<Emphasis Type="Italic">k</Emphasis>-resonance in Toroidal Polyhexes*

This paper considers the k -resonance of a toroidal polyhex (or toroidal graphitoid) with a string (p, q, t) of three integers (p ≥ 2, q ≥ 2, 0 ≤ t ≤ p − 1). A toroidal polyhex G is said to be k-resonant if, for 1≤ i ≤ k, any i disjoint hexagons are mutually resonant, that is, G has a Kekulé structure (perfect matching) M such that these hexagons are M-alternating (in and off M). Characterizations for 1, 2 and 3-resonant toroidal polyhexes are given respectively in this paper. *This work is supported by FRG, Hong Kong Baptist University; NSFC and TRAPOYT.     

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).     

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.      

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.     

The first and second largest Merrifield–Simmons indices of trees with prescribed pendent vertices

The Merrifield–Simmons index of a graph G is defined as the number of subsets of the vertex set, in which any two vertices are non-adjacent, i.e., the number of independent-vertex sets of G. By T(n,k) we denote the set of trees with n vertices and with k pendent vertices. In this paper, we investigate the Merrifield–Simmons index for a tree T in T(n,k). For all trees in T(n,k), we determined unique trees with the first and second largest Merrifield–Simmons index, respectively.     

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.     

k-Resonance of Open-Ended Carbon Nanotubes

An open-ended carbon nanotube or a tubule is a part of some regular hexagonal tessellation of a cylinder. A tubule T is said to be k-resonant if for every k (or fewer) pairwise disjoint hexagons, the subgraph obtained from T by deleting all the vertices of these hexagons must have a Kekule structure (perfect matching) or must be empty. The 1-resonant tubules can be constructed by an approach provided in H. Zhang and F. Zhang, Discrete Appl. Math. 36 (1992) 291. In this paper, we give the construction method of k(k3)-resonant tubules. The lower bound of its Clar number of k(k3)-resonant tubules is also given. Note that the present paper does not consider the capped species.     

Unicyclic Hückel molecular graphs with minimal energy

The minimal energy of unicyclic Hückel molecular graphs with Kekulé structures, i.e., unicyclic graphs with perfect matchings, of which all vertices have degrees less than four in graph theory, is investigated. The set of these graphs is denoted by such that for any graph in , n is the number of vertices of the graph and l the number of vertices of the cycle contained in the graph. For a given n(n ≥ 6), the graphs with minimal energy of have been discussed. MSC 2000: 05C17, 05C35     

On the Ordering of a Class of Graphs with Respect to their Matching Numbers

Based on the number of k-matching m(G,k) of a graph G, Gutman and Zhang introduced an order relation of graphs: for graphs G ₁ and G ₂, if m(G ₁,k) ≥ m(G ₂,k) for all k. The order relation has important applications in comparing Hosaya indices and energies of molecular graphs and has been widely studied. Especially, Gutman and Zhang gave complete orders of six classes of graphs with respect to the relation . In this paper, we consider a class of graphs with special structure, which is a generalization of a class of graphs studied by Gutman and Zhang. Some order relations in the class of graphs are given, and the maximum and minimum elements with respect to the order relation are determined. The new results can be applied to order some classes of graphs by their matching numbers.     

Thermochemical properties of the Fe-analogue of cryolite, Na3FeF6

Relative enthalpies for low-and high-temperature modifications of Na₃FeF₆ and for the Na₃FeF₆ melt have been measured by drop calorimetry in the temperature range 723–1318 K. Enthalpy of modification transition at 920 K, δ_trans H(Na₃FeF₆, 920 K) = (19 ± 3) kJ mol⁻¹ and enthalpy of fusion at the temperature of fusion 1255 K, δ_fusH(Na₃FeF₆, 1255 K) = (89 ± 3) kJ mol⁻¹ have been determined from the experimental data. Following heat capacities were obtained for the crystalline phases and for the melt, respectively: C _p(Na₃FeF₆, cr, α) = (294 ± 14) J (mol K)⁻¹, for 723 = T/K ≤ 920, C _p(Na₃FeF₆, cr, β) = (300 ± 11) J (mol K)⁻¹ for 920 ≤ T/K = 1233 and C _p(Na₃FeF₆, melt) = (275 ± 22) J (mol K)⁻¹ for 1258 ≤ T/K ≤ 1318. The obtained enthalpies indicate that melting of Na3FeF6 proceeds through a continuous series of temperature dependent equilibrium states, likely associated with the production of a solid solution.      

