Estudo teórico QTAIM e DFT dos compostos de coordenação: efeito quelato, titanocenos e ligação química

This work is a study of coordination compounds by quantum theory of atoms in molecules (QTAIM), based on the topological analysis of the electron density of molecular systems, both theoretically and experimentally obtained. The coordination chemistry topics which were studied are the chelate effect, bent titanocene and chemical bond in coordination complexes. The chelate effect was investigated according to topological and thermodynamic parameters. The exchange of monodentate ligands on polydentate ligands from same transition metal increases the stability of the complex both from entropy and enthalpy contributions. In some cases, the latter had a higher contribution to the stability of the complex in comparison with entropy. This enthalpic contribution is explained according to topological analysis of the M-ligand bonds where polidentate complex had higher values of electron density of bond critical point, Laplacian of electron density of bond critical point and delocalization index (number of shared electrons between two atoms). In the second chapter, was studied bent titanocenes with bulky cyclopentadienyl derivative π-ligand. The topological study showed the presence of secondary interactions between the atoms of π-ligands or between atoms of π-ligand and -ligand. It was found that, in the case of titanocenes with small difference in point group symmetry and with bulky ligands, there was an nearly linear relationship between stability and delocalization index involving the ring carbon atoms (Cp) and the titanium. However, the titanocene stability is not only related to the interaction between Ti and C atoms of Cp ring, but secondary interactions also play important role on the stability of voluminous titanocenes. The third chapter deals with the chemical bond in coordination compounds by means of QTAIM. The quantum theory of atoms in molecules so far classifies bonds and chemical interactions in two categories: closed shell interaction (ionic bond, hydrogen bond, van der Waals interaction, etc) and shared interaction (covalent bond). Based on topological parameters such as electron density, Laplacian of electron density, delocalization index, among others, was classified the chemical bond in coordination compounds as an intermediate between closed shell and shared interactions

Estudo teórico aplicado ao petróleo: estabilidade dos alcenos e recuperação terciária

The present work aims to study the theoretical level of some processes employed in the refining of petroleum fractions and tertiary recovery of this fluid. In the third chapter, we investigate a method of hydrogenation of oil fractions by QTAIM (Quantum Theory of Atoms in Molecules) and thermodynamic parameters. The study of hydrogenation reactions, and the stability of the products formed, is directly related to product improvement in the petrochemical refining. In the fourth chapter, we study the theoretical level of intermolecular interactions that occur in the process of tertiary oil recovery, or competitive interactions involving molecules of non-ionic surfactants, oil and quartz rock where oil is accumulated. Calculations were developed using the semiempirical PM3 method (Parametric Model 3). We studied a set of ten non-ionic surfactants, natural and synthetic origin. The study of rock-surfactant interactions was performed on the surface of the quartz (001) completely hydroxylated. Results were obtained energetic and geometric orientations of various surfactants on quartz. QTAIM was obtained through the analysis of the electron density of interactions, and thus, providing details about the formation of hydrogen bonds and hydrogen-hydrogen systems studied. The results show that the adsorption of ethoxylated surfactants in the rock surface occurs through the hydrogen bonding of the type CH---O, and surfactants derivatives of polyols occurs by OH---O bonds. For structures adsorption studied, the large distance of the surfactant to the surface together with the low values of charge density, indicate that there is a very low interaction, characterizing physical adsorption in all surfactants studied. We demonstrated that surfactants with polar group comprising oxyethylene units, showed the lowest adsorption onto the surface of quartz, unlike the derivatives of polyols

Estudo comparativo entre a pirólise do polietileno de baixa densidade, utilizando vermiculita modificada e SBA-15

