Matching energy of unicyclic and bicyclic graphs with a given diameter

Gutman and Wagner proposed the concept of matching energy (ME) and pointed out that the chemical applications of ME go back to the 1970s. Let G be a simple graph of order n and be the roots of its matching polynomial. The ME of G is defined to be the sum of the absolute values of . In this article, we characterize the graphs with minimal ME among all unicyclic and bicyclic graphs with a given diameter d. © 2014 Wiley Periodicals, Inc. Complexity 21: 224–238, 2015     

Optimal Groomings with Grooming Ratios Six and Seven

Grooming uniform all‐to‐all traffic in optical (SONET) rings with grooming ratio C requires the determination of a decomposition of the complete graph into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The determination of optimal C‐groomings has been considered for , and completely solved for . For , it has been shown that the lower bound for the drop cost of an optimal C‐grooming can be attained for almost all orders with 5 exceptions and 308 possible exceptions. For , there are infinitely many unsettled orders; especially the case is far from complete. In this paper, we show that the lower bound for the drop cost of a 6‐grooming can be attained for almost all orders by reducing the 308 possible exceptions to 3, and that the lower bound for the drop cost of a 7‐grooming can be attained for almost all orders with seven exceptions and 16 possible exceptions. Moreover, for the unsettled orders, we give upper bounds for the minimum drop costs.     

Formation of lithium,sodium and potassium complexes with 1,3,5‐triamino‐1,3,5‐trideoxy‐cis‐inositol (taci)

Single crystals of (1,3‐diamino‐5‐azaniumyl‐1,3,5‐trideoxy‐cis‐inositol‐κ³O²,O⁴,O⁶)(1,3,5‐triamino‐1,3,5‐trideoxy‐cis‐inositol‐κ³O²,O⁴,O⁶)lithium(I) diiodide dihydrate, [Li(C₆H₁₆N₃O₃)(C₆H₁₅N₃O₃)]I₂·2H₂O or [Li(Htaci)(taci)]I₂·2H₂O (taci is 1,3,5‐triamino‐1,3,5‐trideoxy‐cis‐inositol), (I), bis(1,3,5‐triamino‐1,3,5‐trideoxy‐cis‐inositol‐κ³O²,O⁴,O⁶)sodium(I) iodide, [Na(C₆H₁₅N₃O₃)₂]I or [Na(taci)₂]I, (II), and bis(1,3,5‐triamino‐1,3,5‐trideoxy‐cis‐inositol‐κ³O²,O⁴,O⁶)potassium(I) iodide, [K(C₆H₁₅N₃O₃)₂]I or [K(taci)₂]I, (III), were grown by diffusion of MeOH into aqueous solutions of the complexes. The structures of the Na and K complexes are isotypic. In all three complexes, the taci ligands adopt a chair conformation with axial hydroxy groups, and the metal cations exhibit exclusive O‐atom coordination. The six O atoms of the resulting MO₆ unit define a centrosymmetric trigonal antiprism with approximate D_3d symmetry. The interligand O...O distances increase significantly in the order Li < Na < K. The structure of (I) exhibits a complex three‐dimensional network of R—NH₂—H...NH₂—R, R—O—H...NH₂—R and R—O—H...O(H)—H...NH₂—R hydrogen bonds. The structures of the Na and K complexes consist of a stack of layers, in which each taci ligand is bonded to three neighbours via pairwise O—H...NH₂ interactions between vicinal HO—CH—CH—NH₂ groups.     

Acetylene Derivatives of Cationic Diazaoxatriangulenes and Diaza [4]Helicenes - Access to Red Emitters and Planar Chiral Stereochemical Traits

Cationic triangulenes, and related helicenes, constitute a rich class of dyes and fluorophores, usually absorbing and emitting light at low energy, in the orange to red domains. Recently, to broaden the scope of applications, regioselective late-stage functionalizations on these core moieties have been developed. For instance, with the introduction of electron-donating groups (EDGs), important bathochromic shifts are observed pushing absorptions towards or in the near-infrared (NIR) spectral domain while emissive properties disappear essentially completely. Herein, to upset this drawback, acetylene derivatives of cationic diazaoxa triangulenes (DAOTA) and [4]helicenes are prepared (16 examples). Contrary to other EDG-functionalized derivatives, C≡C− functionalized products remain broadly fluorescent, with red-shifted absorptions (Δλ_abs up to 25 nm) and emissions (Δλ_em up to 73 nm, Φ_PL up to 51 %). Quite interestingly, a general dynamic stereoisomerism phenomenon is evidenced for the compounds derived from achiral DAOTA cores. At low temperature in ¹H NMR spectroscopy (218 K), N−CH₂ protons become diastereotopic with chemical shifts differences (Δδ) as high as +1.64 ppm. The signal coalescence occurs around 273 K with a barrier of ∼12 kcal mol⁻¹. This phenomenon is due to planar chiral conformations (S_p and R_p configurations), induced by the geometry of the alkyl (n-propyl) side-chains next to the acetylenic substituents. Ion pairing studies with Δ-TRISPHAT anion not only confirm the occurrence of the chiral conformations but evidence a moderate but definite asymmetric induction from the chiral anion onto the cations. Finally, DFT calculations offer a valuable insight on the geometries, the corresponding stereodynamics and also on the very large difference in NMR for some of the diastereotopic protons.     

