The Extremal Generalized Theta Graph with Respect to the Hosoya Index

The Members of Scientific Committee

A.R. Ashrafi University of Kashan, Iran M.R. Darafsheh University of Tehran, Iran M.V. Diudea Babes-Bolyai University, Romania T. Doslic University of Zagreb,Croatia M. Ghoran Nevis University Of Islamic Azad I. Gutman University of Kragujevac, Serbia A. Graovac Rugjer Boskovic Institute, Croatia A. Hoseinzade Lofty Islamic Azad University A. Iranmanesh Tarbiat Modares University, Iran M.A. Iranmanesh University of Yazd, Iran B. Ranjbar Tarbiat Modares University, Iran H. Sabzyan University of Isfahan, Iran B. Taeri Isfahan University of Tecnology, Iran A. Tehranian Islamic Azad University H. Yousefi University of Tehran, Iran K. Zare University of Shahid Beheshty, Iran

The Members of Organizing Committee

Y. Alizadeh A. Moeini V. Aram A. Mohammadpour A.R. Ashrafi S. Naghdi E. Babaei S. M. Najibi A.S. Baghsahi K. Nikzad A. Bahreininejad H. Omid S.N. Banihashemi S.Pouyandeh M.R.Eslahchi B. Ranjbar Y. Fatholahi M.Sabouri A. Giahtazeh N. Sari R. Hooshmand M.Sokhani A. Iranmanesh S. J. Tabatabaei L. Jadidoleslam F. Tavasolian K.S. Kazemi M.H. Yadegari M .Khaksar M. Zeratkar S. Mirzaei

Content Invited Talk

H. Abdollahi Calculation and Visualizing of the Ambiguities in Self‐Modeling Multivariate Curve Resolution Methods ...... 1 M. V. Diudea Nanostructures: design and topological characterization ...... 3 T. Doslic Structural and enumerative aspects of maximal matchings in nanostructures ...... 5 M. Eliasi , G. Raeisi and B. Taeri some graph operations and their topological indices ...... 6 M. Ghorbani and A. R. Ashrafi A Note on Markaracter Tables of Finite Groups ...... 7 A. Graovac On applications of the wiener index un nano sciences ...... 8 I. Gutman Introduction to the Theory of Fluoranthenes ...... 10 Contributed Talk

S. Ahmadian, S. Mah Abadi, M. Sadeghi, H. Pezeshk, C. Eslahchi Construction of Random Perfect Phylogeny Matrix ...... 13

S. Alikhani and M. A. Iranmanesh

Chromatic Polynomials of Some Nanostars ...... 16 M. Arezoomand , B. Taeri First and second Zagreb indices of generalized hierarchical product of graphs ...... 20 A. Asghari, M. Ghazaghi, M. Bagheri-roochi, M. Rajabi Application of experimental design based on Full factorial & Central Composite… ...... 23 A. Behmaram, H. Yousefi-Azari and A. R. Ashrafi MATCING IN FULLERENES ...... 24

A. Dolati and S. Golalizadeh The Extremal Generalized Theta Graph with Respect to the Hosoya Index ...... 25 C. Eslahchi , H. Pezeshk, M. Sadeghi, N Afzaly, A Katanforoush Haplotype block partitioning and tagSNP selection under the perfect phylogeny model ...... 26 M. Golalizadeh and S. M. Najibi On Matching in Structural Bioinformatics via Statistical Shape Analysis ...... 31 A. Heydari Some results on the Wiener and terminal Wiener index of balanced trees ...... 35

A. Ilić, M. V. Diudea,F. Gholami-Nezhaad and A. R. Ashrafi Some polynomials and topological indices in nanocones with even or odd edges in the apex ...... 37 S. Z. Imani Golafshani, A. Iranmanesh , M. V. Diudea Computing the Cluj Index of TUC4C8(S) ...... 43 A. Iranmanesh and Y. Alizadeh