Catalytic processes are widely present in everyday life. This results in large number of studies seeking materials that may combine the low cost catalytic efficiency. Based on this assumption, the clays have long been used as catalysts, with its huge availability, diversity and possibility of improving their properties from structural changes, primarily responsible for this great use. Among the natural clays, vermiculite due to their characteristic properties (high cation exchange capacity and expansion), is suitable for various applications including as catalysts and catalyst supports. In this work, the acid leaching of clay vermiculite was performed, coming from Santa Luzia-PB, with nitric acid (2, 3 and 4 mol / L) and subsequent calcination of the materials obtained. The materials were named as Vx/400, where x is the acid concentration employed and 400 used in calcination temperature. The effectiveness of changes made was determined by XRD techniques, FT-IR, EDS, TG/DTG, nitrogen physisorption and DTP of n-butylamine. Acid leaching has improved some properties of the clay - specific area and acidity - but the control of the acid concentration used is of vital importance, since the highest concentration caused the partial destruction of vermiculite entailing a decline in their properties. For analysis of the catalytic activity of the modified clay was made a comparative study with the SBA -15 mesoporous materials, synthesized via hydrothermal method, using the pyrolysis of low density polyethylene (LDPE). The results showed that the acid plays a fundamental role in the conversion of the polymer into smaller molecules, the material V3/400 was more selective for the source monomer (ethylene) due to their increased acidity, which promotes more breaks bonds in the polymeric chain, while materials and V0/400 V2/400, lower acidity, showed higher selectivity to light hydrocarbons, the range of fuel (41.96 and 41.23%, respectively), due to less breakage and secondary condensation reactions chains; already V4/400 SBA-15/550 and resulted in lower percentages of light hydrocarbons and the partial destruction of the structure and low acidity, respectively, responsible for the inefficiency of materials

Obtenção e caracterização do sistema compósito alginato de sódio-dióxido de titânio em formas de pó e de membrana

The alginates are copolymers of 1→4-linked β-D-mannuronic acid (M) and α-Lguluronic acid (G) residues that are arranjed in a block structure along a linear chain. Titanium dioxide, TiO2, is a ceramic material and can exist in three distinct crystallography forms: anatase, brookite and rutile. composites of organic and inorganic materials have better properties than the components alone. Thus, this study aims to synthesize, characterize and analyze the composite NaAlg-TiO2 in the form of powder and film. The synthesis of composite powders was performed using the sol-gel process and obtain the composite film was performed using the slow evaporation process, then the composites were analyzed by infrared spectroscopy, fluorescence x ray, thermal analysis, attenuated total reflection (ATR), x ray diffraction and impedance spectroscopy. The X ray diffraction patterns of composite powders show that with increasing calcination temperature, there were no complete transition of rutile-anatase crystalline phase, since at all temperatures studied (300, 500, 700, 900 and 1100ºC) were observed peaks of anatase phase. Thermal analysis shows that at 400°C caused the decomposition of sodium alginate in sodium carbonate and above 600°C, we observe an exothermic peak related to the decomposition of sodium carbonate and in the presence of titanium dioxide becomes sodium titanate. The XRD results confirm the formation of sodium carbonate at 700ºC and the formation sodium titanate in the temperature range 900-1100ºC. The sodium titanate influenced the electrical properties of the material, because with increasing temperature there was a decrease in conductivity, probably due to the creation of Ti vacancies, since the sodium can induce the reduction of surface Ti4+ ions into Ti3+ species. The infrared spectra of the composites in the form of powder and film showed a small shift in the bands compared to the spectrum of pure alginate, indicating that these shifts, even small ones, have evidence of miscibility between the polymer and ceramic material

Análise das avaliações realizadas na disciplina de arquitetura atômica e molecular no curso de química, modalidade a distância da UFRN

The field of education is very rich and allows us to research in various aspects. The area of chemical education has been growing more and more, and an important aspect that has been researching this area is about the learning difficulties of students. The approach of the themes atomic structure and chemical bonds are developed in high school and have many problems that are often brought to higher education becoming an obstacle to the advancement of learning. It is necessary for these initial themes - the atomic structure and chemical bonds - are well understood by the student to the other contents of Chemistry will be understood more easily. This paper aims to describe, analyze errors and difficulties presented in the assessments of the discipline Atomic and Molecular Architecture, the students of the degree course in Chemistry - EAD, with respect to the contents of " Atomic Structure and Chemical Bonding ", by of the assessments made by the students and the Virtual Learning Environment (VLE), taking into account the activities , discussion forum and access to materials . AVA allows obtaining reports which were used to analyze regarding access / participation to assess their contribution to learning and its relation to the final result (pass / fail). It was observed that the most frequent errors in the assessments are related to the early part of the chemistry that is the understanding of atomic structure and evolution models. Students who accessed the extra material and participated in the activities and forums were students who achieved success in the course. Ie, the difficulties were emerging and the use of available teaching strategies, students could minimize such difficulties, making their performance in activities and assessments were better. Was also observed by attending the AVA, the discipline began with a large withdrawal from the page access as well as the frequency of face- evidence from observation in Listing presence of classroom assessments