大气环境下飞秒激光对铝靶烧蚀过程的研究

利用时间分辨的光阴影成像技术研究了在大气环境下飞秒激光烧蚀铝靶的动态过程. 在入射激光能量为4 mJ, 激光光斑超过1 mm时, 激光烧蚀区表面物质以近似平面冲击波形式向外喷射; 在同样激光能量下、激光光斑较小时(约0.6 mm), 激光烧蚀区以近似半球型冲击波形式向外喷射. 当激光能量比较大时(7 mJ), 发现空气的电离对于激光烧蚀靶材有着重要影响. 在光轴附近烧蚀产生的喷射物具有额外的柱状和半圆型的结构, 叠加在平面冲击波结构上.     

Incompatibility networks as models of scale-free small-world graphs

We make a mapping from Sierpinski fractals to a new class of networks, the incompatibility networks, which are scale-free, small-world, disassortative, and maximal planar graphs. Some relevant characteristics of the networks such as degree distribution, clustering coefficient, average path length, and degree correlations are computed analytically and found to be peculiarly rich. The method of network representation can be applied to some real-life systems making it possible to study the complexity of real networked systems within the framework of complex network theory.     

[2]Catenane Synthesis via Covalent Templating

After earlier unsuccessful attempts, this work reports the application of covalent templating for the synthesis of mechanically interlocked molecules (MiMs) bearing no supramolecular recognition sites. Two linear strands were covalently connected in a perpendicular fashion by a central ketal linkage. After subsequent attachment of the first strand to a template via temporary benzylic linkages, the second was linked to the template in a backfolding macrocyclization. The resulting pseudo[1]rotaxane structure was successfully converted to a [2]catenane via a second macrocyclization and cleavage of the ketal and temporary linkages.     

Large‐acceptance diamond planar refractive lenses manufactured by laser cutting

For the first time, single‐crystal diamond planar refractive lenses have been fabricated by laser micromachining in 300 µm‐thick diamond plates which were grown by chemical vapour deposition. Linear lenses with apertures up to 1 mm and parabola apex radii up to 500 µm were manufactured and tested at the ESRF ID06 beamline. The large acceptance of these lenses allows them to be used as beam‐conditioning elements. Owing to the unsurpassed thermal properties of single‐crystal diamond, these lenses should be suitable to withstand the extreme flux densities expected at the planned fourth‐generation X‐ray sources.     

Cycle-maximal triangle-free graphs

We conjecture that the balanced complete bipartite graph _{K⌊n/2⌋,⌈n/2⌉}$K_{\lfloor n/2\rfloor ,\lceil n/2\rceil }$ contains more cycles than any other n$n$-vertex triangle-free graph, and we make some progress toward proving this. We give equivalent conditions for cycle-maximal triangle-free graphs; show bounds on the numbers of cycles in graphs depending on numbers of vertices and edges, girth, and homomorphisms to small fixed graphs; and use the bounds to show that among regular graphs, the conjecture holds. We also consider graphs that are close to being regular, with the minimum and maximum degrees differing by at most a positive integer k$k$. For k=1$k=1$, we show that any such counterexamples have n≤91$n\le 91$ and are not homomorphic to _C5$C_5$; and for any fixed k$k$ there exists a finite upper bound on the number of vertices in a counterexample. Finally, we describe an algorithm for efficiently computing the matrix permanent (a #P$\#P$-complete problem in general) in a special case used by our bounds.     

Perfect 1‐Factorizations of a Family of Cayley Graphs

A 1‐factorization of a graph G is a decomposition of G into edge‐disjoint 1‐factors (perfect matchings), and a perfect 1‐factorization is a 1‐factorization in which the union of any two of the 1‐factors is a Hamilton cycle. We consider the problem of the existence of perfect 1‐factorizations of even order 4‐regular Cayley graphs, with a particular interest in circulant graphs. In this paper, we study a new family of graphs, denoted , which are Cayley graphs if and only if k is even or . By solving the perfect 1‐factorization problem for a large class of graphs, we obtain a new class of 4‐regular bipartite circulant graphs that do not have a perfect 1‐factorization, answering a problem posed in 7 . With further study of graphs, we prove the nonexistence of P1Fs in a class of 4‐regular non‐bipartite circulant graphs, which is further support for a conjecture made in 7 .