Some results on the Laplacian eigenvalues of unicyclic graphs

In this paper, we provide the smallest value of the second largest Laplacian eigenvalue for any unicyclic graph, and find the unicyclic graphs attaining that value. And also give an “asymptotically good” upper bounds for the second largest Laplacian eigenvalues of unicyclic graphs. Using this results, we can determine unicyclic graphs with maximum Laplacian separator. And unicyclic graphs with maximum Laplacian spread will also be determined.     

Nanofluidic system for the studies of single DNA molecules

Here, we describe a simple and low-cost lithographic technique to fabricate size-controllable nanopillar arrays inside the microfluidic channels for the studies of single DNA molecules. In this approach, nanosphere lithography has been employed to grow a single layer of well-ordered close-packed colloidal crystals inside the microfluidic channels. The size of the polymeric colloidal nanoparticles could be trimmed by oxygen plasma treatment. These size-trimmed colloidal nanoparticles were then used as the etching mask in a deep etching process. As a result, well-ordered size-controllable nanopillar arrays could be fabricated inside the microfluidic channels. The gap distance between the nanopillars could be tuned between 20 and 80 nm allowing the formation of nanofluidic system where the behavior of a single lambda-phage DNA molecule has been investigated. It was found that the lambda-phage DNA molecule could be fully stretched in the nanofluidic system formed by nanopillars with 50 nm gap distance at a field of 50 V/cm.     

Versatile 2,5‐Pyridinedicarboxylate in Linking Transition‐Metal Atoms into 1D and 2D Coordination Polymers and Concomitant Polymorphs

By employing one bridging ligand, 2,5‐pyridinedicarboxylate (2,5‐pda^2?), in the presence or absence of another bridging ligand, 4,4′‐bipyridine (4,4′‐bpy), one one‐dimensional (1D) {[Co₂(2,5‐pda)(2,5‐Hpda)₂(4,4′‐bpy)(H₂O)₃]·6H₂O}_∞ ( 1 ) and two two‐dimensional (2D) coordination polymers, {[Cu₃(2,5‐pda)₃(H₂O)₃]·6H₂O}_∞ ( 2 ) and {[Co(2,5‐pda)(H₂O)]·2H₂O}_∞ ( 3 ) were synthesized. Complexes 2 and 3 are characterized as concomitant polymorphs from a one‐pot reaction at ambient temperature. A comparison of the coordination geometries of all neutral and anionic coordination polymers containing {M_x(2,5‐pda)_y(H₂O)_z}_∞ available to date is presented.     

On the spectral radius of graphs with connectivity at most <Emphasis Type="Italic">k</Emphasis>

In this paper, we study the spectral radius of graphs of order n with κ(G) ≤ k. We show that among those graphs, the maximal spectral radius is obtained uniquely at , which is the graph obtained by joining k edges from k vertices of K _n-1 to an isolated vertex. We also show that the spectral radius of will be very close to n − 2 for a fixed k and a sufficiently large n.     

Preparation of cobalt-containing bulky monodentate phosphines with electron-withdrawing/donating substituents on bridged arylethynyl and their applications in Suzuki coupling reactions

Several cobalt-containing bulky monodentate phosphines (μ-PPh₂CH₂PPh₂)Co₂(CO)₄(μ,η-(^tBu)₂PCC(C₆H₄R)) (4a: R=H; 4b: R=p-F; 4cp: R=p-CF₃; 4cm: R=m-CF₃; 4d: R=p-OMe) were prepared from the reactions of (^tBu)₂PCC(R-C₆H₄) (2a: R=H; 2b: R=p-F; 2cp: R=p-CF₃; 2cm: R=m-CF₃; 2d: R=p-OMe) with Co₂(CO)₆(μ-PPh₂CH₂PPh₂) 3. Further reactions of 4a, 4b, 4cp, 4cm, and 4d with Pd(OAc)₂ yielded unique palladium complexes (μ-PPh₂CH₂PPh₂)Co₂(CO)₄(μ,η-(^tBu)₂PCC(C₆H₃R)-κC¹)Pd(μ-OAc) (5a: R=H; 5b: R=p-F; 5cp: R=p-CF₃; 5cm: R=m-CF₃; 5d: R=p-OMe, respectively). The strong electron-withdrawing substituents, -F and -CF₃, assist the ortho-metalation process during the formation of 5b, 5cp, and 5cm. The more positively charged palladium center in 5b (or 5cp, 5cm) enhances the probability for PhB(OH)₃⁻ to attack the metal center and the rate of reduction thereafter. DFT studies on the charges of these palladium centers support this assumption. The enhancement of the reaction rates of the Suzuki-Miyaura cross-coupling reactions using 5b, 5cp, and 5cm is thereby attributed to this effect.     

