Median Sets and Median Number of a Graph

A profile is a finite sequence of vertices of a graph. The set of all vertices of the graph which minimises the sum of the distances to the vertices of the profile is the median of the profile. Any subset of the vertex set such that it is the median of some profile is called a median set. The number of median sets of a graph is defined to be the median number of the graph. In this paper, we identify the median sets of various classes of graphs such as Kp − e, Kp,q forP > 2, and wheel graph and so forth. The median numbers of these graphs and hypercubes are found out, and an upper bound for the median number of even cycles is established.We also express the median number of a product graph in terms of the median number of their factors.

Almost Self-Centered Median And Chordal Graphs

Almost self-centered graphs were recently introduced as the graphs with exactly two non-central vertices. In this paper we characterize almost selfcentered graphs among median graphs and among chordal graphs. In the first case P4 and the graphs obtained from hypercubes by attaching to them a single leaf are the only such graphs. Among chordal graph the variety of almost self-centered graph is much richer, despite the fact that their diameter is at most 3. We also discuss almost self-centered graphs among partial cubes and among k-chordal graphs, classes of graphs that generalize median and chordal graphs, respectively. Characterizations of almost self-centered graphs among these two classes seem elusive

The median function on graphs with bounded profiles

The median of a profile = (u1, . . . , uk ) of vertices of a graph G is the set of vertices x that minimize the sum of distances from x to the vertices of . It is shown that for profiles with diameter the median set can be computed within an isometric subgraph of G that contains a vertex x of and the r -ball around x, where r > 2 − 1 − 2 /| |. The median index of a graph and r -joins of graphs are introduced and it is shown that r -joins preserve the property of having a large median index. Consensus strategies are also briefly discussed on a graph with bounded profiles.

On The Center Sets and Center Numbers of Some Graph Classes

For a set S of vertices and the vertex v in a connected graph G, max x2S d(x, v) is called the S-eccentricity of v in G. The set of vertices with minimum S-eccentricity is called the S-center of G. Any set A of vertices of G such that A is an S-center for some set S of vertices of G is called a center set. We identify the center sets of certain classes of graphs namely, Block graphs, Km,n, Kn −e, wheel graphs, odd cycles and symmetric even graphs and enumerate them for many of these graph classes. We also introduce the concept of center number which is defined as the number of distinct center sets of a graph and determine the center number of some graph classes

Raman and infrared spectral analysis of thallium niobyl phosphates: Tl2NbO2PO4, Tl3NaNb4O9(PO4)2 and TlNbOP2O7

Raman and infrared spectra of Tl2NbO2PO4, Tl3NaNb4O9(PO4)2 and TlNbOP2O7 are reported. The observed bands are assigned in terms of vibrations of NbO6 octahedra and PO4 tetrahedra in the first two compounds and in terms of NbO6 octahedra and P2O7 4− anion in the third compound. The NbO6 octahedra in all the title compounds are found to be corner-shared and distorted. The higher wavenumber values of the ν1 (NbO6) mode and other stretching modes indicate that the NbO6 octahedra in them are distorted in the order TlNbOP2O7 > Tl2NbO2PO4 > Tl3NaNb4O9(PO4)2. The splitting of the ν3 (PO4) mode indicates that PO4 tetrahedra is distorted more in Tl2NbO2PO4 than in Tl3NaNb4O9(PO4)2. The symmetry of P2O7 4− anion in TlNbOP2O7 is lowered. Bands indicate that the P–O–P bridge in the above compound has a bent P–O–P bridge configuration

Photoluminescence studies of spray pyrolytically grown nanostructured tin oxide semiconductor thin films on glass substrates

SnO2 nanocrystalline thin films were deposited on glass substrates by the spray pyrolysis technique in air atmosphere at 375, 400, 425, 450 and 500 ◦C substrate temperatures. The obtained films were characterized by using XRD. The room temperature photoluminescence (PL) spectra of these films have near band edge (NBE) and deep level emission under the excitation of 325 nm radiation. NBE PL peak intensity decreased consistently with temperatures for samples prepared at 400, 450 and 500 ◦C, while a sudden reduction in intensity is observed for the sample prepared at 425 ◦C. A similar effect was observed for the optical transmittance spectra. These effects can be explained on the basis of the change in population of oxygen vacancies as indicated by the change in a values

Raman and FTIR studies of the structural aspects of Nasicon-type crystals; AFeTi(PO4)3 [A ¼ Ca, Cd]