Extremal hexagonal chains concerning k-matchings and k-independent sets

Denote by _n the set of the hexagonal chains with n hexagons. For any B _{n n} , let m _k (B _n ) and i _k (B _n ) be the numbers of k-matchings and k-independent sets of B _n , respectively. In the paper, we show that for any hexagonal chain B _{n n} and for any k0, m _k (L _n )m _k (B _n )m _k (Z _n ) and i _k (L _n )i _k (B _n )i _k (Z _n ), with left equalities holding for all k only if B _n =L _n , and the right equalities holding for all k only if B _n =Z _n , where L _n and Z _n are the linear chain and the zig-zag chain, respectively. These generalize some related results known before.     

Determining sulfamonomethoxine and its acetyl/hydroxyl metabolites in chicken plasma under organic solvent-free conditions

A quantitative technique is described for a sample preparation followed by high performance liquid chromatography method for the simultaneous determination of sulfamonomethoxine and its metabolites, N ⁴-acetyl SMM and 2,6-dihydroxy SMM, in chicken plasma. The average recoveries, analytical total time, and limits of quantitation were ≥80% (relative standard deviations (SD) ≤6%), <30 min sample-1 (12 samples in 2 h), and ≤0.09 μg ml⁻¹, respectively. The procedure, performed under 100% aqueous conditions, uses no organic solvents and toxic reagents at all and is, therefore, harmless to the environment and humans.      

Embedding on alphabet overlap digraphs

Alphabet overlap digraphs can be viewed as a generalization of directed de Bruijn graphs. Given three integers α ≥ 1, k ≥ 2 and 1 ≤ i < k, the alphabet overlap digraph O(α, k ; i) is a digraph: the set of all words of length k over a certain alphabet with cardinality α is vertex set, and there is an arc from a vertex u to a vertex v if and only if the word of last k−i letters of u coincides with the word of first k−i letters of v. In this paper, we consider whether O(α, k ; i) can be embedded in O(α, k ; j) for given integers 1 ≤ i < j < k. In order to resolve this problem, we give an O(1)-time algorithm to decide whether there exists a permutation on {1, . . . ,k} from O(α, k ; i) to O(α, k ; j). If such a permutation exists, for any vertex of O(α, k ; i), we apply the permutation to change its label’s position and map it to a vertex of O(α, k ; j). Furthermore, we obtain an embedding from O(α, k ; i) to O(α, k ; j). Hence, we solve partly the problem. As a consequence, we show that every directed de Bruijn graph can be embedded in all alphabet overlap digraphs with the same parameters α and k.     

Integral invariants for high-symmetry open-shell molecules

A method for calculating the energies of electronic states arising from a degenerate open shellγ ^N in terms of integral invariants H^k(γ,γ) is presented. The calculation proceeds from expansions for the electron repulsion integrals 〈mm/nn〉 on degenerate orbitals ofγ symmetry in terms of H^k(γ,γ). The energies of states for theγ ^N electronic configurations with dimγ≤3 (e^N and t^N configurations) are tabulated. Institute of Catalysis, Siberian Branch, Russian Academy of Sciences. Translated fromZhurnal Strukturnoi Khimii, Vol. 39, No. 2, pp. 183–195, March–April, 1998. This work was supported by RFFR grants No. 96-03-01167 and 96-03-34035.     

Crystal structure of [Mn(1,10-C12H8N2)3](B6H7)2

A new compound [MN^II(Phen)₃]²⁺(B₆H₇)₂⁻ is synthesized; its crystal structure is studied by XRD at 100 K. Crystallographic data: C₃₆H₃₈B₁₂N₆Mn, M = 739.39, triclinic symmetry, space group P , unit cell parameters: a = 10.3131(3) ?, b = 13.4839(4) ?, c = 15.1132(4) ?; α = 97.696(1)°, β = 108.324(1)°, γ = 102.211(1)°; V = 1903.9(1) ?³, Z = 2, d _calc = 1.290 g/cm³. The structure is solved by direct and Fourier methods and refined by full-matrix LSM in the anisotropic (isotropic for hydrogen atoms) approximation to the final factor R ₁ = 0.036 for 10169 I _hkl ≥ 2σ _I (Bruker-Nonius X8 APEX CCD diffractometer, λMoK _α). The structure contains two crystallographically different anions. Original Russian Text Copyright ? 2009 by T. M. Polyanskaya, M. K. Drozdova, V. V. Volkov, and K. G. Myakishev __________ Translated from Zhurnal Strukturnoi Khimii, Vol. 50, No. 2, pp. 381–385, March–April, 2009.     

