On the clique number of the square of a line graph and its relation to maximum degree of the line graph

In 1985, Erdős and Nešetřil conjectured that the square of the line graph of a graph , that is, , can be colored with colors. This conjecture implies the weaker conjecture that the clique number of such a graph, that is, , is at most . In 2015, Śleszyńska-Nowak proved that . In this paper, we prove that . This theorem follows from our stronger result that where .     

Photochemical synthesis of cyclic peptide models from phthalimido acetamides and phthaloyl dipeptide esters

The photochemistry of several phthalimido acetamides and phthaloyl dipeptide esters has been investigated. Their photocyclization ability strongly depended on the substitution pattern of the amide linker group. While secondary amide-derived starting materials were largely unreactive, the corresponding tertiary amide-linked derivatives furnished the desired cyclic peptide model compounds in acceptable to good yields (41–80%). The structurally related ester-linked model derivatives also remained unreactive upon irradiation. Preferential hydrogen-abstraction from the E-cis-substituent is suggested to explain the observed differences in cyclization ability.     

Hydrogen-Release Reaction of a Complex Transition Metal Hydride with Covalently Bound Hydrogen and Hydride Ions

The hydrogen-release reaction of a complex transition metal hydride, LaMg₂NiH₇, composed of La³⁺, 2×Mg²⁺, [NiH₄]⁴⁻ and 3×H⁻, was studied by thermal analyses, powder X-ray, and neutron diffraction and inelastic neutron scattering. Upon heating, LaMg₂NiH₇ released hydrogen at approximately 567 K and decomposed into LaH₂₋₃ and Mg₂Ni. Before the reaction, covalently bound hydrogen (H^c°^v.) in [NiH₄]⁴⁻ exhibited a larger atomic displacement than H⁻, although a weakening of the chemical bonds around [NiH₄]⁴⁻ and H⁻ was observed. These results indicate the precursor phenomenon of a hydrogen-release reaction, wherein there is a large atomic displacement of H^c°^v. that induces the hydrogen-release reaction rather than H⁻. As an isothermal reaction, LaMg₂NiH₇ formed LaMg₂NiH_2.4 at 503 K in vacuum for 48 h, and LaMg₂NiH_2.4 reacted with hydrogen to reform LaMg₂NiH₇ at 473 K under 1 MPa of H₂ gas pressure for 10 h. These results revealed that LaMg₂NiH₇ exhibited partially reversible hydrogen-release and uptake reactions.     

Cyclodextrin‐Based Bimodal Fluorescence/MRI Contrast Agents: An Efficient Approach to Cellular Imaging

A novel bimodal fluorescence/MRI probe based on a cyclodextrin scaffold has been synthesized and characterized. The final agent employs the fluorescein (F) functionality as a fluorescence marker and the Gd^III complex of a macrocyclic DOTA‐based ligand (GdL) having one aminobenzyl‐phosphinic acid pendant arm as an MRI probe, and has a statistical composition of (GdL)_6.9‐F_0.1‐β‐CD. Slow rotational dynamics (governed by a very rigid cyclodextrin scaffold) combined with fast water exchange (ensured by the chosen macrocyclic ligand) resulted in a high relaxivity of ～22 s^?1 mM ^?1 per Gd^III or ～150 s^?1 mM ^?1 per molecule of the final conjugate (20 MHz, 25 °C). In vitro labelling of pancreatic islets (PIs) and rat mesenchymal stem cells has been successfully performed. The agent is not cytotoxic and is easily internalized into cells. The labelled cells can be visualized by MRI, as proved by the detection of individual labelled PIs. A fluorescence study performed on mesenchymal stem cells showed that the agent stays in the intracellular space for a long time.     

Stereoselective formation of fused tricyclic amines from acyclic aldehydes by a cascade process involving condensation, cyclization, and dipolar cycloaddition

The preparation of tricyclic amines from acyclic precursors is described using a cascade of tandem reactions involving condensation of an aldehyde with a primary amine, cyclization (with displacement of a halide), and then in situ deprotonation or decarboxylation to give an azomethine ylide or nitrone followed by intramolecular dipolar cycloaddition. The methodology is straightforward, and the aldehyde precursors are prepared easily and quickly in high yield using nitrile alkylations followed by DIBAL-H reduction. The relative ease of reaction of various substrates with different tether lengths between the aldehyde and the halide or dipolarophile has been studied. Several primary amines including simple amino acids such as glycine, alanine, and phenylalanine and derivatives such as glycine ethyl ester and also hydroxylamine have been investigated. High yields are obtained in the formation of different tricyclic ring sizes; the dipolar cycloaddition necessarily creates a five-membered ring, and we have investigated the formation of five- and six-membered rings for the other two new ring sizes. In all cases, yields are high (except when using glycine when the tether to the terminal alkene dipolarophile leads to a six-membered ring), and most efficient is the formation of the tricyclic product in which all five-membered rings are formed. Examples with an alkyne as the dipolarophile were also successful. In all the reactions studied, the products are formed with complete regioselectivity and remarkably with complete stereoselectivity. The key step involves the formation of three new rings and potentially up to four new stereocenters in a single transformation. The power of the chemistry was demonstrated by the synthesis of the core ring systems of the alkaloids (+/-)-scandine and (+/-)-myrioneurinol and the total syntheses of the alkaloids (+/-)-aspidospermine, (+/-)-quebrachamine, and (+/-)-aspidospermidine.     

Vibrational properties of the gallium monohydrides SrGaGeH, BaGaSiH, BaGaGeH, and BaGaSnH

Vibrational properties of the gallium monohydrides SrGaGeH, BaGaSiH, BaGaGeH, and BaGaSnH (AeGaTtH) have been investigated by means of inelastic neutron scattering (INS) and first principles calculations. The compounds contain separated Ga–H units being part of a two dimensional polyanionic layer, [TtGaH]²⁻ (Tt=Si, Ge, Sn). The INS spectra show internal Ga–H bending and stretching modes at frequencies around 900 and 1200 cm⁻¹, respectively. While the stretching mode is virtually invariant with respect to the variable chemical environment of the Ga–H unit, the bending mode frequency varies and is highest for BaGaSiH and lowest for BaGaSnH. The stretching mode is a direct measure of the Ga–H bond strength, whereas the bending mode reflects indirectly the strength of alkaline earth metal–hydrogen interaction. Accordingly, the terminal Ga–H bond in solid state AeGaTtH is distinct, but—compared to molecular gallium hydrides—very weak.     

Distribution theory on P.C.F. fractals

We construct a theory of distributions in the setting of analysis on post-critically finite self-similar fractals, and on fractafolds and products based on such fractals. The results include basic properties of test functions and distributions, a structure theorem showing that distributions are locally-finite sums of powers of the Laplacian applied to continuous functions, and an analysis of the distributions with point support. Possible future applications to the study of hypoelliptic partial differential operators are suggested.