Estudo da imobilização de proteases para a síntese de oligolisinas

Enzymatic synthesis of peptides using proteases has attracted a great deal of attention in recent years. One key challenge in peptide synthesis is to find supports for protease immobilization capable of working in aqueous medium at high performance, producing watersoluble oligopeptides. At present, few reports have been described using this strategy. Therefore, the aim of this thesis was to immobilize proteases applying different methods (Immobilization by covalent bound, entrapment onto polymeric gels of PVA and immobilization on glycidil metacrylate magnetic nanoparticles) in order to produce water-soluble oligopeptides derived from lysine. Three different proteases were used: trypsin, α-chymotrypsin and bromelain. According to immobilization strategies associated to the type of protease employed, trypsin-resin systems showed the best performance in terms of hydrolytic activity and oligopeptides synthesis. Hydrolytic activities of the free and immobilized enzymes were determined spectrophotometrically based on the absorbance change at 660 nm at 25 °C (Casein method). Calculations of oligolysine yield and average degree of polymerization (DPavg) were monitored by 1H-NMR analysis. Trypsin was covalently immobilized onto four different resins (Amberzyme, Eupergit C, Eupergit CM and Grace 192). Maximum yield of bound protein was 92 mg/g, 82 mg/g and 60 mg/g support for each resin respectively. The effectiveness of these systems (Trypsin-resins) was evaluated by hydrolysis of casein and synthesis of water-soluble oligolysine. Most systems were capable of catalyzing oligopeptide synthesis in aqueous medium, albeit at different efficiencies, namely: 40, 37 and 35% for Amberzyme, Eupergit C and Eupergit CM, respectively, in comparison with free enzyme. These systems produced oligomers in only 1 hour with DPavg higher than free enzyme. Among these systems, the Eupergit C-Trypsin system showed greater efficiency than others in terms of hydrolytic activity and thermal stability. However, this did not occur for oligolysine synthesis. Trypsin-Amberzyme proved to be more successful in oligopeptide synthesis, and exhibited excellent reusability, since it retained 90% of its initial hydrolytic and synthetic activity after 7 reuses. Trypsin hydrophobic interactions with Amberzyme support are responsible for protecting against strong enzyme conformational changes in the medium. In addition, the high concentration of oxirane groups on the surface promoted multi-covalent linking and, consequently, prevented the immobilized enzyme from leaching. The aforementioned results suggest that immobilized Trypsin on the supports evaluated can be efficiently used for oligopeptides synthesis in aqueous media

Síntese e caracterização de compostos de coordenação homolépticos e heterolépticos com o ligante ácido kójico

The present work has as main objective to contribute to the coordination chemistry of the ligand kojic acid, with the synthesis and characterization of the homoleptic compounds [Al(kj)3], [Fe(kj)3], [Fe(kj)2], [Cu(kj)2] e [Ru(kj)3], and the new heteroleptic complexes, trans- K2[Fe(kj)2(CN)2] and trans-Na2[Ru(kj)2(CN)2]. The obtained compounds were characterized by vibrational spectroscopy in the infrared region (IV) and Electronic spectroscopy in the ultraviolet and visible region (Uv-Vis). The infrared results indicated the coordination of the bidentate ligand kojic acid, due to reductions in the values of the stretching frequencies of the carbonyl and double bonds, compared to the free ligand for all complexes obtained. The presence of new vibrational modes indicated the change of symmetry of the molecules in the new compounds synthesized. Additionally, the presence of vibrational modes assigned to metal-oxygen also contributed to confirm the ligand coordinating to the metal ions. Through this technique, was also possible to perform correlations of the numbers of vibrational modes, in the region 1400-900 cm-1 and the compounds geometry. The heteroleptic compounds exhibited υC≡N in 2065 and 2053 cm-1, respectively, for the trans-K2[Fe(kj)2(CN)2] and trans-Na2[Ru(kj)2(CN)2], indicating coordination of the cyano ligand to metal ions FeII e RuII. Comparing the obtained values with literature data was possible to identify the complex isomerism as trans. In relation to the results of electronic spectroscopy, studies of pH variation of kojic acid provided information on the distribution of electron density in the molecule, showing characteristic spectral profile of kojic ion and its protonated form (Hkj, kojic acid), with two bands at 215 and 269 nm, or deprotonated (kj-), with bands at 226 and 315 nm. The electronic spectra obtained for all complexes in aqueous medium, in the ultraviolet region, exhibited variations of the energies assigned to kojic acid intraligand transitions while in the visible region, only transitions assigned to charge transfer of iron and ruthenium complex have been identified