Eccentric connectivity Index of HAC5C7[p,q] and HAC5C6C7[p,q] Nanotubes ...... 45 S. Lotfei, M. Mazloum-Ardakani, G.Ghasemi, S. Ghahri Saremi Application of chemometric Method and spectrophotometric tech… ...... 46 A. Madanshekaf, M. Ghaneei, M. Moradi THE 1‐ CONNECTIVITY INDEX OF DENDRIMER NANOSTARS ...... 51

K. Maleknejad, B. Sohrabi, M. Alizadeh, R. Mollapourasl A Computational Scheme for Conductor‐like Screening Model by Sinc Quadrature Technique ...... 54

A. Mohajeri and M. Alipour Cusp at the nucleus for the Dunning's correlation consistent basis sets ...... 55 S. Moradi Eccentricity, Radius and Diameter of Tensor Product of Graph ...... 59 M. Saheli, A. R. Ashrafi and M. V. Diudea The Omega Polynomial of a new type of Nanostructures ...... 63

Z. Yarahmadia, T. Došlićb, A. R. Ashrafia A Note on The Bipartite Edge Frustration ...... 66

Posters

M.B.AHMADI, M. SADEGHIMEHR Third‐order Connectivity Index of an Infinite Class of Dendrimers ...... 72 M.B.AHMADI, Z. SEIF Computing frustration index using a mathematical programming model ...... 75 M.B.AHMADI, M. SALAMI The Neural Network Approach for Computing the Topological Index Am1 of C4C8[p,q] Nanotours .. 79 A. Afzal Shahidi, S. Rahimi Sharebaf Proposing an Upper and a Lower Bound for the Energy of Spider Graph ...... 83 H. Aminikhah An Analytical Approximation to the Solution of Chemical Kinetics System ...... 88 M. Arezoomand , A. Zare Chavoushi Reverse Wiener Index of Cartesian Product of Graphs ...... 94 E. Babaei and A. Iranmanesh Computation of the edge‐Wiener index of the rooted product of graphs ...... 97 J. Beheshtian, F. Zanjanchi, H.Behzadi, N. L. Hadipour A computational study of ammonia adsorption on boron nitride nanotube ...... 99 R. Behjatmanesh-Ardakani, F. Dehghan-Chenari MLR modeling for non‐nucleoside HIV‐1 reverse transcriptase inhibitors ...... 103 A. Dolati1, M. Haghighat, and M. Safari The Hosoya and Merri_eld‐Simmons indices of a denderimer nanostars ...... 108 A. Dolati and S. TAROMI COMPUTING THE ZAGREB INDICES OF A TYPE OF DENDRIMER NANOSTARS ...... 111 M. Eliasi and A.Iranmanesh Harary index of Armchair polyhex nanotubes ...... 113 M. Faghani and A. R. Ashrafi Topological Symmetry of Fullerenes ...... 114 P. Farhami and A. R. Ashrafi Some Remarks on DNA Sequences ...... 116 G. H. Fath-Tabar and A. R. Ashrafi The Hyper‐Wiener Polynomial of Graphs ...... 118