Reactions of the cationic diruthenium carbonyl complex [Ru2(μ-dppm)2(CO)4(μ,η-O2CMe)] with bidentate ligands; intramolecularly assisted stereospecific synthesis via the second-sphere face-to-face π–π stacking interactions

The reactions of the diruthenium carbonyl complexes [Ru₂(μ-dppm)₂(CO)₄(μ,η²-O₂CMe)]X (X=BF₄⁻ (1a) or PF₆⁻ (1b)) with neutral or anionic bidentate ligands (L,L) afford a series of the diruthenium bridging carbonyl complexes [Ru₂(μ-dppm)₂(μ-CO)₂(η²-(L,L))₂]X_n ((L,L)=acetate (O₂CMe), 2,2′-bipyridine (bpy), acetylacetonate (acac), 8-quinolinolate (quin); n=0, 1, 2). Apparently with coordination of the bidentate ligands, the bound acetate ligand of [Ru₂(μ-dppm)₂(CO)₄(μ,η²-O₂CMe)]⁺ either migrates within the same complex or into a different one, or is simply replaced. The reaction of [Ru₂(μ-dppm)₂(CO)₄(μ,η²-O₂CMe)]⁺ (1) with 2,2′-bipyridine produces [Ru₂(μ-dppm)₂(μ-CO)₂(η²-O₂CMe)₂] (2), [Ru₂(μ-dppm)₂(μ-CO)₂(η²-O₂CMe)(η²-bpy)]⁺ (3), and [Ru₂(μ-dppm)₂(μ-CO)₂(η²-bpy)₂]²⁺ (4). Alternatively compound 2 can be prepared from the reaction of 1a with MeCO₂H–Et₃N, while compound 4 can be obtained from the reaction of 3 with bpy. The reaction of 1b with acetylacetone–Et₃N produces [Ru₂(μ-dppm)₂(μ-CO)₂(η²-O₂CMe)(η²-acac)] (5) and [Ru₂(μ-dppm)₂(μ-CO)₂(η²-acac)₂] (6). Compound 2 can also react with acetylacetone–Et₃N to produce 6. Surprisingly [Ru₂(μ-dppm)₂(μ-CO)₂(η²-quin)₂] (7) was obtained stereospecifically as the only one product from the reaction of 1b with 8-quinolinol–Et₃N. The structure of 7 has been established by X-ray crystallography and found to adopt a cis geometry. Further, the stereospecific reaction is probably caused by the second-sphere π–π face-to-face stacking interactions between the phenyl rings of dppm and the electron-deficient six-membered ring moiety of the bound quinolinate (i.e. the N-included six-membered ring) in 7. The presence of such interactions is indeed supported by an observed charge-transfer band in a UV–vis spectrum.     

On a Family of Diophantine Difference Equations

The family of Diophantine difference equations considered byJ. H. Conway, H. T. Croft, P. Erds and M. J. T. Guy can be solvedin closed form when the parameter is a prime. The associatedcharacterisation problem is shown to be related to the determinationof certain constants, which also happen to be coefficients inthe asymptotic formula for the associated exponential sum.     

Mode correlation of product pairs in the reaction OH+CD4-->HOD+CD3

The hydrogen abstraction reaction from methane by a hydroxyl radical produces two polyatomic molecules. Each product has several vibrational modes that characterize distinct, concerted motions of the constituent atoms of the molecule. This communication describes the first measurement that maps out the coincident information on how the mode of excitation of one product varies with that of the other co-product. Such information on mode correlation of product pairs is particularly appealing in that it provides intuitively a glimpse of the reaction paths by which the chemical transformation occurs.     

Toward chiral sum-frequency spectroscopy

Chiral sum-frequency (SF) spectroscopy that measures both the real and the imaginary components of the SF spectral response was demonstrated for the first time. It was based on interference of the SF signal with a dispersionless SF reference. Solutions of 1,1'-bi-2-naphthol (BN) were used as model systems, and their chiral SF spectra over the first exciton-split transitions were obtained. Chiral spectra are useful for determination of absolute configuration and conformation of chiral molecules.