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).      

Radiation effects of paraffins as polymer model compounds—I. Evolved gas analysis

Radiation yields of gases from n-paraffins of n-C₂₀H₄₂ to n-C₂₄H₅₀ and squalane (C₃₀H₆₂), as polymer model compounds, in the liquid and solid phases were analyzed by gas chromatography. G(H₂) in the liquid phase was 3.2–3.3 for all samples and found to be almost independent of the chemical structure and molecular weight; G(H₂) in the crystalline state of n-paraffins was 2.2–2.5 at -77 to 25°C. G(CH₄) was about 1% of G(H₂) for n-paraffins and increased with the methyl content of the branched chain for squalane. G(C₂H₆) in the liquid phase was about 0.05 for n-paraffins, but G(C₂H₆) in the crystalline state was found to depend on the crystal structure; that is, nearly zero for triclinic of an even number of carbons and about 0.02 for orthorhombic of an odd number. C₃H₈ and C₂H₄(C₃H₆ in squalane) were observed only in the liquid phase of n-paraffins and in glass and the liquid phase of squalane; G(C₃H₈) = 0.03–0.05 and G(C₂H₄ or C₃H₆) = 0.01–0.03. But the C₄-compounds were not detected in any phase of any of the samples.Chain scission by radiation is supposed to proceed mainly at chain end carbons until the third carbon in the liquid phase of n-paraffins, only at the chain end carbon of the crystalline surface in triclinic crystals and at chain end carbons until the second carbon in orthorhombic crystals. These chain scission phenomena in the liquid phase and crystalline state of n-paraffins and in the liquid phase of squalane would be analogous to those in the amorphous and crystalline states of polyethylene, and in amorphous ethylene-propylene copolymer, respectively.     

Gas-phase solvation of protonated aliphatic amines: methyl,ethyl, n-propyl,and iso-propylamine

The thermodynamic quantities K_{n?1 n}, ΔG⁰_{n?1, n} and ΔS⁰_n?1, n for the gas phase equilibrium reactions RNH⁺₃(RNH₂)_n?1 + RNH₂ = RNH⁺₃(RNH₂)_n, where n ? 3 and R indicates an alkyl group (CH₃, C₂H₅, n-C₃H₇ and iso-C₃H₇), have been determined.     

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.     

Microdetermination of aliphatic amines and polyamines with cupric ion: Relationship between pK1 and half-complexation potential

The first equilibrium constant for the combination of ligand with cupric ion, K₁, is related to the half-complexation potential, E_HC, by log K₁ = a + b[E_HC]. The coefficients differ for n ? 2 and n ? 2, where n is the number of nitrogen atoms in the molecule. The correlation coefficients, r, based on data available from the literature were ?0.98. The equations can be used for estimating log K₁ of aliphatic amines. Successful titration of aliphatic amines vs cupric ion requires log K₁ ? 4 and an initial pH of a 10^?3M solution of ?9.     

Theoretical study on the small carbon nano-ladders

1,3-Butadiyne, 1,3,5-hexatriyne, 1,3,5,7-octatetrayne, and 1,3,5,7,9-decapentayne are small oligomeric forms of acetylene. These oligomers participate in cyclization reactions to form ladder-like structures. Enthalpies, ?H, and Gibbs free energies, ?G, of the cyclization reactions were calculated employing MP2 and B3LYP methods. The calculated ?H and ?G were positive, and their variation versus carbon atoms number, n, was fitted in linear functions as ?H(n) = a+bn. The calculations were performed on the structures with carbon numbers up to 20. Also, consecutive cyclization reactions between acetylene molecules were studied. During these consecutive reactions, two different structures, zigzag-ladder-like and cyclic molecules with tetragonal rings, were produced. Among the cyclic structures, the hexagonal form was the most stable structure. The calculated ?H and ?G of formation of zigzag-ladder-like molecules were excellently fitted in linear functions. The obtained functions for ?H and ?G calculated by MP2 method are ?H(n) = 139.67?126.44n and ?G(n) = 80.987?75.684n, respectively.     

Evaluation of the stability of dye heteroassociates formed in aqueous solutions

The additive tetraphenylarsonium-tetraphenylborate model of interactions was found to be applicable to the problem of “preexperimental” evaluation of the stability of associates formed by dye cations (Ct⁺) and anions (An^?) in aqueous solutions. The possibility of predicting equilibrium association constants K _as from preliminarily calculated ΔG(Ct⁺) and ΔG(An^?) and of solving the inverse problem was analyzed. The invariability of the ΔG(Ct⁺) and ΔG(An^?) values and the problem of bringing calculation results in consistency with the experimental K _as values are discussed.     