Interações entre a carboximetilcelulose, carbonato de cálcio e bentonita: repercussões sobre as propriedades dos fluidos de perfuração aquosos

The role of carboxymethylcellulose (CMC) in association to calcium carbonate particles (CaCO3) in most water-based drilling fluids is to reduce the fluid loss to the surrounding formation. Another essential function is to provide rheological properties capable of maintaining in suspension the cuttings during drilling operation. Therefore, it is absolutely essential to correlate the polymer chemical structure (degree of substitution, molecular weight and distribution of substituent) with the physical-chemical properties of CaCO3, in order to obtain the better result at lower cost. Another important aspect refers to the clay hydration inhibitive properties of carboxymethylcellulose (CMC) in drilling fluids systems. The clay swelling promotes an undesirable damage that reduces the formation permeability and causes serious problems during the drilling operation. In this context, this thesis consists of two main parts. The first part refers to understanding of interactions CMC-CaCO3, as well as the corresponding effects on the fluid properties. The second part is related to understanding of mechanisms by which CMC adsorption occurs onto the clay surface, where, certainly, polymer chemical structure, ionic strength, molecular weight and its solvency in the medium are responsible to affect intrinsically the clay layers stabilization. Three samples of carboximetilcellulose with different molecular weight and degree of substitution (CMC A (9 x 104 gmol DS 0.7), CMC B (2.5 x 105 gmol DS 0.7) e CMC C (2.5 x 105 gmol DS 1.2)) and three samples of calcite with different average particle diameter and particle size distribution were used. The increase of CMC degree of substitution contributed to increase of polymer charge density and therefore, reduced its stability in brine, promoting the aggregation with the increase of filtrate volume. On the other hand, the increase of molecular weight promoted an increase of rheological properties with reduction of filtrate volume. Both effects are directly associated to hydrodynamic volume of polymer molecule in the medium. The granulometry of CaCO3 particles influenced not only the rheological properties, due to adsorption of polymers, but also the filtration properties. It was observed that the lower filtrate volume was obtained by using a CaCO3 sample of a low average size particle with wide dispersion in size. With regards to inhibition of clay swelling, the CMC performance was compared to other products often used (sodium chloride (NaCl), potassium chloride (KCl) and quaternary amine-based commercial inhibitor). The low molecular weight CMC (9 x 104 g/mol) showed slightly lower swelling degree compared to the high molecular weight (2.5 x 105 g/mol) along to 180 minutes. In parallel, it can be visualized by Scanning Electron Microscopy (SEM) that the high molecular weight CMC (2.5 x 105 g/mol e DS 0.7) promoted a reduction in pores formation and size of clay compared to low molecular weight CMC (9.0 x 104 g/mol e DS 0.7), after 1000 minutes in aqueous medium. This behavior was attributed to dynamic of interactions between clay and the hydrodynamic volume of CMC along the time, which is result of strong contribution of electrostatic interactions and hydrogen bounds between carboxylate groups and hydroxyls located along the polymer backbone and ionic and polar groups of clay surface. CMC adsorbs on clay surface promoting the skin formation , which is responsible to minimize the migration of water to porous medium. With the increase of degree of substitution, it was observed an increase of pores onto clay, suggesting that the higher charge density on polymer is responsible to decrease its flexibility and adsorption onto clay surface. The joint evaluation of these results indicate that high molecular weight is responsible to better results on control of rheological, filtration and clay swelling properties, however, the contrary effect is observed with the increase of degree of substitution. On its turn, the calcite presents better results of rheological and filtration properties with the decrease of average viii particle diameter and increase of particle size distribution. According to all properties evaluated, it has been obvious the interaction of CMC with the minerals (CaCO3 and clay) in the aqueous medium

Obtenção de dispersões de complexos polieletrolíticos à base de quitosana e poli(ácido metacrílico) e análise de adsorção de albumina bovina sérica

