Characterizations for Some Types of DNA Graphs

Vertex induced subgraphs of directed de Bruijn graphs with labels of fixed length k and over α letter alphabet are (α,k)-labelled. DNA graphs are (4,k)-labelled graphs. Pendavingh et al. proved that it is NP-hard to determine the smallest value α _k (D) for which a directed graph D can be (α _k (D),k)-labelled for any fixed . In this paper, we obtain the following formulas: and for cycle C _n and path P _n . Accordingly, we show that both cycles and paths are DNA graphs. Next we prove that rooted trees and self-adjoint digraphs admit a (Δ,k)-labelling for some positive integer k and they are DNA graphs if and only if Δ ≤ 4, where Δ is the maximum number in all out-degrees and in-degrees of such digraphs.     

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).      

The Hydrolysis of AuCl<Stack><Subscript>4</Subscript><Superscript>−</Superscript></Stack> and the Stability of Aquachlorohydroxocomplexes of Gold(III) in Aqueous Solution

The equilibria AuCl₄⁻+jOH⁻+kH₂O⇌AuCl_4−j−k (OH) _j (H₂O) _k ^k−1+(j+k)Cl⁻, β _jk (0≤j,k≤4) have been studied spectrophotometrically at 20 °C in aqueous solution. For I=2 mol⋅dm⁻³(HClO₄) the conventional constants, β _i ^*, of the equilibria, Au^*+iCl⁻ ⇌AuCl _i ^*, are equal to log ₁₀ β ₁^*=(6.98±0.08); log ₁₀ β ₂^*=(13.42±0.05); log ₁₀ β ₃^*=(19.19±0.09); and log ₁₀ β ₄^*=(24.49±0.07), where [AuCl _i ^*]=∑[AuCl _i (OH) _j (H₂O)_4−i−j ] at i=const. The hydrolysis and other transformations of AuCl₄⁻ in aqueous solution are discussed. On the basis of new and known data, a full set of equilibrium constants, β _jk , or their estimates has been obtained.     

Dependence of the relative retention of compounds on the average pressure of helium as a carrier gas in capillary GLC

The dependence of retention factork _i , relative retention time α _i , and retention indexI _i of organic compounds on the average pressure (p _av) of the carrier gas (helium) was studied experimentally using a long narrow-bore capillary column with the SE-30 nonpolar phase at 120°C. The linear dependencesk _i =f(p _av), α _i =φ(p _av), andI _i =φ(p _av) obtained previously were found to be in good agreement with experimental data. Invariant relative retention valuesk _0,i , α _0,i , andI _0,i , which do not depend on the helium pressure, were determined for some organic compounds of various chemical classes. The dependence of the relative retention on the carrier gas pressure needs to be taken into account in precision measurements and in experiments with narrow-bore capillary columns. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 2, pp. 314–316, February, 1998.     

Phenomenological kinetics of irreversible electrochemical dissolution of metal-oxide microparticles

The electrochemical dissolution of metal oxides and other stable solid phases composed of nano- to micro-crystalline particles is generally a complex process. It can be simplified by distinguishing two main contributions to the reactivity of the solid: the potential-dependent rate coefficient k(E), and the conversion-dependent function f(y). These contributions can be evaluated by a combination of potentiostatic and potentiodynamic experiments. Both k(E) and f(y) were obtained experimentally for the dissolution of iron and chromium oxides, and theoretical consequences of their particular forms are discussed. A peak-shaped function k(E) was observed in the case of γ-Fe₂O₃, whereas, for α-Cr₂O₃ and CrO₂, a different model based on intermediate surface complexes is proposed. This model also explains the complete electrochemical dissolution of metal oxides regardless of their low intrinsic electric conductivity. Received: 2 July 1997 / Accepted: 3 November 1997     

Solution Stoichiometry Control for Pure LiLaMo2O8Phases in Sol-Gel Preparation

The hydrolysis of solutions, containing either LiMo₂O₄(OPr ⁱ )₅( ⁱ PrOH) and “La(OPr ⁱ )₃” or LiOPr ⁱ and La₂Mo₄O₈(OPr ⁱ )₁₄ in 1∶1 ratio, in toluene or ⁱ PrOH, by water solutions in isopropanol, leads to formation of monolythic gels. The latter can be converted by drying and heat treatment at 600°C into a fine powder of pure α-LiLaMo₂O₈ phase, which is transformed into β-LiLaMo₂O₈ powder by annealing at 800°C. The sintering of the pellets pressed under 4 bar pressure of α-LiLaMo₂O₈ powder at 800°C for 2 h converts them into pieces of β-LiLaMo₂O₈ transparent ceramics, thus providing a route to LiLnMo₂O₈ laser waveguide materials.     

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.      