Raman and FTIR spectra of CaFeTi(PO4)3 and CdFeTi(PO4)3 are recorded and analyzed. The observed bands are assigned in terms of vibrations of TiO6 octahedra and PO4 tetrahedra. The symmetry of TiO6 octrahedra and PO4 tetrahedra is lowered from their free ion symmetry. The presence of Fe3+ ion disrupts the Ti–O–P–O–Ti chain and leads to the distortion of TiO6 octrahedra and PO4 tetrahedra. The PO4 3 tetrahedra in both crystals are linearly distorted. The covalency bonding factor of PO4 3 polyanion of both the crystals are calculated from the Raman spectra and compared to that of other Nasicon-type systems. The numerical values of covalency bonding factor indicates that there is a reduction in redox energy and cell voltage and is attributed to strong covalency of PO4 3 polyanionin

Vibrational spectroscopic studies of Guanidinium metal (MII) sulphate hexahydrates [MII = Co, Fe, Ni]

The Raman and FTIR spectra of [C(NH2)3]2M(SO4)2 ·6H2O (withM= Co, Fe, Ni) were recorded and analysed. The observed spectral bands are assigned in terms of vibrations of guanidinium ions, sulphate groups and water molecules. The analysis shows that the sulphate tetrahedra are distorted from their free state symmetry Td to C1. This is attributed to the presence of hydrogen bonds from water molecules. The order of distortion of the metal oxygen octahedra influenced the distortion of the sulphate tetrahedra. The appearance of 1– 3 modes of water molecules above 3300 cm−1 indicates the presence of weak hydrogen bonds

FT-IR and FT-Raman study of Nasicon type phosphates, ASnFe(PO4)3 [A=Na2, Ca, Cd]

FT-Raman and FT-IR spectra of ASnFe(PO4)3 [A=Na2, Ca, Cd] were recorded and analyzed. The bands were assigned in terms of the vibrational group frequencies of SnO6 octahedral and PO4 tetrahedral. The spectral analysis shows that the symmetry of corner shared octahedral (SnO6) and the tetrahedral (PO4) are lowered from their free ion symmetry state. The presence of Fe3+ ions disrupts the S–N–O–S–N chain in the structure. This causes distortion of SnO6 and PO4 in the structure of all the compounds. Also it is seen that there are two distinct PO4 tetrahedra of different P–O bond lengths. One of these tetrahedra is linearly distorted in all the title compounds. The PO4 frequencies and bond lengths are calculated theoretically and are in agreement with the experimental values. The presence of PO4 polyanion in the structure can reduce the redox energy and hence reduce the metal oxygen covalency strength in the structure

Vibrational spectroscopic studies of FeClMoO4, Na2MoO4 and Na2MoO4·2H2O:D2O

FTIR and Raman spectra of FeClMoO4 single crystal and polycrystalline Na2MoO4, Na2MoO4·2H2O and Na2MoO4·2D2O are recorded and analysed. The band positions for different modes suggest that MoO4 tetrahedron is more distorted in FeClMoO4. The larger splitting observed for the bending modes and partial retention of degeneracy of the asymmetric stretching mode indicate that angular distortion is greater than liner distortion in MoO4 2 ion in FeClMoO4 confirming x-ray data. The non-appearance of the n1 and n2 modes in the IR and partial retention of the degeneracies of various modes show that MoO4 2 ion retains Td symmetry in Na2MoO4. Wavenumber values of the n1 mode indicate that the distortion of MoO4 tetrahedra in the four crystals are in the order FeClMoO4\ Na2MoO4·2H2O\Na2MoO4·2D2O\Na2MoO4. The water bands suggest the presence of two crystallographically distinct water molecules in Na2MoO4·2H2O. They form strong hydrogen bonds

Versatile binding properties of a di-2-pyridyl ketone nicotinoylhydrazone ligand: Crystal structure of a Cu(II) complex

Six new copper complexes of di-2-pyridyl ketone nicotinoylhydrazone (HDKN) have been synthesized. The complexes have been characterized by a variety of spectroscopic techniques and the structure of [Cu(DKN)2]·H2O has been determined by single crystal X-ray diffraction. The compound [Cu(DKN)2]·H2O crystallized in the monoclinic space group P21 and has a distorted octahedral geometry. The IR spectra revealed the presence of variable modes of chelation for the investigated ligand. The EPR spectra of compounds [Cu2(DKN)2( -N3)2] and [Cu2(DKN)2( -NCS)2] in polycrystalline state suggest a dimeric structure as they exhibited a half field signal, which indicate the presence of a weak interaction between two Cu(II) ions in these complexes