S.Garmroudi-Sabet, B.Sohrabi, S.M.Mousavi-khoshdel, S. M. Hashemianzadeh Theoretical study of the ternary oil‐water‐asymmetric mixed surfactants systems by lattice Monte Carlo simulation ...... 122 M. Ghiasi , A. Malekzadeh Estimating kinetic parameters of the Oxidative Coupling of Methane… ...... 127 M. Ghoranneviss, M. Eshghabadi, A. Mahmoodi Different Time of sputtering Effect on the Synthesis of Cu nano particles on AISI 304 by cylindrical magnetron sputtering system ...... 131 M. Ghorbani, M. Jalali and A. R. Ashrafi Counting Numbers of Permutational Isomers of some Hetero Fullerenes ...... 132 M. Ghorbani and M. Jalali On Omega and Sadhana Polynomials of Some Fullerenes ...... 133 M. Ghorbani, A. R. Ashrafi and M. Saheli On the Geometric Arithmetic GA5 Index ...... 134 M. Ghorbani and H. Maimani A New Type of Geometric Arithmetic Index ...... 135 A. Hamzeh, S. Hossein-Zadeh and A. R. Ashrafi COUNTING THE NUMBER OF (3,6)‐FULLERENES ...... 136 F. Hassani , O. Khormali and A. Iranmanesh Computation of Co‐PI index of TUC4C8(R) nanotubes ...... 140 S. Hossein-Zadeh, A. Hamzeh and A. R. Ashrafi The Study of Splices and Links of Graphs by Topological Index ...... 140 A. Iranmanesh, S. Mirzaie

Computing Schultz and modified Schultz polynomials of C100Fullerene by GAP program ...... 143 M. Keshavarz ,A. kazemi Babaheydari, K. Tavakoli Investigation and Study of Chemical Adsorption of Biatomic Gases on the… ...... 144 E. Keshmirzadeh Activity Calculations Based on A New Theory For Binary Mixtures Of Polymer/Solve…...... 145 Z.Khalaj , M.Ghoranneviss, S.Nasiri lahegi THE GROWTH OF DIAMOND NANO STRUCTURE ON SILICON SUBSTRATE USING DIFFERENT ETCHING GASSES BY HFCVD ...... 150 O. Khormali, A. Iranmanesh and A. Ahmadi On Zagreb Polynomials and indices ...... 151

V. Kiarostami, M. Davallo, M. vahidpur, A. kianmehr Optimization of the NBR blends for hardness property by Mixture Design ...... 152 S. Lotfei, M. Mazloum-Ardakani, G. Ghasemi Using algorithm at potentiometeric study of binary and mixed complex formation of some … ...... 153 M. Malekzadeh , M. A. Karimi A Kinetic‐Potentiometric Method with Partial least squares (PLS) and principle … ...... 157 M. Mazloum-Ardakani, P. Pourhakak Application of factorial design in digestion of ore samples by microwave and determination of some metals by AAS ...... 161 M. Mirzargar Automorphism Group Symmetry of Nanostars ...... 163 M. J. Najafi-Arani, H. Khodashenas and A. R. Ashrafi The Relationship between Szeged and Wiener Indices of Graphs ...... 167 A. Taherpour and A. M. Hashemi Theoretical Study of Free Energy of Electrontransfer Properties of Metal Nitride Cluster… ...... 171 A. Taherpour and R. Jalajerdi Theoretical Study of Free Electrontransfer Energy Properties of… ...... 176 A. Taherpour and N. Mahdizadeh

Polarizability Study of Fullerene Nano‐structures C20 to C300 by Using Monopole‐Dipole Interactions Theorem ...... 181 S. Rahimi Sharebaf , A. Afzal Shahidi A Classification of Upper and Lower Bounds of Energy in Graphs ...... 185 A. Seyed Mirzaei, A. R. Ashrafi and G. H. Fath-Tabar Szeged Index of some Dendrimers ...... 190 H. Shabani and A. R. Ashrafi Balaban Index of Dendrimers ...... 194 Z.Yarahmadi and S. Moradi Computing Some Topological Indices of Tensor Product of Graphs ...... 196 M. Zabihi Automorphism Group and Topological Indices of the Chemical Graph of Fullerenes ...... 200

Invited Talk

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Calculation and Visualizing of the Ambiguities in Self-Modeling Multivariate Curve Resolution Methods

Hamid Abdollahi Faculty of Chemistry, Institute for Advanced Studies in Basic Sciences 45195-159, Zanjan, Iran,

Multivariate Curve Resolution (MCR) has emerged and consolidated as a powerful chemometrics tool to investigate multivariate data with the goal of resolving, describing, interpreting and quantifying the true sources of data variance present in a particular data set. In Chemistry, it is rather usual to have large data sets of measured variables and samples. This is for instance the case of spectroscopy when multiwavelength measurements are performed on a set of different samples, or when chemical or processes evolve with time, chromatography or with any other physical or chemical variable like temperature. MCR is specially suited for the resolution of unknown mixtures and for the improvement of the selectivity properties in mixture analysis problems. In many of these problems, the initial knowledge of the system is partial or even non existent. MCR methods address this type of situations.

