Narrow Results

For a graph $G$ with $n$ vertices and $m$ edges, the energy (or the adjacency energy) is the sum of the absolute values of the eigenvalues (or adjacency eigenvalues) of $G$, that is, ${ℰ}(G)={\sum }_{i=1}^n|{\lambda }_i(G)|$, where ${\lambda }_i(G)$ is the $i\mathrm{\text{th}}$ adjacency eigenvalue of $G$. In this paper, we discuss the characteristic polynomial and the energy of dendrimers. We first show the formula obtained for the characteristic polynomial and the energy of tree dendrimers considered in Bokhary and Tabassum [The energy of some tree dendrimers, J. Appl. Math. Comput.68 (2022) 1033–1045] are incorrect. Using a different algorithmic procedure we will derive the correct formula for the characteristic polynomial and the energy of dendrimer $d(3,k)$. Further, we obtain the energy of $d(l,k)$ for $k=4$ and $k=5$, which partially answers a problem raised in Bokhary and Tabassum [The energy of some tree dendrimers, J. Appl. Math. Comput.68 (2022) 1033–1045].

We study percolation as a critical phenomenon on a random multifractal support. The scaling exponent β related to the mass of the infinite cluster and the fractal dimension of the percolating cluster d_f are quantities that have the same value as the ones from the standard two-dimensional regular lattice percolation. The scaling exponent ν related to the correlation length is sensitive to the local anisotropy and assumes a value different from standard percolation. We compare our results with those obtained from the percolation on a deterministic multifractal support. The analysis of ν indicates that the deterministic multifractal is more anisotropic than the random multifractal. We also analyze connections with correlated percolation problems and discuss some possible applications.

A survey of recent studies of biaxial liquid crystals (LCs), whose nematic and/or smectic-A phases do not possess optical uniaxiality (viz., more than one optical axis exists), is given in this review. In particular, we emphasize on how Nuclear Magnetic Resonance (NMR) spectroscopy can help to advance the understanding of phase biaxiality in general, and to examine recent debates on the existence of biaxial nematic phase reported in low molecular mass bent-core or V-shaped mesogens. A general discussion of orientational order parameters is detailed, particularly in smectic-C (SmC) and biaxial nematic phases. How these orientational order parameters can be determined by various techniques such as NMR, IR absorbance and Raman scattering studies, will be mentioned. Recent X-ray observations of SmC clusters in the nematic phase of V-shaped mesogens are highlighted and contrasted with probable theory. Moreover, deuterium and carbon-13 NMR techniques are briefly reviewed, and their possible utilization to identify phase biaxiality in these biaxial LC systems is explored.

This is an invited overview of a lecture presented at the American Physical Society (APS) Meeting, Boston, USA (March 1, 2012). The primary focus of this APS lecture was to trace the historical emergence of Hard and Soft nanoscale superatoms (i.e. nano-element categories) as well as a recent merging of these concepts/entities by chemists/physicists into a unified system and framework for defining nanoscience. The convergence of these quantized, organic/inorganic superatom entities involved the application of traditional "first principles" and their nanoscale "atom mimicry" features as a criteria for evolving a roadmap of quantized nano-elemental categories, nano-compound/assemblies and nano-periodic patterns, etc., much as was observed in traditional chemistry. This simple perspective was used to define a nanoscale taxonomy of hard/soft superatom/nano-element categories, as well as to explain the dependency of a broad range of nano-periodic properties/features on one or more of six Critical Nanoscale Design Parameters (CNDPs) associated with these nano-building blocks, namely: (1) size, (2) shape, (3) surface chemistry, (4) rigidity/flexibility, (5) architecture and (6) elemental composition. Validation and support of this systematic nano-periodic perspective has appeared in many recent publications describing CNDP dependent nano-periodic property patterns/trends, rules and Mendeleev-like nano-periodic tables which may unify and provide first steps toward a "central paradigm" for nanoscience.

This paper describes some of our efforts in the area of nanostructured thin film architectures. The resulting interfacial hybrid assemblies are built from (1) organic/polymeric objects based on dendrimer systems, from (2) surface-functionalized Au nanoparticles, and (3) from a variety of semiconducting quantum dots. Dendrimers as polymeric building blocks with a strictly monodisperse particle size distribution in the nanometer range can be functionalized in the core, the scaffold, or at the periphery, thus offering interesting hybrid materials for a wide range of applications. The combination with Au clusters and their local surface plasmon resonances suggests new strategies for optoelectronic devices or unconventional bio-sensor platforms. The possibility of tuning the luminescent properties of semiconducting nanoparticles by size or compositional bandgap engineering complements the assembly kit with building blocks for supramolecular thin film nanocomposite materials.

We report here the nonlinear absorption properties of dendritic molecules of oligofluorenes. Measurements were performed in chloroform between 450 and 650 nm for nanosecond time duration pulses. The nonlinear absorption is attributed to a three-photon absorption process involving first a two-photon absorption (TPA) step followed by an excited state absorption (ESA) process. The corresponding three-photon absorption coefficient α₃ presents at resonance high values (α₃ > 10000 cm³/GW² for 2G1 the first dendrimer generation bearing a bifluorene at a concentration of 50 g/L). These values are strongly enhanced with respect to those obtained in the same conditions for analog linear oligofluorenes.