Synthesis, Characterization and Applications of Modified TiO2 Systems

This thesis is divided in to 9 chapters and deals with the modification of TiO2 for various applications include photocatalysis, thermal reaction, photovoltaics and non-linear optics. Chapter 1 involves a brief introduction of the topic of study. An introduction to the applications of modified titania systems in various fields are discussed concisely. Scope and objectives of the present work are also discussed in this chapter. Chapter 2 explains the strategy adopted for the synthesis of metal, nonmetal co-doped TiO2 systems. Hydrothermal technique was employed for the preparation of the co-doped TiO2 system, where Ti[OCH(CH3)2]4, urea and metal nitrates were used as the sources for TiO2, N and metals respectively. In all the co-doped systems, urea to Ti[OCH(CH3)2]4 was taken in a 1:1 molar ratio and varied the concentration of metals. Five different co-doped catalytic systems and for each catalysts, three versions were prepared by varying the concentration of metals. A brief explanation of physico-chemical techniques used for the characterization of the material was also presented in this chapter. This includes X-ray Diffraction (XRD), Raman Spectroscopy, FTIR analysis, Thermo Gravimetric Analysis, Energy Dispersive X-ray Analysis (EDX), Scanning Electron Microscopy(SEM), UV-Visible Diffuse Reflectance Spectroscopy (UV-Vis DRS), Transmission Electron Microscopy (TEM), BET Surface Area Measurements and X-ray Photoelectron Spectroscopy (XPS). Chapter 3 contains the results and discussion of characterization techniques used for analyzing the prepared systems. Characterization is an inevitable part of materials research. Determination of physico-chemical properties of the prepared materials using suitable characterization techniques is very crucial to find its exact field of application. It is clear from the XRD pattern that photocatalytically active anatase phase dominates in the calcined samples with peaks at 2θ values around 25.4°, 38°, 48.1°, 55.2° and 62.7° corresponding to (101), (004), (200), (211) and (204) crystal planes (JCPDS 21-1272) respectively. But in the case of Pr-N-Ti sample, a new peak was observed at 2θ = 30.8° corresponding to the (121) plane of the polymorph brookite. There are no visible peaks corresponding to dopants, which may be due to their low concentration or it is an indication of the better dispersion of impurities in the TiO2. Crystallite size of the sample was calculated from Scherrer equation byusing full width at half maximum (FWHM) of the (101) peak of the anatase phase. Crystallite size of all the co-doped TiO2 was found to be lower than that of bare TiO2 which indicates that the doping of metal ions having higher ionic radius into the lattice of TiO2 causes some lattice distortion which suppress the growth of TiO2 nanoparticles. The structural identity of the prepared system obtained from XRD pattern is further confirmed by Raman spectra measurements. Anatase has six Raman active modes. Band gap of the co-doped system was calculated using Kubelka-Munk equation and that was found to be lower than pure TiO2. Stability of the prepared systems was understood from thermo gravimetric analysis. FT-IR was performed to understand the functional groups as well as to study the surface changes occurred during modification. EDX was used to determine the impurities present in the system. The EDX spectra of all the co-doped samples show signals directly related to the dopants. Spectra of all the co-doped systems contain O and Ti as the main components with low concentrations of doped elements. Morphologies of the prepared systems were obtained from SEM and TEM analysis. Average particle size of the systems was drawn from histogram data. Electronic structures of the samples were identified perfectly from XPS measurements. Chapter 4 describes the photocatalytic degradation of herbicides Atrazine and Metolachlor using metal, non-metal co-doped titania systems. The percentage of degradation was analyzed by HPLC technique. Parameters such as effect of different catalysts, effect of time, effect of catalysts amount and reusability studies were discussed. Chapter 5 deals with the photo-oxidation of some anthracene derivatives by co-doped catalytic systems. These anthracene derivatives come underthe category of polycyclic aromatic hydrocarbons (PAH). Due to the presence of stable benzene rings, most of the PAH show strong inhibition towards biological degradation and the common methods employed for their removal. According to environmental protection agency, most of the PAH are highly toxic in nature. TiO2 photochemistry has been extensively investigated as a method for the catalytic conversion of such organic compounds, highlighting the potential of thereof in the green