Ionic solvation in water + co-solvent mixtures. Part 8. Total free energies of transfer and free energies of transfer with the “neutral” component removed of single ions from water into water + acetone

The acid dissociation constants of a wide range of acids in water+acetone mixtures have been combined with values for the free energy of transfer of the proton. ΔG⁰_t(H⁺ to calculate values for the free energy of transfer of ions which derive only from the charge on the ion. ΔG⁰_t(i)_c. As the values of ΔG⁰_t(H⁺) have been revised, revised values for the total free energies of transfer of cations and anions, ΔG⁰_t(M⁺) and ΔG^o_t(X^-), are given. New data for ΔG^o_t(MX_n) is also split into values for ΔG⁰_t(Mⁿ⁺) (where n=1 and 2) and ΔG⁰_t(X^?). These free energies of transfer, both total and those deriving from the charge alone, are compared with similar free energies in other mixtures water+co-solvent. Values for ΔG^o_t(i)_c do not conform to a Born-type relationship and show the importance of structural effects in the solvent even when only the transfer of the charge is involved.     

Experimental study of the D + H2(ν = 1) reaction

The D + H₂(ν = 1) reaction, D + H₂(ν = 1) → K_a HD(ν = 1) + H, → K_n HD(ν = 0) + H, → K_r D + H₂(ν = 0) has been studied. The measurements were made in a flow-tube apparatus at 300 K. Vibrationally excited H₂ was generated in a furnace and D atoms in a microwave discharge. EPR and thermometric techniques were used for the detection of D and H atoms and H₂(ν = 1). The product branching rate constants (in CM³/Molecule s) were found to be K_a = (10.7 ± 4.1) × 10^?13. K_n = (5.4 ± 2.7) × 10^?13, K_r, < 2.7 × 10^?13.     

On the connectivity index of trees

The connectivity index χ₁(G) of a graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. Let T(n, r) be the set of trees on n vertices with diameter r. In this paper, we determine all trees in T(n, r) with the largest and the second largest connectivity index. Also, the trees in T(n, r) with the largest and the second largest connectivity index are characterized. Mei Lu is partially supported by NNSFC (No. 10571105).     

PI indices of pericondensed benzenoid graphs

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., , 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 a class of pericondensed benzenoid graphs consisting of three rows of hexagonal of various lengths.     

On Extremal Unicyclic Molecular Graphs with Prescribed Girth and Minimal Hosoya Index

Let G be an n-vertex unicyclic molecular graph and Z(G) be its Hosoya index, let F _n be the nth Fibonacci number. It is proved in this paper that if G has girth l then Z(G) ≥ F _l+1+(n−l)F _l +F _l-1, with the equality holding if and only if G is isomorphic to , the unicyclic graph obtained by pasting the unique non-1-valent vertex of the complete bipartite graph K _1,n-l to a vertex of an l-vertex cycle C _l . A direct consequence of this observation is that the minimum Hosoya index of n-vertex unicyclic graphs is 2n−2 and the unique extremal unicyclic graph is. The second minimal Hosoya index and the corresponding extremal unicyclic graphs are also determined.     

Sharp Bounds for the Second Zagreb Index of Unicyclic Graphs

The second Zagreb index M ₂(G) of a (molecule) graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we give sharp upper and lower bounds on the second Zagreb index of unicyclic graphs with n vertices and k pendant vertices. From which, and C _n have the maximum and minimum the second Zagreb index among all unicyclic graphs with n vertices, 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.     

Gamma-radiolysis of hydrogen—Oxygen-mixtures— Part I. Influences of temperature,vessel wall,pressure and added gases (N2, Ar,H2) on the reactivity of H2/O2-mixtures

The influence of pressure, temperature, of “matrix gases” N₂, Ar, H₂ and of the pretreatment of the vessel wall on the rate of reaction from ⁶⁰Co γ-radiolysis of hydrogen—oxygen-mixtures, in the region of slow reaction, was investigated. The G(-H₂)-value2 of H₂/O₂-mixtures (H₂:O₂ = 1:9−2:1) ranges from 1 to 14 with only slight dependence on pressure, temperature, H₂/O₂-ratio, and surface/volume ratio (S/V). The temperature has little influence (35–210°C). Replacing most of the O₂ in the H₂:O₂ (1:9)-mixtures with N₂, Ar or excess H₂ at higher temperature, causes the G(-H₂)-values to increase. The influence of these matrix gases increases with increasing temperature (35–210°C) and decreasing S/V ratio (0.59 and 3.8 cm^-1) of the reaction vessel; it depends also on the pretreatment of the wall surface. Varying the total pressure, the G(-H₂)-values show a temperature and gas mixture dependent maximum between about 20 and 200 mb. At higher temperature (210°C) we observed an influence of dose for 50 mb H₂/air-mixtures, whereas at 1 b and 35–90°C no influence of the dose on the rate of reaction of such mixtures was found.The activation by N₂, Ar, H₂ is discussed on the base of the H₂/O₂-reaction being a radical-chain reaction, built up by at least 38 coupled elementary steps (Ref⁽¹⁾ or see part 2). O₂ reacts with H₂, at increased rates of conversions (> 25%), in the expected stoichiometric ratio of 2:1. Oxygen may however also be converted in non-stiochiometric amounts under certain conditions.     