Iron oxide nanoparticles were prepared and functionalized by succinamide based dendrimer. The resultant particles were characterized by XRD, VSM, and FTIR spectroscopy. The results indicate that the dendrimers has effectively functionalized the magnetite nanoparticles which remain dispersive and exhibited super-paramagnetism with a magnetization value of 33.2 emu/g in a field of 2T. Mean particle size as calculated from the AFM was found to be ~ 23 nm. Bovine serum albumin was immobilized on the magnetic nanoparticles as was confirmed by the FTIR results.

Studies of excited singlet and triplet states of porphyrins and phthalocyanines in organized media of reverse micelles, vesicles, monolayers, and Langmuir-Blodgett films along with more complex supramolecular organizates with proteins and dendrimers, are reported. Self-assembly in these systems was followed by imaging and temporal fluorescence techniques.

A series of new zinc phthalocyanine dendrimers with an increasing branched shell of oligoethylene glycol end groups has been synthesized. Owing to the presence of these hydrophilic termini at the dendrimer surface, all compounds are highly soluble in a wide range of solvents including water. The monodispersity of the amphiphilic dendrimers has been unambiguously evidenced by both NMR and MS techniques. UV-vis experiments point out that the tendency of the central phthalocyanine chromophore to aggregate in polar protic solvents is not significantly reduced regardless of the dendritic shell surrounding the macrocyclic chromophore.

A series of ruthenium phthalocyanine centered dendrimers has been prepared with up to 32 ethyl ester end groups. The photophysical properties have been assessed in THF to reveal a correlation between the degree of dendritic environment and the ability to generate singlet oxygen. In the attempt to prepare the correponding highly water-soluble polyelectrolyte dendrimers upon saponification we observed the cleavage of the central esters moieties adjacent to the pyridine unit in all cases. However, despite the lack of shielding by a growing branched shell surrounding the photoactive center, the obtained tetracarboxylate-terminated ruthenium phthalocyanine derivative was still able to generate singlet oxygen in aqueous medium.

Nanotechnology, a separate field of knowledge since 1980s, involves utilization of nanomaterials not only in electronics and catalysis, but also in biomedical research including drug delivery, bioimaging, biomedical-diagnostics and tissue engineering. Multidisciplinary of this science has led to the development of different areas of technology and might contribute to innovations that will, as a final consequence, help humanity. Dendrimers are large and complex molecules that are characterized by well-defined nanoscale architecture, monodispersity and structural versatility. These highly interesting polymers consist of three elements: core, branches and peripheral groups. There is a wide variety of potential applications of dendritic polymers. One of the most promising is utilization of polyamidoamine (PAMAM) dendrimers as drug delivery devices. Among pharmaceuticals that have been connected with different types of dendrimers are nonsteroidal anti-inflammatory drugs (NSAIDs), anticancer drugs and other. Dendrimers application as drug carriers improves pharmacokinetic properties of drug particles, decreases drugs' side effects and, by possibility of surface modification with different ligands, enables to target specific tissues and tumor cells. Dendrimers might be also utilized as devices for delivery of genetic material and contrast agents for magnetic resonance imaging (MRI).

Dendrimers are highly branched monodisperse, macromolecules and are considered in nanotechnology with a variety of suitable applications. In this paper, the matching polynomial and some results of the matchings for three classes of nanostar dendrimers are obtained. Furthermore, we express the recursive formulas of the Hosoya index for these structures of dendrimers by their matching polynomials.

In the past two decades dendrimers have emerged as a distinctive branch of macromolecular chemistry. Tailoring of dendrimer structure yields precise placement of chromophores that can serve as energy harvesters, mimicking photosynthesis. The unique architecture afforded by dendrimers allows for multiple energy harvesters which can transfer their energy to a single core. These materials can have impact in optoelectronic applications such as organic light emitting diodes (OLEDs). This chapter reviews artificial energy-harvesting systems that employ a dendritic architecture, emphasizing the energy transfer characteristics that these dendrimers provide rather then their synthesis.

Since the discovery of cisplatin, drugs based on platinum, have made a significant impact on the treatment of various cancers. The administration of platinum drugs is however accompanied by significant side effects. This chapter discusses the types of drug delivery systems that have been developed in order to enable the targeted delivery while maintaining controlled temporal supply of the drug. The sizes of carriers range from nanometer to micrometer sized particles. The most common types of drug carriers are micelles, liposomes, nanoparticles, and dendrimers, but also a few microspheres have been developed. Most striking aspect of the delivery of platinum drugs is the possibility of physical encapsulation but also the binding of the drug to the polymer carrier coordinate covalent bond. Since platinum drugs have typically two permanent and two leaving ligands, the polymer can be part of either ligand. As the leaving ligand, the platinum drug is released often as cisplatin. If the polymer provides the functionality for the permanent ligand, a new macromolecular drug has been formed. In addition to the attachment of pt(II) drugs, recent offorts are devoted to the conjugation via the Pt((IV) prodrug.