chemistry. There are actually two methods for the removal of pollutants from the ecosystem. Complete mineralization is the one way to remove pollutants. Conversion of toxic compounds to another compound having toxicity less than the initial starting compound is the second way. Here in this chapter, we are concentrating on the second aspect. The catalysts used were Gd(1wt%)-N-Ti, Pd(1wt%)-N-Ti and Ag(1wt%)-N-Ti. Here we were very successfully converted all the PAH to anthraquinone, a compound having diverse applications in industrial as well as medical fields. Substitution of 10th position of desired PAH by phenyl ring reduces the feasibility of photo reaction and produced 9-hydroxy 9-phenyl anthrone (9H9PA) as an intermediate species. The products were separated and purified by column chromatography using 70:30 hexane/DCM mixtures as the mobile phase and the resultant products were characterized thoroughly by 1H NMR, IR spectroscopy and GCMS analysis. Chapter 6 elucidates the heterogeneous Suzuki coupling reaction by Cu/Pd bimetallic supported on TiO2. Sol-Gel followed by impregnation method was adopted for the synthesis of Cu/Pd-TiO2. The prepared system was characterized by XRD, TG-DTG, SEM, EDX, BET Surface area and XPS. The product was separated and purified by column chromatography using hexane as the mobile phase. Maximum isolated yield of biphenyl of around72% was obtained in DMF using Cu(2wt%)-Pd(4wt%)-Ti as the catalyst. In this reaction, effective solvent, base and catalyst were found to be DMF, K2CO3 and Cu(2wt%)-Pd(4wt%)-Ti respectively. Chapter 7 gives an idea about the photovoltaic (PV) applications of TiO2 based thin films. Due to energy crisis, the whole world is looking for a new sustainable energy source. Harnessing solar energy is one of the most promising ways to tackle this issue. The present dominant photovoltaic (PV) technologies are based on inorganic materials. But the high material, low power conversion efficiency and manufacturing cost limits its popularization. A lot of research has been conducted towards the development of low-cost PV technologies, of which organic photovoltaic (OPV) devices are one of the promising. Here two TiO2 thin films having different thickness were prepared by spin coating technique. The prepared films were characterized by XRD, AFM and conductivity measurements. The thickness of the films was measured by Stylus Profiler. This chapter mainly concentrated on the fabrication of an inverted hetero junction solar cell using conducting polymer MEH-PPV as photo active layer. Here TiO2 was used as the electron transport layer. Thin films of MEH-PPV were also prepared using spin coating technique. Two fullerene derivatives such as PCBM and ICBA were introduced into the device in order to improve the power conversion efficiency. Effective charge transfer between the conducting polymer and ICBA were understood from fluorescence quenching studies. The fabricated Inverted hetero junction exhibited maximum power conversion efficiency of 0.22% with ICBA as the acceptor molecule. Chapter 8 narrates the third order order nonlinear optical properties of bare and noble metal modified TiO2 thin films. Thin films were fabricatedby spray pyrolysis technique. Sol-Gel derived Ti[OCH(CH3)2]4 in CH3CH2OH/CH3COOH was used as the precursor for TiO2. The precursors used for Au, Ag and Pd were the aqueous solutions of HAuCl4, AgNO3 and Pd(NO3)2 respectively. The prepared films were characterized by XRD, SEM and EDX. The nonlinear optical properties of the prepared materials were investigated by Z-Scan technique comprising of Nd-YAG laser (532 nm,7 ns and10 Hz). The non-linear coefficients were obtained by fitting the experimental Z-Scan plot with the theoretical plots. Nonlinear absorption is a phenomenon defined as a nonlinear change (increase or decrease) in absorption with increasing of intensity. This can be mainly divided into two types: saturable absorption (SA) and reverse saturable absorption (RSA). Depending on the pump intensity and on the absorption cross- section at the excitation wavelength, most molecules show non- linear absorption. With increasing intensity, if the excited states show saturation owing to their long lifetimes, the transmission will show SA characteristics. Here absorption decreases with increase of intensity. If, however, the excited state has strong absorption compared with that of the ground state, the transmission will show RSA characteristics. Here in our work most of the materials show SA behavior and some materials exhibited RSA behavior. Both these properties purely depend on the nature of the materials and alignment of energy states within them. Both these SA and RSA have got immense applications in electronic devices. The important results obtained from various studies are presented in chapter 9.

