A Linear Time Algorithm for Constructing Maximally Symmetric Straight Line Drawings of Triconnected Planar Graphs

Symmetry is one of the most important aesthetic criteria in graph drawing because it reveals structure in the graph. To draw graphs symmetrically, we employ two steps. The first step is to find appropriate automorphisms. The second step is to draw the graph to display the automorphisms. Our aim in this paper is to construct maximally symmetric straight line drawings of triconnected planar graphs in linear time. Previously known algorithms run in quadratic time. We show that an algorithm of Fontet can be used to find an embedding in the plane with the maximum number of symmetries, and present a new algorithm for finding a straight line drawing that achieves that maximum. Both algorithms run in linear time.     

The asymptotic numbers of regular tournaments,Eulerian digraphs and Eulerian oriented graphs

LetRT(n), ED(n) andEOG(n) be the number of labelled regular tournaments, labelled loop-free simple Eulerian digraphs, and labelled Eulerian oriented simple graphs, respectively, onn vertices. Then, asn,, for any>0. The last two families of graphs are also enumerated by their numbers of edges. The proofs use the saddle point method applied to appropriaten-dimensional integrals.     

Equilibrium studies of the sorption of Cu(II) ions onto chitosan

Chitosan is a polymer that can be obtained from the shells of seafood such as prawns, crabs, and lobsters. Chitosan has free amino groups, which can attract metal ions, and has been used as an adsorbent for the removal of metal ions from effluents. In this research, the sorption of copper ions from solution onto chitosan at two pHs has been investigated. DSC, TGA, surface area, SEM, and NMR studies have been used to report the pure physical states of chitosan and the chitosan-copper complex. The experimental isotherm data were analyzed using the Langmuir, Freundlich, and Redlich-Peterson equations. Correlation coefficients were determined for each isotherm analysis. Error functions have been used to determine the alternative single component parameters by nonlinear regression due to the inherent bias in using the correlation coefficient resulting from linearization. The error function method provided the best parameters for the isotherm equations in this system and is demonstrated for error comparison purposes.     

A Branched Pore Model Analysis for the Adsorption of Acid Dyes on Activated Carbon

A new branched-pore adsorption model has been developed using an external mass transfer coefficient, K _f, an effective diffusivity, D _eff, a lumped micropore diffusion rate parameter, K _b, and the fraction of macropores, f, to describe sorption kinetic data from initial adsorbent-adsorbate contact to the long-term adsorption phase. This model has been applied to an environmental pollution problem—the removal of two dyes, Acid Blue 80 (AB80) and Acid Red 114 (AR114), by sorption on activated carbon. A computer program has been used to generate theoretical concentration-time curves and the four mass transfer kinetic parameters adjusted so that the model achieves a close fit to the experimental data. The best fit values of the parameters have been determined for different initial dye concentrations and carbon masses. Since the model is specifically applicable to fixed constant values of these four parameters, a further and key application of this project is to see if single constant values of these parameters can be used to describe all the experimental concentration-time decay curves for one dye-carbon system.The error analysis and best fit approach to modeling the decay curves for both dye systems show that the correlation between experimental and theoretical data is good for the fixed values of the four fitted parameters. A significantly better fit of the model predictions is obtained when K _f, K _b and f are maintained constant but D _eff is varied. This indicates that the surface diffusivity may vary as a function of surface coverage.     

Study of the cerium(IV)-picrate system in acetonitrile

A potentiometric and spectrophotometric study has been made of the reaction between hexanitratocerate and picrate in dry acetonitrile. Several cerium(IV)-picrate complexes are formed; the formation constant for the first is estimated to be 4 from spectrophotometric measurements. The catalytic effect of picrate on hydroquinone oxidation by nitratocerate is postulated to be due to more rapid electron transfer by cerium picrate complexes.     

Composition and thermal stability of SiO2-based hybrid materials TEOS-MTEOS system

Hybrid materials with different amounts of organics permanently bound on the inorganic network obtained in the TEOS-MTEOS (tetraethoxysilan-methyltriethoxysilan) system are used for obtaining coatings with different optical and mechanical properties. To study the thermal stability of the mentioned materials, compositions with different molar ratios of the precursors were prepared. The influence of the solvent and water amounts on the gelation process was also investigated. The gels obtained were characterised by IR spectrometry and their decomposition temperatures were determined by DTA/TG. Thermal stability of the gels is rather influenced by their composition than the conditions of the gelation process. This revised version was published online in July 2006 with corrections to the Cover Date.     

Separation of rhodium from iridium with sodium borohydride and standardization of rhodium(III) solutions

Of the platinum group metal separations, that of rhodium from iridium is the most difficult. The existing gravimetric methods are too lengthy or make use of organic reagents which ultimately need to be removed before iridium can be determined. The proposed method of separation is rapid, needs no pH control, and easy to carry out. Rh(III) ions are quantitatively reduced to Rh(0) by the action of aqueous sodium borohydride. The separation is best achieved in perchlorate medium in the presence of hydroxylamine. The separation is dependent on the concentration ratio of iridium to rhodium; if this is high, some iridium is co-precipitated; if low, the rhodium obtained is free from even spectrographic traces of iridium. A new method for standardization of Rh(III) solutions with sodium borohydride is proposed.     

Reaction studies of carbon clusters

Reaction studies of carbon clustersC _n in the rangen=8–37, produced by laser vaporisation in a supersonic nozzle, have been investigated using time-of-flight mass spectrometry. Clear differences in reaction products formed on hydrogenation are detected which show that even and odd clusters behave quite differently and furthermore thatat least two different types of even cluster appear to exist. The reactivity patterns for clusters C _n withn=16, 18 and 22 are in a different class from those withn=20, 24, 26 ..., a behaviour consistent with the existence of closed cage fullerene structures for even clusters with 20 or more carbon atoms (other thann=22).