The vertex-cover polynomial of a graph

In this paper we define the vertex-cover polynomial Ψ(G,τ) for a graph G. The coefficient of τ^r in this polynomial is the number of vertex covers V′ of G with |V′|=r. We develop a method to calculate Ψ(G,τ). Motivated by a problem in biological systematics, we also consider the mappings f from {1, 2,…,m} into the vertex set V(G) of a graph G, subject to f⁻¹(x)f⁻¹(y)≠ for every edge xy in G. Let F(G,m) be the number of such mappings f. We show that F(G,m) can be determined from Ψ(G,τ).     

A tight upper bound for the number of intersections between two rectangular paths

The problem of finding the number of intersections between two geometric figures in the plane has been studied extensively in literature. In this paper, the geometric figure comprising a continuous rectilinear path (called rectangular path) is considered, and a tight (least) upper bound onI(P, Q), the number of intersections between two rectangular pathsP andQ, is given.Editors' Note: One of our referees has reported that the main result of this paper has recently been given independently by a Chinese researcher at the University of Science and Technology, Hefei, P. R. of China. His paper is under publication in the Chinese Science Bulletin. However, since this journal may not be easily accessible to our readers, and further the two papers are obviously independent of each other, theBIT Editors have decided to accept the present paper.     

Packing two graphs of ordern having total size at most 2n − 2

Two graphsG andH of the same order are packable ifG can be embedded in the complement ofH. In this paper we give a complete characterization of two graphs of ordern having total size at most 2n – 2 which are packable. This result extends an earlier result of B. Bollobás and S.E. Eldridge.     

Crystallinity of ethylene oxide oligomers

Nonaethylene glycol and pentadecaethylene glycol and their dimethyl ethers have been prepared and characterized, with respect to crystallinity by wide- and small-angle X-ray scattering, Raman scattering, i.r. spectroscopy and differential scanning calorimetry. Wide-angle X-ray scattering is similar to that from high molecular weight poly(ethylene oxide). The crystal habit is lamella. The lamellae are highly crystalline and the surface layers are ordered. Comparison with crystalline poly(ethylene oxide) prepared conventionally, and having a distribution of chain lengths, shows that such samples crystallize into lamellae with disordered surface layers.     

Synthesis and Structure of [DB24C8)Na][Cd(SCN)3. Formation of a novel linear Cd...Cd...Cd chain with a mer-CdN3S3 coordination confirmation and new coiled [(DB24C8)Na]+ cation

We report herein a novel coordination solid, [(DB24C8)Na][Cd(SCN)3] (6) (DB24C8 denotes dibenzo-24-crown-8), which exhibits a new type of [Cd(SCN)3-]infinity chain with two unusual stereochemical characteristics: (1) a mer-CdN3S3 coordination and (2) a linear Cd chain with a Cd...Cd...Cd angle of 180 degrees. In addition, the [(DB24C8)Na]+ monocation adopts a new structural type-a coiled structure-for the combination of crown ether DB24C8 and alkali metal Na+. The title compound crystallizes in a monoclinic unit cell of C2/c space group symmetry with lattice parameters a = 16.110(8) A, b = 20.380(5) A, c = 11.01(1) A, beta = 119.87(3) degrees, and Z = 4. The arrangement of the [Cd(SCN)3-](infinity) chains in the crystal lattice in the title compound is approximately hexagonal, creating triangular channels which are filled with [(DB24C8)Na]+ monocations. It was previously reasoned by us that the coiled [(DB24C8)Na]+ monocation, which lacks inversion or mirror symmetries, should enhance the tendency for the formation of the noncentrosymmetric space group of the title crystal, making it a potential second-order nonlinear optical crystal. Interestingly, however, the title compound crystallizes in a centrosymmetric space group (C2/c) and gives rise to no second harmonic generation (SHG). Previously known [Cd(SCN)3-](infinity) chains adopt fac-CdN3S3 coordination and a zigzag Cd chain configuration with a Cd...Cd...Cd angle of 165 degrees. The zigzag chains can align in either parallel or antiparallel fashion, resulting in efficient or no SHG effects, respectively. The linear Cd.Cd.Cd chain configuration observed in the title compound, on the other hand, makes it indistinguishable between parallel and antiparallel alignments. It is concluded that, to ensure the formation of noncentrosymmetric space groups, it is necessary to employ optically pure chiral cations as spacers and/or controllers. Furthermore, to enhance the nonlinear optical responses, [Cd(SCN)3-]infinity chains with fac-CdN3S3 coordination and parallel alignments of the zigzag Cd chains should be used.     