Preparation and Study of Optical Properties of CdS, CdS-TiO2 and CdS-Au Nanocomposites for Photonic Applications

Nanophotonics can be regarded as a fusion of nanotechnology and photonics and it is an emerging field providing researchers opportunities in fundamental science and new technologies. In recent times many new methodsand techniques have been developed to prepare materials at nanoscale dimensions. Most of these materials exhibit unique and interesting optical properties and behavior. Many of these have been found to be very useful to develop new devices and systems such as tracers in biological systems, optical limiters, light emitters and energy harvesters. This thesis presents a summary of the work done by the author in the field by choosing a few semiconductor systems to prepare nanomaterials and nanocomposites. Results of the study of linear and nonlinear optical properties of materials thus synthesized are also presented in the various chapters of this thesis. CdS is the material chosen here and the methods and the studies of the detailed investigation are presented in this thesis related to the optical properties of CdS nanoparticles and its composites. Preparation and characterization methods and experimental techniques adopted for the investigations were illustrated in chapter 2 of this thesis. Chapter 3 discusses the preparation of CdS, TiO2 and Au nanoparticles. We observed that the fluorescence behaviour of the CdS nanoparticles, prepared by precipitation technique, depends on excitation wavelength. It was found that the peak emission wavelength can be shifted by as much as 147nm by varyingthe excitation wavelengths and the reason for this phenomenon is the selective excitation of the surface states in the nanoparticles. This provided certain amount of tunability for the emission which results from surface states.TiO2 nanoparticle colloids were prepared by hydrothermal method. The optical absorption study showed a blue shift of absorption edge, indicating quantum confinement effect. The large spectral range investigated allows observing simultaneously direct and indirect band gap optical recombination. The emission studies carried out show four peaks, which are found to be generated from excitonic as well as surface state transitions. It was found that the emission wavelengths of these colloidal nanoparticles and annealed nanoparticles showed two category of surface state emission in addition to the excitonic emission. Au nanoparticles prepared by Turkevich method showed nanoparticles of size below 5nm using plasmonic absorption calculation. It was also found that there was almost no variation in size as the concentration of precursor was changed from 0.2mM to 0.4mM.We have observed SHG from CdS nanostructured thin film prepared onglass substrate by chemical bath deposition technique. The results point out that studied sample has in-plane isotropy. The relative values of tensor components of the second-order susceptibility were determined to be 1, zzz 0.14, xxz and 0.07. zxx These values suggest that the nanocrystals are oriented along the normal direction. However, the origin of such orientation remains unknown at present. Thus CdS is a promising nonlinear optical material for photonic applications, particularly for integrated photonic devices. CdS Au nanocomposite particles were prepared by mixing CdS nanoparticles with Au colloidal nanoparticles. Optical absorption study of these nanoparticles in PVA solution suggests that absorption tail was red shifted compared to CdS nanoparticles. TEM and EDS analysis suggested that the amount of Au nanoparticles present on CdS nanoparticles is very small. Fluorescence emission is unaffected indicating the presence of low level of Au nanoparticles. CdS:Au PVA and CdS PVA nanocomposite films were fabricated and optically characterized. The results showed a red-shift for CdS:Au PVA film for absorption tail compared to CdS PVA film. Nonlinear optical analysis showed a huge nonlinear optical absorption for CdS:Au PVA nanocomposite and CdS:PVA films. Also an enhancement in nonlinear optical absorption is found for CdS:Au PVA thin film compared to the CdS PVA thin film. This enhancement is due to the combined effect of plasmonic as well as excitonic contribution at high input intensity. Samples of CdS doped with TiO2 were also prepared and the linear optical absorption spectra of these nanocompositeparticles clearly indicated the influence of TiO2 nanoparticles. TEM and EDS studies have confirmed the presence of TiO2 on CdS nanoparticles. Fluorescence studies showed that there is an increase in emission peak around 532nm for CdS nanoparticles. Nonlinear optical analysis of CdS:TiO2 PVA nanocomposite films indicated a large nonlinear optical absorption compared to that of CdS:PVA nanocomposite film. The values of nonlinear optical absorption suggests that these nanocomposite particles can be employed for optical limiting applications. CdSe-CdS and CdSe-ZnS core-shell QDs with varying shell size were characterized using UV–VIS spectroscopy. Optical absorption and TEM analysis of these QDs suggested a particle size around 5 nm. It is clearly shown that the surface coating influences the optical properties of QDs in terms of their size. Fluorescence studies reveal the presence of trap states in CdSe-CdS and CdSe- ZnS QDs. Trap states showed an increase as a shell for CdS is introduced and increasing the shell size of CdS beyond a certain value leads to a decrease in the trap state emission. There is no sizeable nonlinear optical absorption observed. In the case of CdSe- ZnS QDs, the trap state emission gets enhanced with the increase in ZnS shell thickness. The enhancement of emission from trap states transition due to the increase in thickness of ZnS shell gives a clear indication of distortion occurring in the spherical symmetry of CdSe quantum dots. Consequently the nonlinear optical absorption of CdSe-ZnS QDs gets increased and the optical limiting threshold is decreased as the shell thickness is increased in respect of CdSe QDs. In comparison with CdSe-CdS QDs, CdSe-ZnS QDs possess much better optical properties and thereby CdSe-ZnS is a strong candidate for nonlinear as well as linear optical applications.