Transport numbers and ionic conduction of eutectic methacomposites {(1 ? x )MeWO4 · x WO3}) (Me = Sr, Ba; x = 0?0.55)

Transport numbers of oxygen ions, , in methacomposites (1 − x)MeWO₄ · xWO₃, where Me = Sr and Ba and x = 0−0.55, are determined in the temperature interval 600 to 900°C by a method of the emf of an oxygen-air galvanic cell. It is demonstrated that the region of small contents of the additive (x ≤ 0.2) is predominantly characterized by oxygen-ion conduction ( = 1), which gives way to electronic conduction (t _e = 1) at x > 0.35. It is confirmed once again that subeutectic compositions (1 − x)MeWO₄ · xWO₃ where Me = Sr and Ba and x = 0−0.2 belong in the class of ion-conducting methacomposites. The threshold of percolation of electronic conduction (t _e ≥ 0.5, < 0.5) occurs at x _t ≥ 0.3. Dependences of the transport numbers of the oxygen ions on the volume ratio between components in both composites resemble one another; specifically, the threshold composition contains nearly 20 vol % of WO₃. The dramatic amplification (by 1–1.5 orders of magnitude) of the ionic conductivity in the methacomposites occurs at small contents of tungstic oxide (x ≤ 0.01). A chemical transport removal of excess tungstic oxide, which is segregated in the form of the surface compound MeW-s, from the surface of the MeWO₄ grains destroys MeW-s, leading to a 10–15-fold drop of the ionic conductance. At x ≥ 0.05, the oxygen-ion conductance in the methacomposites is practically independent of their composition. A model for the formation and architecture of the methacomposites is qualitatively modified. The modified model takes into account doubled surface activity and mobility of the MeW-s phase with respect to MeWO₄ and WO₃. Original Russian Text ? N.N. Pestereva, A.Yu. Zhukova, A. Ya. Neiman, 2007, published in Elektrokhimiya, 2007, Vol. 43, No. 11, pp. 1379–1386.     

Interactions in the subsolidus region of the Na2MoO4-NiMoO4-Fe2(MoO4)3 system

Phase relations have been investigated in the subsolidus region of the Na₂MoO₄-NiMoO₄-Fe₂(MoO₄)₃ system by X-ray diffraction, differential thermal analysis, and vibrational spectroscopy. The phase of variable composition Na_1−x Ni_1−x Fe_1+x (MoO₄)₃(0≤x≤0.5) with the NASICON structure (space group R c) and the NaNi₃Fe(MoO₄)₅ ternary molybdate crystallizing in the triclinic crystal system (space group P ) have been obtained. A high conductivity was found in Na_1−x Ni_1−x Fe_1+x (MoO₄)₃, which allows one to consider this phase of variable composition as a promising solid electrolyte with sodium ion conduction. Original Russian Text ? N.M. Kozhevnikova, A.V. Imekhenova, 2009, published in Zhurnal Neorganicheskoi Khimii, 2009, Vol. 54, No. 4, pp. 695–700.     

Synthesis and properties of iron(II) and iron(III) complexes with 3-(α,γ-dicarboxy-n-propylidene-hydrazino)-5,6-diphenyl-1,2,4-triazine (DCPT)

Summary Proton-ligand formation constants ofDCPT and metal-ligand formation constants of its complexes with Fe(II) and Fe(III) have been determined potentiometrically at 10, 20, 30, and 40°C in 75% (v/v) dioxane-water at 0.10M ionic strength (KNO₃). The thermodynamic parameters G, H, and S have also been evaluated. The possibility of formingM(HL),M(HL)₂, andM(HL)₃ species was substantiated from potentiometric and electronic absorption measurements. The values of the stability constants logK _M(HL), log , and log derived from the spectrophotometric method are in good agreement with those obtained from potentiometric data. The use ofDCPT as an analytical reagent for the spectrophotometric determination of iron is discussed. The solid complexes have been characterized by chemical analysis and magnetic susceptibility, IR, NMR, and electronic spectral data.

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Synthese und Eigenschaften von Eisen(II)- und Eisen(III)-Komplexen mit 3-(,-Dicarboxy-n-propylidene-hydrazino)-5,6-diphenyl-1,2,4-triazin (DCPT)

Zusammenfassung Die Bildungskonstanten sowohl vonDCPT als auch seiner Komplexe mit Fe(II) und Fe(III) wurden bei 10, 20, 30 und 40°C in Dioxan-Wasser (75% (v/v)) bei einer lonenstärke von 0.1M KNO₃ potentiometrisch bestimmt. Zusätzlich wurden die thermodynamischen Parameter G, H und S ermittelt. Die Wahrscheinlichkeit des Auftretens von Komplexen der ArtM(HL),M(HL)₂ undM(HL)₃ wurde mittels potentiometrischer und absorptionsspektroskopischer Messungen erhärtet. Nach beiden Methoden ermittelte Stabilitätkonstanten stimmen gut überein. Die Verwendung vonDCPT als analytisches Reagens zur spektrophotometrischen Eisenbestimmung wird diskutiert. Die Komplexe konnten in Substanz hergestellt werden und wurden durch chemische Analyse und über ihre magnetischen und spektroskopischen Eigenschaften charakterisiert.