Hosoya polynomials of armchair open‐ended nanotubes

For a connected graph G we denote by d(G,k) the number of vertex pairs at distance k. The Hosoya polynomial of G is H(G,x) = ∑_k≥0 d(G,k)x^k. In this paper, analytical formulae for calculating the polynomials of armchair open‐ended nanotubes are given. Furthermore, the Wiener index, derived from the first derivative of the Hosoya polynomial in x = 1, and the hyper‐Wiener index, from one‐half of the second derivative of the Hosoya polynomial multiplied by x in x = 1, can be calculated. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

On reciprocal reverse Wiener index

We report some properties, especially bounds for the reciprocal reverse Wiener index of a connected (molecular) graph. We find that the reciprocal reverse Wiener index possesses the minimum values for the complete graph in the class of n-vertex connected graphs and for the star in the class of n-vertex trees, and the maximum values for the complete graph with one edge deleted in the class of n-vertex connected graphs and for the tree obtained by attaching a pendant vertex to a pendant vertex of the star on n − 1 vertices in the class of n-vertex trees. These results are compared with those obtained for the ordinary Wiener index.     

Terminal Wiener index

Motivated by some recent research on the terminal (reduced) distance matrix, we consider the terminal Wiener index (TW) of trees, equal to the sum of distances between all pairs of pendent vertices. A simple formula for computing TW is obtained and the trees with minimum and maximum TW are characterized.     

Trees with the minimum Wiener number

The Wiener number (𝒲) of a connected graph is the sum of distances for all pairs of vertices. As a graphical invariant, it has been found extensive application in chemistry. Considering the family of trees with n vertices and a fixed maximum vertex degree, we derive some methods that can strictly reduce 𝒲 by shifting leaves. And then, by a process, we prove that the dendrimer on n vertices is the unique graph reaching the minimum Wiener number. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 331–340, 2000     

Theory of reverse scan square-wave voltammetry influenced by the kinetics of reactant adsorption

A model of electrode reaction complicated by slow adsorption of the reactant is developed for square-wave voltammetry with inverse scan direction. The relationship between the dimensionless net peak current and the logarithm of dimensionless rate constant of adsorption is a curve with a minimum and a maximum. For this reason the ratio of real net peak current and the square-root of frequency is a non-linear function of the logarithm of frequency and exhibits either a maximum or a minimum. The frequency of extreme serves for the estimation of the rate constant: log(k _ads /D ^1/2 ) = log(k* _ads )_crit + 0.5 log f _crit , where (k* _ads ) _crit is a critical dimensionless rate constant of adsorption. Square-wave voltammetry is sensitive to the kinetics of adsorption if k _ads < 10² cm s⁻¹      

Conformation of poly(methacrylic acid) in alcohol–water mixtures. III. Dependence on molecular weight

Poly(methacrylic acid) has been studied in 0.002N HCl–ethanol mixtures as a function of molecular weight. A different dependence on molecular weight is noted at different alcohol concentrations. Since the intrinsic viscosity passes through a series of extrema with changes in alcohol concentration, the dependence on molecular weight has been considered in two regions of alcohol concentration in particular. The region of the first minimum and the region of the second minimum (or overall maximum). In the region of the first minimum, intrinsic viscosity is proportional to M^½, just as in 0.002N HCl. The Huggins coefficient k′ is large (ca. 60) but drops to about 10 when the molecular weight exceeds 320,000. In the region of the second minimum the dependence on molecular weight is complex. Intrinsic viscosity is proportional to molecular weight both at low and at high molecular weight and thus indicates freely draining structures. There is a conformational contraction, however, at molecular weight about 320,000 leading from one type of structure to the other. The structure at higher molecular weight may involve a specially strong bond between specifically grouped segments in the chain. The positions of the extrema along the alcohol concentration axis are not molecular weight dependent, particularly above 320,000. Results available for molecular weight dependence in methanol agree well with this picture. The present results confirm prediction inherent in the model of Silberberg and Priel and Silberberg.     

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.     

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).      