Catalytic enantioselective allylation of aldehydes via a moisture-tolerant chiral BINOL-In(III) complex

A moisture-tolerant chiral indium complex has been developed to effect good enantioselectivities in the addition of allyltributylstannanes to aldehydes. The allylation of a variety of aromatic, α,β-unsaturated and aliphatic aldehydes resulted in both moderate to good yields and high enantioselectivities (up to 86% ee).     

Characterization of organometallic coordinative cluster compounds of silver by matrix-assisted laser desorption/ionization time-of-flight mass spectrometry

Matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOFMS) was successfully applied to characterize the organosilver coordinative cluster compounds, silver phenylacetylide and three silver thiolates, namely, silver tertiary butylthiolate, silver 2,6-dimethylbenzenethiolate, and silver 2, 6-dichlorobenzenethiolate. Samples and dithranol matrix were finely dispersed in 1:1 tetrahydrofuran (THF)/chloroform (CHCl(3)) mixed solvent. In most cases the monomer units remained intact during ionization, and the oligomeric molecular ions were produced through silver cationization, with a general molecular ion formula [nM + Ag](+). This was further verified by the relative abundances of the isotopic peaks within the molecular ion clusters, which were in close agreement with those theoretically calculated for nM cationized with one silver ion. In the case of silver 2, 6-dichlorobenzenethiolate, in addition to the dominant [nM + Ag](+) peaks, weak peaks corresponding to the successive losses of hydrogen chloride molecules were observed. Copyright 2000 John Wiley & Sons, Ltd.     

A DSC study of the effects of sugars on thermal properties of rice starch gels before and after aging

Thermal characterization of gelatinized binary rice starch-water and ternary starch-sugar-water gels before and after aging was carried out using differential scanning calorimetry. The glass transition temperature of the maximally freeze-concentrated solution (T_g) in both fresh and aged gels was observed to decrease progressively with increasing sugar concentration. Aging of the gels generally shiftedT_g to higher temperatures, but had little or no effect on the ice melting peak temperature (T _m). The presence of various sugars could either accelerate or retard starch (amylopectin) recrystallization, depending on the type and concentration of sugar, as well as on starch/water ratio. A hypothesis based on the dual antiplasticizing-plasticizing effects of sugars was postulated to explain the observed effects. Of the sugars studied, xylose and fructose appeared to display exceptional retardative and accelerative effects, respectively, on retrogradation.This study was funded by a Sixth Malaysia Plan R&D grant under the Intensification of Research Priority Areas (IRPA) Program of the Ministry of Science, Technology and Environment, Malaysia. We thank Dr. Harry Levine and Dr. Louise Slade for giving us the opportunity to contribute a paper to this special issue of JTA.     

Hollow Versus Hydrogen-Occupied Icosahedral Cages of Coinage–Metal Clusters

It is suggested that the hollow coinage–metal icosahedral cage of the [Ag₄₄(SR)₃₀]^4? tetraanion (1a) may be occupied by two hydrogen atoms, giving rise to a dihydridic cluster [H₂Ag₄₄(SR)₃₀]^4? tetraanion (2b). As a consequence, two series of clusters, with different electron counts, can be formed by chemical means: the 18-electron series [H _x Ag₄₄(SR)₃₀]^(4?x)? via stepwise protonation of 1a and the 20-electron series [H _x Ag₄₄(SR)₃₀]^(6?x)? via stepwise deprotonation of 2b (here x = 0, 1, 2). Both series are closed-shell Jelliumatic clusters and expected to be stable. The corresponding members of these two series (for a given x value) are related by a two-electron reduction. These pairs raise the possibility of the hollow icosahedral metal cages in housing a number of hydrogen atoms, either via stepwise protonations or by absorption of hydrogen molecules.