In many factor analysis methods, like in principal component analysis, the mathematical resolution of the contributions of the different components is performed under well-defined orthogonal constraints, normalization and maximum variance explained by successive resolved components. Using these constraints, PCA solutions are unique and there is no ambiguity in their estimation. However, since orthogonal constraints are not in general fulfilled by the contributions of the true underlying components in their mixtures (since in general these contributions do overlap and they are partially correlated), the component contributions deduced by PCA are completely different to the true ones.

On the other hand, in MCR methods, although they also assume the same bilinear model as in PCA, the mathematical resolution of the components is

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

performed using other type of more natural (with more physical sense) constraints like non-negativity, unimodality or mass balance-closure equations. So, MCR methods attempt to recover the true underlying contributions of the components in the mixture from the mathematical resolution of the original data system. The difficulty or trade off here is that the constraints applied in MCR do not assure a unique solution like in PCA. The mathematical properties of the MCR decomposition are not so well-defined as in PCA and the resolved contributions are more ambiguous [1,2]. In order to ascertain the quality of a particular MCR solution is then necessary to evaluate the amount of rotation ambiguity still present for this particular solution. Therefore the ambiguities in MCR solutions are very important problem in using soft-modeling methods. Visualizing the ambiguities can make sense for understanding of this problem and calculating the amount of these ambiguities is an important challenge in chemometrics area.

References:

[1] Maryam Vosough, Caroline Mason, Roma` Tauler, Mehdi Jalali-Heravi and Marcel Maeder, J. Chemometrics 2006; 20: 302–310.

[2] Hamid Abdollahi, Marcel Maeder, and Roma Tauler, Anal. Chem. 2009, 81, 2115–2122

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Nanostructures: design and topological characterization

Mircea V. Diudea Faculty of Chemistry and Chemical Engineering, “Babes-Bolyai” University, 400028 Cluj, Romania [email protected]

Abstract.

In the Nano-era, the last twenty years period, several new carbon allotropes have been discovered and studied for applications in nano-technology, in view of reducing the dimensions of devices and increasing their performance. Among the carbons structures, fullerenes (zero-dimensional), nanotubes (one dimensional), graphenes (two dimensional), spongy carbon and nano-diamond (three dimensional) represent the novelty [1,2].

Modeling the above structures is possible by using dedicated software based on cutting procedures (Torus), operations on maps (CageVersatile) or composition of networks by repeating units (Nano Studio), as developed at the TOPO Group CLUJ, ROMANIA (see Figure).

Toroidal objects tessellated by (6,,3) (left) an ((4,8)3)R (right) patterns

The attention of scientists was also directed to inorganic compounds, a realm where almost any metal atom can form clusters, tubules or crystal networks, very ordered structures at the nano-level. Recent articles in crystallography promoted the

3 The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

idea of topological description and classification of crystal structures [3]. They present data on real but also hypothetical lattices designed by computer.

In the present lecture, structures belonging to all of the Euclidean dimensions are described by their topology, particularly by Cluj [4], Omega [5,6] and related polynomials and indices. Appropriate examples will be given.