Dispersions composed of polyelectrolyte complexes based on chitosan and poly(methacrylic acid), PMAA, were obtained by the dropping method and template polymerization. The effect of molecular weight of PMAA and ionic strength on the formation of chitosan/poly(methacrylic acid), CS/PMAA, complexes was evaluated using the dropping method. The increase in molecular weight of PMAA inhibited the formation of insoluble complexes, while the increase in ionic strength first favored the formation of the complex followed by inhibiting it at higher concentrations. The polyelectrolyte complexation was strongly dependent on macromolecular dimensions, both in terms of molecular weight and of coil expansion/contraction driven by polyelectrolyte effect. The resultant particles from dropping method and template polymerization were characterized as having regions with different charge densities: chitosan predominating in the core and poly(methacrylic acid) at the surface, the particles being negatively charged, as a consequence. Albumin was adsorbed on templatepolymerized CS/PMAA complexes (after crosslinking with glutardialdehyde) and pH was controlled in order to obtain two conditions: (i) adsorption of positively charged albumin, and (ii) adsorption of albumin at its isoelectric point. Adsorption isotherms and zeta potential measurements showed that albumin adsorption was controlled by hydrogen bonding/van der Waals interactions and that brushlike structures may enhance adsorption of albumin on these particles

Desenvolvimento de catalisadores bifuncionais de óxido de zircônio modificado por óxidos de tungstênio e molibdênio contendo platina para a reação de isomerização de n-parafinas

Bifunctional catalysts based on zircon oxide modified by tungsten (W = 10, 15 and 20 %) and by molybdenum oxide (Mo= 10, 15 e 20 %) containg platinum (Pt = 1%) were prepared by the polymeric precursor method. For comparison, catalysts the tungsten base was also prepared by the impregnation method. After calcinations at 600, 700 and 800 ºC, the catalysts were characterized by X-ray diffraction, fourier-transform infrared spectroscopy, thermogravimetric and differential thermal analysis, nitrogen adsorption and scanning electron microscopy. The profile of metals reduction was determined by temperature programmed reduction. The synthesized catalysts were tested in n-heptane isomerization. X-ray diffractogram of the Pt/WOx-ZrO2 and Pt/MoOx-ZrO2 catalysts revealed the presence of tetragonal ZrO2 and platinum metallic phases in all calcined samples. Diffraction peaks due WO3 and ZrO2 monoclinic also were observed in some samples of the Pt/WOx-ZrO2 catalysts. In the Pt/MoOx-ZrO2 catalysts also were observed diffraction peaks due ZrO2 monoclinic and Zr(MoO4)2 oxide. These phases contained on Pt/WOx-ZrO2 and Pt/MoOx-ZrO2 catalysts varied in accordance with the W or Mo loading and in accordance with the calcination temperature. The infrared spectra showed absorption bands due O-W-O and W=O bonds in the Pt/WOx-ZrO2 catalysts and due O-Mo-O, Mo=O and Mo-O bonds in the Pt/MoOx-ZrO2 catalysts. Specific surface area for Pt/WOx-ZrO2 catalysts varied from 30-160 m2 g-1 and for the Pt/MoOx-ZrO2 catalysts varied from 10-120 m2 g-1. The metals loading (W or Mo) and the calcination temperature influence directly in the specific surface area of the samples. The reduction profile of Pt/WOx-ZrO2 catalysts showed two peaks at lower temperatures, which are attributed to platinum reduction. The reduction of WOx species was evidenced by two reduction peak at high temperatures. In the case of Pt/MoOx-ZrO2 catalysts, the reduction profile showed three reduction events, which are attributed to reduction of MoOx species deposited on the support and in some samples one of the peak is related to the reduction of Zr(MoO4)2 oxide. Pt/WOx-ZrO2 catalysts were active in the n-heptane isomerization with high selectivity to 3-methyl-hexane, 2,3- dimethyl-pentane, 2-methyl-hexane among other branched hydrocarbons. The Pt/MoOx-ZrO2 catalysts practically didn't present activity for the n-heptane isomerization, generating mainly products originating from the catalytic cracking

Generalizações do conceito de distância, i-distâncias, distâncias intervalares e topologia

