonal squeeze.     

On p -nilpotency and minimal subgroups of finite groups

We call a subgroup H of a finite group G c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ core(H). In this paper it is proved that a finite group G is p-nilpotent if G is S ₄-free and every minimal subgroup of P ∩ G ^N is c-supplemented in N _G(P), and when p = 2 P is quaternion-free, where p is the smallest prime number dividing the order of G, P a Sylow p-subgroup of G. As some applications of this result, some known results are generalized.     

On topological indices of carbon nanocones and nanotori

Let G be a graph with n vertices and d _i is the degree of its ith vertex (d _i is the degree of v _i). In this article, we compute the redefined first Zagreb index, redefined second Zagreb index, redefined third Zagreb index, augmented Zagreb index of graphs carbon nanocones CNC _k[n], and nanotori [C₄C₆C₈(p,q)]. Also, compute the multiplicative redefined first Zagreb index, multiplicative redefined second Zagreb index, multiplicative redefined third Zagreb index, multiplicative augmented Zagreb index of carbon nanocones CNC _k[n], and nanotori [C₄C₆C₈(p,q)].     

Chiral Carbon Nanorings: Synthesis,Properties and Hierarchical Self-assembly of Chiral Ternary Complexes Featuring a Narcissistic Chiral Self-Recognition for Chiral Amines

We report the synthesis and chiroptical properties of novel chiral carbon nanorings S p-/Rp-[12]PCPP containing a planar chiral [2.2]PCP unit, and demonstrate that S p-/Rp-[12]PCPP can not only host crown ether 18-Crown-6 to form ring-in-ring complexes with a binding constant 3.35×10³ M⁻¹, but also accommodate the complexes of 18-Crown-6 and S/R-protonated amines to form homochiral S @Sp -/ R @Rp - and heterochiral S @Rp -/ R @Sp - ternary complexes, displaying significantly larger binding constants of up to 3.31×10⁵ M⁻¹ depending on the chiral guests. Importantly, homochiral S @Sp -/ R @Rp - ternary complexes exhibit an enhanced CD signal, while the heterochiral S @Rp -/ R @Sp - ones have a constant CD signal compared with the chiral carbon nanorings, respectively, which suggests that homochiral S @Sp -/ R @Rp - ternary complexes display a highly narcissistic chiral self-recognition for S/R-protonated chiral amines, respectively. Finally, the chiral ternary complexes can be further applied to determine the ee values of chiral guests. The findings highlight a new application of carbon nanorings in supramolecular sensors, beyond the common recognition of π-conjugated molecules.     

Dirac equation with a Coulomb plus scalar potential in D + 1 dimensions

We generalize the Dirac equation to D + 1‐dimensional spacetime. The exact solutions of the D‐dimensional radial equations with a Coulomb plus scalar potential taking the form 1/r are analytically presented by studying the Tricomi equations. The energies E(n, l, D) are exactly presented. The dependences of the energies E(n, l, D) on the dimension D are analyzed in some detail. The energies E(n, 0, D) first decrease and then increase when increasing dimension D, but the energies E(n, l, D) (l ≠ 0) increase when increasing dimension D. The energies E(n, 0, D) are symmetric with respect to D = 1 for D ∈ (0, 2). It is shown that the energies E(n, l, D) (l ≠ 0) are almost independent of the quantum number l for large D and are completely independent of it if the Coulomb potential is equal to the scalar one. The energies E(n, l, D) almost overlap for large D. The dependences of the energies E(n, l, v) and E(n, l, s) on the vector potential parameter v and scalar potential one s are also studied for D = 3. All are found to decrease when these parameters are increased. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Neutral Pentacoordinate Silicon(IV) Complexes with a Tridentate Dianionic O,N,O or N,N,O Ligand,an Anionic PhX Ligand (X = O,S, Se), and a Phenyl Group: Synthesis and Structural Characterization in the Solid State and in Solution

A series of neutral pentacoordinate silicon(IV) complexes with a SiO₃NC, SiO₂SNC, SiO₂SeNC, SiO₂N₂C, SiOSN₂C, or SiOSeN₂C skeleton was synthesized and structurally characterized by multinuclear NMR spectroscopy in the solid state and in solution and by single‐crystal X‐ray diffraction. The compounds studied contain a tridentate dianionic O,N,O or N,N,O ligand, an anionic PhX ligand (X = O, S, Se), and a phenyl group. The structures, NMR spectroscopic parameters, and chemical properties of these silicon(IV) complexes were compared with those of related compounds that contain a tridentate dianionic S,N,O ligand instead of the O,N,O or N,N,O ligand.     

k-resonant toroidal polyhexes

A toroidal polyhex H(p, q, t) is a cubic bipartite graph embedded on the torus such that each face is a hexagon, which can be described by a string (p, q, t) of three integers (p≥ 1, q≥ 1, 0≤ t≤ p−1). A set of mutually disjoint hexagons of H(p, q, t) is called a resonant pattern if H(p, q, t) has a prefect matching M such that all haxgons in are M-alternating. A toroidal polyhex H(p, q, t) is k-resonant if any i (1 ≤ i ≤ k) mutually disjoint hexagons form a resonant pattern. In [16], Shiu, Lam and Zhang characterized 1, 2 and 3-resonant toroidal polyhexes H(p, q, t) for min(p, q)≥ 2. In this paper, we characterize k-resonant toroidal polyhexes H(p, 1, t). Furthermore, we show that a toroidal polyhex H(p, q, t) is k-resonant (k≥ 3) if and only if it is 3-resonant.      

