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).     

Ionic nanocomposites based on XNBR-OMg filled with layered nanoclays

Ionic elastomers based on carboxylated nitrile rubber (XNBR) crosslinked with magnesium oxide (OMg) and filled with layered nanoclays have been prepared. Two types of nanofillers were employed: a commercial one (Cloisite 15 A) and the other prepared in our laboratory (BCA-ODA). The properties of raw and vulcanized compounds have been studied in order to elucidate the effect of layered nanoclays. X-ray diffraction analysis shows that during compounding and crosslinking the macromolecular chains of rubber were intercalated in a commercial nanofiller, whilst our nanoclay was exfoliated. However, this fact does not affect the properties of compounds homogeneously. Vulcanization parameters, Mooney viscosity and decay and rheological properties were affected depending on the filler type. Mechanical properties were not significantly varied, but glass transition temperature and ionic transition temperature were shifted in comparison with those of the unfilled compound.     

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.     

Long-Range proton-proton coupling constants. I. Propanic coupling involving a methyl group 4JMeH

An equation is formulated on the basis of theoretical INDO/FPT calculations which describes the angular dependence of the propanic long-range coupling constant ⁴J_MeH in substituted HCCCH₃ fragments. This equation is a truncated Fourier series in the torsion angle ?, HCCMe, which takes into account the dependence of the Fourier coefficients on the bond angle θ, CCMe. The substituent effects are assumed to be additive. Some parameters in the equation may be obtained from the ⁴J_MeH couplings in propane and neopentane derivatives. The calculated effect upon ⁴J_MeH of changes in the bond angle θ is significant and it seems to be in part the cause of some effects which have been attributed to conformational dependence.     

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.