Influence of Substance on the Equilibrium Translation Rate for the Isothermal and Isobaric Chemical Reactions of Ideal gases

To increase inert substance i will make the equilibrium translation rate α _j of reactant j decrease if ∑ _i ν _i < 0 or increase if ∑ _i ν _i > 0. When or , to increase non-inert substance i will make α _j increase if i is reactant (i ≠ j) or decrease if i is resultant. When has maximum if i is reactant (i≠ j) or minimum if i is resultant. If i is reactant, (x _r ⁰ is “optimum proportion” of reactant)     

On the spectral radius of graphs with connectivity at most <Emphasis Type="Italic">k</Emphasis>

In this paper, we study the spectral radius of graphs of order n with κ(G) ≤ k. We show that among those graphs, the maximal spectral radius is obtained uniquely at , which is the graph obtained by joining k edges from k vertices of K _n-1 to an isolated vertex. We also show that the spectral radius of will be very close to n − 2 for a fixed k and a sufficiently large n.     

On Unicycle Graphs with Maximum and Minimum Zeroth-order Genenal Randić Index

Let G be a graph and d _v denote the degree of the vertex v in G. The zeroth-order general Randić index of a graph is defined as R _α⁰(G) = ∑ _v∈V(G) d _v ^α where α is an arbitrary real number. In this paper, we obtained the lower and upper bounds for the zeroth-order general Randić index R _α⁰(G) among all unicycle graphs G of order n. We give a clear picture for R _α⁰(G) of unicycle graphs according to real number α in different intervals.     

Acetate-Functionalized Zirconium-Substituted Tungstogermanate, [Zr4O2(OH)2(CH3COO)2(α-GeW10O37)2]12−

The two acetate-functionalized zirconium(IV)-substituted tungstogermanates, Na₈K₄[Zr₄O₂(OH)₂(CH₃COO)₂(α-GeW₁₀O₃₇)₂] · 33H₂O and Na₈Cs₄[Zr₄O₂(OH)₂(CH₃COO)₂(α-GeW₁₀O₃₇)₂] · 32H₂O, were synthesized by the reaction of ZrOCl₂ with [A-α-GeW₉O₃₄]¹⁰⁻ in pH = 4.8 buffer and their structures were determined by single-crystal X-ray analysis. Both of them contain a centrosymmetric polyanion [Zr₄O₂(OH)₂(CH₃COO)₂(α-GeW₁₀O₃₇)₂]¹²⁻ consisting of two {α-GeW₁₀O₃₇} units sandwiching an inorganic–organic hybrid {Zr₄O₂(OH)₂(CH₃COO)₂} cluster. The polyanion contains a mixing of seven- and eight-coordinate Zr centres. The two compounds were also characterized by elemental analysis, IR spectroscopy, UV–vis and TG–DSC analysis.     

On the Randić Index

The Randić index of an organic molecule whose molecular graph G is defined as the sum of (d(u)d(v))^−1/2 over all pairs of adjacent vertices of G, where d(u) is the degree of the vertex u in G. In Delorme et al., Discrete Math. 257 (2002) 29, Delorme et al gave a best-possible lower bound on the Randić index of a triangle-free graph G with given minimum degree δ(G). In the paper, we first point out a mistake in the proof of their result (Theorem 2 of Delorme et al., Discrete Math. 257 (2002) 29), and then we will show that the result holds when δ(G)≥ 2.     

Sandwich-type Uranium-Substituted of Bismuthotungstate: Synthesis and Structure Determination of [Na(UO2)2(H2O)4(BiW9O33)2]13?

The sandwich-type [Na(UO₂)₂(H₂O)₄(BiW₉O₃₃)₂]¹³⁻ uranium (VI) has been synthesized by reacting the trivacant species of B-α-[BiW₉O₃₃]⁹⁻ with and investigated by IR and UV–Vis spectroscopy, and elemental analysis. The X-ray single crystal analysis was carried out on Na₁₃[Na(UO₂)₂(H₂O)₄(BiW₉O₃₃)₂] · 33H₂O (I) which crystallizes in the orthorhombic system, space group Pna2₁ with a = 33.8454(19) ?, b = 21.1484(12) ?, c = 13.2403(7) ?, α = 90°, β = 90°, γ = 90°, and Z = 4. The polyanion consists of two lacunary B-α-[BiW₉O₃₃]⁹⁻ groups which sandwich two uranyl cations and one sodium cation. The uranium atoms adopt distorted pentagonal–bipyramidal coordination, achieved by two equatorial bonds to each BiW₉O₃₃ unit and one external water ligand. The coordination of each uranium atom is evident by the shift of ν_as(W–O_b–W) and ν_as(Bi–O) stretching vibrational bonds. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Minimum general sum-connectivity index of unicyclic graphs

The general sum-connectivity index of a graph G is defined as χ _α (G) = ∑_edges (d _u + d _v ) ^α , where d _u denotes the degree of vertex u in G and α is a real number. In this report, we determine the minimum and the second minimum values of the general sum-connectivity indices of n-vertex unicyclic graphs for non-zero α ≥ −1, and characterize the corresponding extremal graphs.     