Hosoya polynomials of TUC4C8(R) nanotubes

For a connected graph G, the Hosoya polynomial of G is defined as H(G, x) = ∑_{u,v}?V(G)x^{d(u, v)}, where V(G) is the set of all vertices of G and d(u,v) is the distance between vertices u and v. In this article, we obtain analytical expressions for Hosoya polynomials of TUC₄C₈(R) nanotubes. Furthermore, the Wiener index and the hyper‐Wiener index can be calculated. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Characterisation by rotational spectroscopy of a hydrogen-bonded dimer formed between H2O and HCN

The rotational spectra in the vibrational ground states of (H₂O, HC¹⁴N) and (H₂O, HC¹⁵N) have been assigned in the frequency range 6–19 GHz. Values of rotational constants (B_O, C_O) and centrifugal distortion constants (Δ_J, Δ_JK) have been determined for both species, while the ¹⁴N-nuclear quadrupole coupling constants x_aa and x_bb have been established for the first. Observations concerning additional hyperfine structure arising from H,H nuclear spin-nuclear spin coupling in the H₂O subunit suggest that (H₂O,HCN) has a pair of equivalent protons and is effectively planar in the zero-point state. Observed spectroscopic constants are consistent only with the arrangement H₂O…HCN, with r(O…C) = 3.1387 Å.     

On molecular topological properties of hex‐derived networks

Topological indices are numerical parameters of a molecular graph, which characterize its topology and are usually graph invariant. In quantitative structure–activity relationship/quantitative structure–property relationship study, physico‐chemical properties and topological indices such as Randić, atom–bond connectivity (ABC), and geometric–arithmetic (GA) index are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this area of research. In this paper, we study hex‐derived networks HDN1(n) and HDN2(n), which are generated by hexagonal network of dimension n and derive analytical closed results of general Randić index R_α(G) for different values of α, for these networks of dimension n. We also compute the general first Zagreb, ABC, GA, ABC₄, and GA₅ indices for these hex‐derived networks for the first time and give closed formulae of these degree‐based indices for hex‐derived networks. Copyright © 2016 John Wiley & Sons, Ltd.     

Synthese und Kristallstruktur von K2(HSO4)(H2PO4), K4(HSO4)3(H2PO4) und Na(HSO4)(H3PO4)

Synthesis and Crystal Structure of K₂(HSO₄)(H₂PO₄), K₄(HSO₄)₃(H₂PO₄), and Na(HSO₄)(H₃PO₄) Mixed hydrogen sulfate phosphates K₂(HSO₄)(H₂PO₄), K₄(HSO₄)₃(H₂PO₄) and Na(HSO₄)(H₃PO₄) were synthesized and characterized by X‐ray single crystal analysis. In case of K₂(HSO₄)(H₂PO₄) neutron powder diffraction was used additionally. For this compound an unknown supercell was found. According to X‐ray crystal structure analysis, the compounds have the following crystal data: K₂(HSO₄)(H₂PO₄) (T = 298 K), monoclinic, space group P 2₁/c, a = 11.150(4) Å, b = 7.371(2) Å, c = 9.436(3) Å, β = 92.29(3)°, V = 774.9(4) ų, Z = 4, R₁ = 0.039; K₄(HSO₄)₃(H₂PO₄) (T = 298 K), triclinic, space group P 1, a = 7.217(8) Å, b = 7.521(9) Å, c = 7.574(8) Å, α = 71.52(1)°, β = 88.28(1)°, γ = 86.20(1)°, V = 389.1(8)ų, Z = 1, R₁ = 0.031; Na(HSO₄)(H₃PO₄) (T = 298 K), monoclinic, space group P 2₁, a = 5.449(1) Å, b = 6.832(1) Å, c = 8.718(2) Å, β = 95.88(3)°, V = 322.8(1) ų, Z = 2, R₁ = 0,032. The metal atoms are coordinated by 8 or 9 oxygen atoms. The structure of K₂(HSO₄)(H₂PO₄) is characterized by hydrogen bonded chains of mixed H_nS/PO₄^– tetrahedra. In the structure of K₄(HSO₄)₃(H₂PO₄), there are dimers of H_nS/PO₄^– tetrahedra, which are further connected to chains. Additional HSO₄^– tetrahedra are linked to these chains. In the structure of Na(HSO₄)(H₃PO₄) the HSO₄^– tetrahedra and H₃PO₄ molecules form layers by hydrogen bonds.