In this dissertation we present some generalizations for the concept of distance by using more general value spaces, such as: fuzzy metrics, probabilistic metrics and generalized metrics. We show how such generalizations may be useful due to the possibility that the distance between two objects could carry more information about the objects than in the case where the distance is represented just by a real number. Also in this thesis we propose another generalization of distance which encompasses the notion of interval metric and generates a topology in a natural way. Several properties of this generalization are investigated, and its links with other existing generalizations

Avaliação de uma fração polissacaridica da alga Lobophora variegata (Lamouroux) em modelo de artrite induzida em ratos por Zymosan

Fucans seaweed Lobophora variegata estructures are known for their chemical and biological properties. In this study, we analyzed, the action of fucans L. variegata and the fractions purified with acetone in Zymosan-induced arthritis. After differential fractionation with acetone, six fractions were obtained and named F0.3, F0.5, F0.8, F1, F1.5 and F2. The results showed that the fraction F1 showed high yield (51.9%) and was chosen for studies of antioxidant activity and induced arthritis. Nuclear magnetic resonance (NMR) of 13C showed signals at 103.3 and 15.78 ppm that are assigned to links β13 galactose and of the C6 methyl fucose, respectively. The infrared (IR) showed absorbance at 1238 and 850 cm-1 which are attributed to sulfate. The fraction F1 showed antioxidant activities in vitro. For analysis of inflammatory parameters chosen the polysaccharide was administered in different doses (25, 50 and 75 mg / kg ip, per body weight) and diclofenac sodium (5 mg / kg ip) and L-NAME (25 mg / kg ip) in groups of animals (n = 6). After 6 h, were analyzed for cellular influx and levels of nitrite. In experiment five days, were made analysis of swelling and serum TNF-α. Histopathological analysis were performed for confirmation of results. The fraction F1 (25, 50 and 75 mg / kg ip) reduced the cellular influx (52.1 to 96.7%) and nitric oxide levels (27.2 - 39%) compared to control group. The reduction of edema (63.4 - 100%) and serum TNF-α (p <0.001) were observed when the polysaccharide F1 administered at a dose (50 mg / kg) These results suggest that these heterofucanas of Lobophora variegata have besides the activity antioxidant and potential anti-inflammatory activity in arthritis induced by zymosan

Purificação e caracterização estrutural de homogalactanas sulfatadas anticoagulantes da alga verde Codium isthmocladum

Conselho Nacional de Desenvolvimento Científico e Tecnológico

Histerese térmica de sistemas magnéticos nanoestruturados

We report a theoretical investigation of thermal hysteresis in magnetic nanoelements. Thermal hysteresis originates in the existence of meta-stable states in temperature intervals which may be tuned by small values of the external magnetic field, and are controlled by the systems geometric dimensions as well as the composition. Two systems have been investigated. The first system is a trilayer consisting of one antiferromagnetic MnF2 film, exchange coupled with two Fe lms. At low temperatures the ferromagnetic layers are oriented in opposite directions. By heating in the presence of an external magnetic field, the Zeeman energy induces a gradual orientation of the ferromagnets with the external field and the nucleation of spin- op-like states in the antiferromagnetic layer, leading eventually, in temperatures close to the Neel temperature, to full alignment of the ferromagnetic films and the formation of frustrated exchange bonds in the center of the antiferromagnetic layer. By cooling down to low temperatures, the system follows a different sequence of states, due to the anisotropy barriers of both materials. The width of the thermal hysteresis loop depends on the thicknesses of the FM and AFM layers as well as on the strength of the external field. The second system consists in Fe and Permalloy ferromagnetic nanoelements exchange coupled to a NiO uncompensated substrate. In this case the thermal hysteresis originates in the modifications of the intrinsic magnetic

Dinâmica e estrutura de redes complexas no modelo de afinidade

In this work we elaborate and discuss a Complex Network model which presents connectivity scale free probability distribution (power-law degree distribution). In order to do that, we modify the rule of the preferential attachment of the Bianconi-Barabasi model, including a factor which represents the similarity of the sites. The term that corresponds to this similarity is called the affinity, and is obtained by the modulus of the difference between the fitness (or quality) of the sites. This variation in the preferential attachment generates very interesting results, by instance the time evolution of the connectivity, which follows a power-law distribution ki / ( t t0 )fi, where fi indicates the rate to the site gain connections. Certainly this depends on the affinity with other sites. Besides, we will show by numerical simulations results for the average path length and for the clustering coefficient