References

1. Diudea, M. V. (Ed.) Nanostructure, Novel Architecture, NOVA, N.Y., 2005. 2. Diudea, M. V.; Nagy, Cs. L. Periodic Nanostructures, SPRINGER, 2007. 3. V. A. Blatov, L. Carlucci, G. Ciani and D. Proserpio, Cryst. Eng. Comm., 2004, 6, 377. 4. Diudea, M. V. Cluj polynomials. J. Math. Chem., 2009, 45, 295 -308. 5. Diudea, M. V., Omega polynomial, Carpath. J. Math., 2006, 22, 43-47 6. John, P. E., Vizitiu, A. E., Cigher, S., Diudea, M. V., CI index in tubular nanostructures, MATCH Commun. Math. Comput. Chem., 2007, 57, 479-484.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Structural and enumerative aspects of maximal matchings in nanostructures

Tomislav Doslic Faculty of Civil Engineering, University of Zagreb, Croatia [email protected]

Since the very beginning of the chemical graph theory various matching- related concepts and invariants have been used to study the properties of wide classes of chemically interesting compounds and structures. This line of research has been vigorously followed over the course of several decades and has lead to accumulation of a vast number of results, mostly concerned with enumerating perfect matchings in benzenoids, and, more recently, also in fullerene graphs. Much less is known about maximal matchings, i.e., the matchings are not included in any larger matching. Here we are concerned with maximal matchings in light of their potential use in modeling adsorption processes on various structured surfaces. We investigate some of their structural properties, in particular the saturation number, and present upper and lower bounds on this quantity for the fullerene graphs. We also look at the enumerative challenges presented by the study of maximal matchings. Exact formulas for the number of maximal matchings of a given size are presented for various classes of graphs of chemical interest. A number of open problems are discussed and some possible directions for future research are suggested.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

some graph operations and their topological indices

Mehdi Eliasi , Ghafar Raeisi and Bijan Taeri Faculty of Mathematical Science, Tarbiat Modares University , Tehran Department of Mathematical Sciences, Isfahan University of Technology

Abstract

A topological index is a numerical value associated with chemical constitution purporting for correlation of chemical structure with various physical properties, chemical reactivity or biological activity.

In this lecture we compute some topological indices, for example, Wiener, hyper Wiener and Zagreb indices, of some class of graphs which constructed by some operations on graphs such as Mycielski's construction, t  th subdivision and generalized hierarchical products of graphs. Also we give some applications of these operations to compute topological indices of well-known graphs. In fact must results in this lecture are extensions of previous results.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

A Note on Markaracter Tables of Finite Groups

M. Ghorbani and A. R. Ashrafi Department of Mathematics,Faculty of Science, Shahid Rajaei Teacher Training University, Tehran, 16785 - 136, I R. Iran

Abstract

The concept of markaracter tables of finite groups was introduced first by a Japanese chemist Shinsaku Fujita. He applied this notion in the context of stereochemistry and enumeration of molecules. In this paper, a simple computational method is described, by means of which it is possible to calculate the markaracter tables of finite groups. A GAP program is also included which is efficient for computing markaracter table of groups of

order  10000. Using this program, the markaracter table of Ih point group symmetry is computed. This group appears as the point group symmetry of a Buckminster fullerene.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

ON APPLICATIONS OF THE WIENER INDEX IN NANOSCIENCES

Ante Graovac Institute "R. Boskovic", Zagreb, Croatia, and Faculty of Science, Split, Croatia [email protected]

Abstract

The Wiener index, W, has found many applications in chemistry and related fields and here we report on its possible applications in nanosciences.

In recent paper: O.Ori, F. Cataldo, and A. Graovac (2009) Fullerenes,

Nanotubes and Carbon Nanostructures, 17: 308-323, topological ranking of C28 fullerene reactivity is proposed and the number of 13C NMR peaks and their relative intensities are determined. The same authors (in: work in preparation), have applied a similar methodology to analyze C60H38 molecules of symmetry C1, C3 and T.

By following their earlier ideas on generalized Stone-Wales (gSW) transformations: D. Babic et al. (1995) Molecular Simulation 14:395-401, the authors have been able to derive a series of new results on the gSW in various nanostructures: F. Cataldo, O. Ori, and A. Graovac (2010) Int. J. Chem. Model., submitted.