Nonstoichiometric polycondensation I. synthesis of polythioether from dibromomethane and 4,4′-thiobisbenzenethiol

High molecular weight poly(phenylene thioether) ( 3 ) was successfully obtained by the polycondensation of 4,4′-thiobisbenzenethiol ( 1 ) and dibromomethane ( 2 ) with a variety of feed ratios in the presence of 1,8-diazabicyclo[5,4,0]undec-7-ene (DBU) in 1-methyl-2-pyrrolidinone (NMP) at 75°C. The resulting polymer showed the maximum inherent viscosity (η_inh) of 0.50 dL/g in 4 h when 1.5 equivalents excess of 2 was used. The model reaction using benzenethiol ( 4 ) and dichloromethane ( 5 ) in the presence of DBU in deuterated dimethylsulfoxide (DMSO-d₆) at 25°C indicated that the rate of the second nucleophilic displacement reaction (k₂) is 61 times faster than that of the first one (k₁). The maximum of theoretical molecular weights calculated at various stoichiometric imbalance (S) under the condition of k₂/k₁ = 61 showed a good agreement with the experimental molecular weights at specific polymerization times.     

Wiener indices of trees and monocyclic graphs with given bipartition

The Wiener index of a connected graph is defined as the sum of distances between all unordered pairs of its vertices. It has found various applications in chemical research. We determine the minimum and the maximum Wiener indices of trees with given bipartition and the minimum Wiener index of monocyclic graphs with given bipartition, respectively. We also characterize the graphs whose Wiener indices attain these values. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

On geometric-arithmetic index

The geometric-arithmetic (GA) index is a newly proposed graph invariant in mathematical chemistry. We give the lower and upper bounds for GA index of molecular graphs using the numbers of vertices and edges. We also determine the n-vertex molecular trees with the minimum, the second and the third minimum, as well as the second and the third maximum GA indices.     

Trees with the same Topological Index JJ

Abstract

In the present paper we investigate the trees with the same JJ index (called JJ-equivalent trees). The topological index JJ is derived from the so called Wiener matrix introduced by Randic et al., in 1994. The Wiener matrix is constructed by generalizing the procedure of Wiener for evaluation of Wiener numbers in alkanes. From such matrices several novel structural invariants of potential interest in structure-property studies were obtained. These include higher Wiener numbers, Wiener sequences, and hyper-Wiener number, etc. The topological index JJ is constructed by considering the row sums of the Wiener matrix. A construction method for a class of JJ-equivalent trees is given. By using this method we construct the smallest pairs of non-isomorphic JJ-equivalent trees which have 18 vertices. In addition we report on groups of 3,4, and 6 non-isomorphic JJ-equivalent trees. The smallest such trees are of size 28 for triples and quadruples, and 34 for the group of 6 elements.     

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.     

Monte Carlo simulation of the chain length distribution in pulsed-laser polymerization experiments in microemulsion

The Monte Carlo method has been used for numerically simulating pulsed-laser polymerization (PLP) in microemulsion, in order to establish if a shift from inflection point to peak maximum as the best measure of the propagation rate constant, k_p, will occur theoretically. Termination is assumed to be instantaneous in the simulations as droplet sizes can be very small in microemulsions. From the results of the simulations it is found that instantaneous termination indeed causes the peak maximum to become the best measure of k_p. From these results it can be deduced that in bulk it is not simply the Poisson-broadening that causes the peak maximum to yield an overestimation of k_p. This overestimation is rather caused by the fact that the termination rate is finite leading to an asymmetrical peak in the molecular weight distribution. In combination with broadening this yields the inflection point to be the best measure of k_p in the bulk.     

Improving kp Data Originating From Higher Order PLP‐SEC Peaks

Summary: Based on certain features, especially the width of the so‐called extra peaks in the simulated chain length distribution (CLD) of polymers prepared by pulsed laser polymerization (PLP), it is calculated by which factor the positions of the higher order points of inflections and maxima deviate from the theoretical L₀ data that are to be used for the evaluation of k_p. These corrections, which can be put into the form of master equations, are for slightly chain length dependent termination by disproportionation or combination and cover a wide range of chain lengths and primary radical production and a reasonable range of axial dispersion σ_ad,k, caused by the chromatographic device used in the evaluation of the chain length distribution. They can be applied either to the point of inflection on the low molecular weight side of the extra peaks as well as to the peak maximum. For usual extents of column broadening (σ_ad,k ≈ 0.05) the mean error that is about 7% for uncorrected data from second order points of inflection is reduced to the order of 1.5% even if no assumption concerning the mode of termination is made. The situation is a little less satisfactory for the correction of the positions of the second order peak maxima. Third order peak data are a priori less falsified and yield still better results after correction. Thus the proper treatment of higher order peaks helps to extend the range of chain lengths for which highly reliable k_p data can be gained from PLP experiments followed by chromatographic analysis.