Theoretical treatment of predissociation of the CO (3sσ) B and (3pσ) C1Σ+ Rydberg states based on a rigorous adiabatic representation

Ab initio multireference configuration interaction calculations for adiabatic potential curves, nonadiabatic couplings 〈φ _i (R,r)|d/dR|φ _j (R,r)〉 and 〈φ _i (R,r)|d²/dR ²|φ _j (R,r)〉, and nuclear kinetic energy corrections 〈dφ _i (R,r)/dR|dφ _i (R,r)/dR〉 for the (3sσ) B and (3pσ) C¹Σ⁺ Rydberg states of the CO molecule have been carried out. The energy positions and predissociation linewidths for the observed vibrational levels of these two states have been determined in a rigorous adiabatic representation by the complex scaling method employing a basis of complex scaled harmonic vibrational functions in conjunction with the Gauss-Hermite quadrature method to evaluate the complex Hamiltonian matrix elements. The present treatment correctly reproduces the observed trends in energies and line broadening for vibrational levels of the B¹Σ⁺ state and represents an improvement over the previous treatment in literature. The errors in the determined spacings of the v = 0–4 vibrational levels of the C¹Σ⁺ state are less than 2% compared with measured data. The predissociation linewidths for the v=3,4 levels of the C¹Σ⁺ state are found to be 4.9 and 8.9 cm⁻¹, respectively, in good agreement with the observed values. Received: 23 March 1998 / Accepted: 27 July 1998 / Published online: 9 October 1998     

Equilibria in the system of aquachlorohydroxogold(III) complexes in aqueous solution

AuCl₄⁻ + jOH⁻ + kH₂O = AuCl_{4 − j − k} OH _j (H₂O) _k ^{k − 1} + (j + k)Cl⁻ equilibria at 20°C were studied spectrophotometrically, and the constants β _jk in acid aqueous solutions were determined for I = 2.0 mol/L (HClO₄).     

Effect of Ionic Strength and Temperature on the Protonation of Oxidized Glutathione

The protonation constants for oxidized glutathione, H _i−1L^(4−i+1)−, K _i ^H=[H _i L⁽⁴⁻ⁱ⁾⁻]/[H _i−1L^(4−i+1)−][H⁺] i=1,2,…,6 have been measured at 5, 25 and 45 °C as a function of the ionic strength (0.1 to 5.4 mol⋅[kg(H₂O)]⁻¹) in NaCl solutions. The effect of ionic strength on the measured protonation constants has been used to determine the thermodynamic values (K _i ^H0) and the enthalpy (ΔH _i ) for the dissociation reaction using the SIT model and Pitzer equations. The SIT (ε) and Pitzer parameters (β ⁽⁰⁾, β ⁽¹⁾ and C) for the dissociation products (L⁴⁻, HL³⁻, H₂L²⁻, H₃L⁻, H₄L, H₅L⁺, H₆L²⁺) have been determined as a function of temperature. These results can be used to examine the effect of ionic strength and temperature on glutathione in aqueous solutions with NaCl as the major component (body fluids, seawater and brines).     

New polyhydroxysteroids and steroid glycosides from the far east starfishCeramaster patagonicus

Two new polyhydroxysteroids and five new glycosides were isolated from the starfishCeramaster patagonicus and their structures were elucidated: 5α-cholestane-3β,6α,15β,16β,26-pentol, (22E)-5α-cholest-22-ene-3β,6α,8,15α,24-pentol, (22E)-28-O-[O-(2-O-methyl-β-d-xylopyranosyl)-(1→2)-β-d-galactofuranosyl]-24-hydroxymethyl-5α-cholest-22-ene-3β,4β, 6α,8,15β,16β,28-heptol (ceramasteroside C₁), (22E)-28-O-[O-(2,4-di-O-methyl-β-d-xylopyranosyl)-(1→2)-β-d-galactofuranosyl]-24-hydroxymethyl-5α-cholest-22-ene-3β, 6α,8,15β,16β,28-hexol (ceramasteroside C₂), (22E)-28-O-[O-methyl-β-d-xylopyranosyl)-(1→2)-β-d-galactofuranosyl]-24-hydroxymethyl-5α-cholest-22-ene-3β,6α,8,15β,16β 28-hexol (eramasteroside C₃), (22E)-28-O-[O-(2-O-methyl-β-d-xylopyranosyl)-(1→2)-β-d-galactofuranosyl]-24-methyl-5α-cholest-22-ene-3β,4β,6α,8, 15β, 26-hexol (ceramasteroside C₄), and (22E)-28-O-[O-(2-O-methyl-β-d-xylopyranosyl)-(1→2)-β-d-xylopyranosyl]-5α-cholest-22-ene-3β,6α,8,15β,24-pentol (ceramasteroside C₅)). Three known polyhydroxysteroids (24-methylene-5α-cholestane-3β,6α,8,15β,16β,26-hexol, 5α-cholestane-3β,6α,8,15β,16β,26-hexol, and 5α-cholestane-3β,6β,15α,16β,26-pentol) were also isolated. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 1, pp. 190–195, January, 1997.