Further, the Wiener index of infinite graphene lattice is computed and used as a sort of topological potential to determine and compare relative chemical stability of graphene layer to that of fullerene-like nanocone. It was shown that the central pentagonal ring becomes chemically reactive when the fullerene fragment reaches a certain size. Our considerations are performed both in dual and direct topological space.

In parallel with the above results related to experimental work in nanosciences, a series of mathematical studies on the Wiener index has been undertaken by the authors in collaboration with T. Doslic, especially on the eccentric connectivity in

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

hexagonal chains: MATCH Commun. Math. Comput. Chem., to appear, and in 2- dimensional square and comb lattices: the same journal, to be submitted.

The talk will be closed with discussion of some novel and intriguing ideas: D.Babic et al., in preparation, on relationship between the Wiener index and dimensionality of fractal lattices.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Introduction to the Theory of Fluoranthenes

Ivan Gutman Faculty of Science, University of Kragujevac, Kragujevac, Serbia [email protected]

Abstract

The theory of benzenoid hydrocarbons is one of the best elaborated parts of Theoretical Organic Chemistry and Mathematical Chemistry. Interestingly, however, until recently almost nothing has been done on the theory of a closely related class of polycyclic conjugated π-electron systems, referred here as fluoranthenes.

A fluoranthene F is obtained by joining two benzenoid fragments (X and Y) so as to form a new five-membered ring. Examples of fluoranthenes are 1, 2, 3, 4, and 5. The fragments X and Y are referred to as female and male, respectively.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Systematic studies of the topological properties of fluoranthenes started only two years ago [I. Gutman, J. Đurđević, Fluoranthene and its congeners - A graph theoretical study, MATCH Commun. Math. Comput. Chem. 60 (2008) 659-670]. In the lecture the following main results obtained so far will be outlined:

1. on Kekulé structures of fluoranthenes and their classification;

2. on fluoranthenes with extremal (minimal and maximal) number of Kekulè

structures;

3. on the validity of the Hall rule;

4. on the relations between the energy of the fragments X and Y and the

energy of F.

5. on cyclic conjugation in the five-membered ring;

6. the PCP and the linear rule for cyclic conjugation;

7. male/female differences for cyclic conjugation in the five-membered ring.

Contributed Talks

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Construction of Random Perfect Phylogeny Matrix

S. Ahmadian 1 , S. Mah Abadi 1 , M. Sadeghi 2,3 , H. Pezeshk 4 , C. Eslahchi 5 1: Department of Computer Engineering, Sharif University of Technology, Tehran, Iran. 2: National Institute of Genetic Engineering and Biotechnology, Tehran, Iran. 3: School of Computer Science, Institute for Studies in Theoretical Physics and Mathematics (IPM), Niavaran Sq., Tehran, Iran. 4: Center of Excellence in Biomathematics, School of Mathematics, Statistics and Computer Sciences, College of Science, University of Tehran, Tehran, Iran. 5: Department of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran.

Abstract

The amount of genome-wide molecular data and the interest in developing methods appropriate for such data are increasing rapidly. There is also an increasing need for methods that are able to efficiently simulate such data. In this paper we introduce a method for generating random perfect phylogeny matrix with k non identical haplotype, n Single Nucleotide Polymorphism(SNP) and population size of m for which the Minimum Allele Frequency(MAF) of each SNP is between two specific numbers a and b . We provide a graph theoretical approach for the problem to construct a tree for a perfect phylogeny matrix and find necessary and sufficient conditions for the existence of such a tree. We then provide an Omaxn((,)2 nm) algorithm for construction of the random tree which satisfies the necessary conditions and construct the desired matrix from this tree. The running time of any algorithm for solving this problem would be ()nm . We have developed a software, RAPPER, based on this algorithm. It is available at http://bioinf.cs.ipm.ir/softwares/RAPPER.http://bioinf.cs.ipm.ir/softwares/RAPPER Keywords: Perfect phylogeny, Minimum Allele Frequency(MAF), Tree, Recursive algorithm.