Plots of lequation/tex2gif-stack-1.gif/(nL₀) versus lg(L₀) obtained from first order (circles), second order (triangles) and third order (squares) peaks showing uncorrected values in the left diagram and corrected values using correction functions X in the right one, both calculated for σ_ad,k = 0.05. (+) and (×) represent ill‐defined peaks.     

New aspects of pseudostationary polymerization and their application

Contrary to the stationary state little thought has been given so far to the general principles of the pseudostationary state. In this discourse an attempt is made to demonstrate that — within wide limits — arbitrary initiation profiles may be used to determine k_p/k_t (k_p = rate constant of chain propagation, k_t = rate constant of chain termination) from the frequency dependence of rate of polymerization (in analogy to the rotating-sector technique) as well as to evaluate k_p from the chain-length distribution (CLD) of samples prepared under pseudostationary conditions. Adverse factors like nonspontaneous transformation of absorbed photons into primary radicals do not invalidate this result. The existence of a universal relationship (independent of the initiation profile) is proved to exist for the second moment of the CLD of samples prepared under pseudostationary initiation conditions for constant (chain-length independent) k_t. Pseudostationarity, however, might be also achieved if not the initiation but the termination is periodically varied. In this case the CLD has a completely different shape but allows determination of k_p likewise. Finally, the case of chain-length dependent k_t is shortly discussed in connection with pulsed-laser initiation. Although the general equation for the second moment of the CLD does not apply any longer for this case some generality appears to exist under these conditions, too.     

On the (im)possibility of evaluating correct individual rate constants of chain propagation kp and chain termination kt by combining kp2/kt and kp/kt data for chain‐length dependent termination

On the basis of simulated data two ways of evaluating individual rate constants by combining k_p²/k_t and k_p /k_t (k_p , k_t = rate constants of chain propagation and termination, respectively) were checked considering the chain‐length dependence of k_t. The first way tried to make use of the fact that pseudostationary polymerization yields data for k_p²/k_t as well as for k_p /k_t referring to the very same experiment, in the second way k_p²/k_t (from steady state experiments) and k_p/k_t data referring to the same mean length of the terminating radical chains were compared. In the first case no meaningful data at all could be obtained because different averages of k_t are operative in the expressions for k_p /k_t and k_p²/k_t. In spite of the comparatively small difference between these two averages (≈15% only) this makes the method collapse. The second way, which can be regarded as an intelligent modification of the “classical” method of determining individual rate constants, at least succeeded in reproducing the correct order of magnitude of the individual rate constants. However, although stationary and pseudostationary experiments independently could be shown to return the same k_t for the same average chain‐length of terminating radicals within extremely narrow limits no reasonable chain‐length dependence of k_t could be derived in this way. The reason is an extreme sensitivity of the pair of equations for k_p/k_t and k_p²/k_t towards small errors and inconsistencies which renders the method unsuccessful even for the high quality simulation data and most probably makes it even collapse for real data. This casts a characteristic light on the unsatisfactory situation with respect to individual rate constants determined in the classical way, regardless of a chain‐length dependence of termination. As a consequence, all efforts of establishing the chain‐length dependence of k_t are recommended to avoid this way and should rather resort to methods based on inserting a directly determined k_p into the equations characteristic of k_p²/k_t or k_p/k_t, properly considering the chain‐length dependent character of k_t.     

The energy of the homogeneous electron gas to second order

The energy of an infinite, homogeneous electron gas is examined by second order perturbation theory using a Hartee-Fock rather than a noninteracting particle unperturbed state. The second order energy still diverges for small promotions k , albert than as ln|ln k| rather than as In k.     

Kinetics of polymer decomposition

Kinetic equations have been developed for polymers that decompose by the zipper mechanism. The usual assumption of steady state kinetics has not been made. Plots of the fraction decomposed α, as a function of time demonstrate a variety of patterns depending upon the relative values of the fraction k₁, of chains becoming activated per second and the fraction k₂ of an activated chain that decomposes per second.