Approaches for on-line coupling of extraction and chromatography

This review provides an overview of the approaches available in order to perform on-line coupling of various extraction techniques with liquid and gas chromatography, for the analysis of semivolatile and nonvolatile analytes in liquid and solid samples. The main focus is on the instrumental set-up of these techniques. Selected real applications are described by way of illustration. The extraction methods suitable for on-line coupling covered in this review are: liquid-liquid extraction, solid-phase extraction, membrane-based techniques, pressurised liquid extraction, supercritical fluid extraction, and microwave- and sonication-assisted extractions. The following systems are not covered in this review: on-line coupled solid-phase extraction-liquid chromatography, purge-and-trap-GC, and membrane extraction with a sorbent interface-GC.Abbreviations DMAE Dynamic microwave-assisted extraction - DSAE Dynamic sonication-assisted extraction - FIA Flow injection analysis - FID Flame ionisation detection - GC Gas chromatography - HGAAS Hydride generation atomic absorption spectroscopy - IC Ion chromatography - IPLC Ion pair liquid chromatography - LC Column liquid chromatography - LLE Liquid-liquid extraction - LVI Large-volume injection - MAE Microwave-assisted extraction - MESI Membrane extraction with a sorbent interface - MMLLE Microporous membrane liquid-liquid extraction - MS Mass spectrometry - NP Normal-phase - OTT Open-tubular trapping - OTTTD Open-tubular trapping with thermal desorption - PAH Polycyclic aromatic hydrocarbon - PHWE Pressurised hot water extraction - PCB Polychlorinated biphenyl - PLE Pressurised liquid extraction - PTV Programmed-temperature vaporizer - RP Reversed-phase - RSD Relative standard deviation - SAE Sonication-assisted extraction - SFE Supercritical fluid extraction - SIM Selective ion monitoring - SLM Supported liquid membrane - SPE Solid-phase extraction - SPE-TD Solid-phase extraction-thermal desorption - SVE Solvent vapour exit - TD Thermal desorption     

Four New Lignan Glycosides from Osmanthus fragrans Lour. var. aurantiacus Makino

Four new tetrahydrofuranoid lignan glycosides, (7S,8R,7′R,8′S)‐4,9,4′,7′‐tetrahydroxy‐3,3′‐dimethoxy‐7,9′‐epoxylignan 9‐O‐β‐D ‐glucopyranoside ( 2 ), (7R,8S,7′S,8′R)‐4,9,4′,7′‐tetrahydroxy‐3,3′‐dimethoxy‐7,9′‐epoxylignan 9‐O‐β‐D ‐glucopyranoside ( 3 ), (7R,8S,7′R,8′S)‐4,9,4′,9′‐tetrahydroxy‐3,3′‐dimethoxy‐7,7′‐epoxylignan 9‐O‐β‐D ‐glucopyranoside ( 4 ), and rel‐(7R,8S,7′S,8′R)‐4,9,4′,9′‐tetrahydroxy‐3,3′‐dimethoxy‐7,7′‐epoxylignan 9‐O‐β‐D ‐glucopyranoside ( 5 ), and ten known lignan glycosides, 1 and 6 – 14 , were isolated from the leaves of Osmanthus fragrans Lour. var. aurantiacus Makino . Their structures were established on the basis of spectral and chemical studies.     

Influence of substituted groups on the binding ability of benzoyl-modified β-cyclodextrins

A series of substituted benzoyl modified β-cyclodextrins, including mono-6-O-(p-methylbenzoyl)-β-CD (1), mono-6-O-(m-methylbenzoyl)-β-CD (2), mono-6-O-(o-methylbenzoyl)-β-CD (3), mono-6-O-(p-methoxylbenzoyl)-β-CD (4), mono-6-O-(m-methoxylbenzoyl)-β-CD (5), mono-6-O-(o-methoxylbenzoyl)-β-CD (6), mono-6-O-(m, p-dimethoxylbenzoyl)]-β-CD (7), mono-6-O-(o,m-dimethoxylbenzoyl)-β-CD (8), and mono-(6-O-benzoyl)-β-CD (9) were synthesized and their inclusion properties were studied by using fluorescence spectroscopy. The binding constants (K_a) of the modified β-CD derivatives with 2-p-toluidinylnaphthalene-6-sulfonate (TNS) were determined on the basis of the fluorescence spectroscopy. The effect of types and location of substituted groups of the benzene ring of the modified β-cyclodextrins on the binding property was discussed. Results indicated that the substituents had significant influences on the binding abilities of modified β-cyclodextrins.