Introduction and Preliminaries

With the widespread availability of molecular data, computational methods for gene mapping are being developed. It is often the case that the statistical properties and the behavior of these methods need to be assessed and tested by simulation. By increasing the number of computational methods for gene mapping, there is an increasing need for tools that can simulate data appropriate for long genomic regions. SNPs are used as markers available for both linkage analysis and family-based association studies (Murray et al. (2004) and Kong et al. (2004)). Once a model for the generating polymorphism data as genotype or haplotype is in place, different studies can be done using the simulated data. One of the most popular model is based on the perfect phylogeny or coalescent property. By using the set of haplotypes which satisfy the coalescent property we can simulate a long genomic regions for which the simulated data does not depend on the data set. The simulated data can also be used as an approximation to the evolutionary processes which produced the real data. However the coalescent has been proved to be a powerful simulation tool in this context (Schaffner et al 2005). By above discussions, finding a method to construct random perfect phylogeny matrix with k non identical haplotypes, SNPs and population size of m where minimum allele

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010. frequency (MAF) of each SNP is between two specific numbers a and b will be very useful for data simulation (We consider MAF of column c in A as the number of 1's in column c). In this paper we take a graph theory approach to the problem and show that there is a one to one correspondence between the set of perfect phylogeny matrices with certain conditions and some rooted trees and then find necessary and sufficient conditions for existing such trees with respect to input parameters. We present an Omaxn((,)2 nm) algorithm for generating a random matrix with above conditions. To find some necessary and sufficient conditions for the existence of a perfect phylogeny matrix, PPM, with m rows, k non identical rows and n columns such that in every column number of 1's are between a and b , we need some definitions. We consider the cases that k  2 .

Definition 1 . Let a and b be two integer numbers and assume ab . The matrix

B mn is called a (k,a,b)-PPM if B is a PPM, the number of 1's in each column is between a and b and B has k non identical rows.

The matrix Akn is called (m,a,b)-extendable if A is a PPM, has k non identical rows and there exists a matrix B mn which is (k,a,b)-PPM and the set of rows of A and B are identical. (In this case we say that Akn is extendable to B mn ). A matrix B is called good if it can be decomposed as follows:

1- The entries of its leftmost column are all 1's.

2- There exist good matrices BB12,,, Bd such that the rest (0 or more) of the columns of B form the block-structure.

A matrix A is canonical if it satisfies the second condition of the good matrix definition. In a good matrix B we consider the leftmost all-one-column as the root of the B.

Matrices and Trees

According to Theorem 8 of Pe'er (2004), every PPM has an ordering of its rows and columns which yields a canonical matrix. We construct a tree T ()A from canonical form of

PPM A with non identical rows. Let Akn consist of BB12,,, Bd good blocks and ci be the corresponding root of B i . We construct the tree T ()A by taking the following steps:

Add an all-one-column to A as the leftmost column. Index this column by 0.

Let vertex 0 be the parent of ci for all 1  id

Construct a tree form canonical form of every B i in a recursive manner.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

The vertex set of T ()A is {0,1,2, ,n } . Now we label some vertices of T ()A as follows:

If the last 1 entry in row r occurs in the column j then we label the vertex j of

T ()A with r .

Let Akn be a ( mab,, )-extendable matrix and let the ( kab,, )-PPM matrix B mn be its extension and T ()A be its corresponding tree. Let w be the repeating time function defined on the labeled vertices of T ()A as w(vt ) = if and only if v is labeled by r and the row r repeated t times in B. We call (,)Tw()A the extended Tree of A and w(v ) as repeating label of v .

Theorem 2 . (T ,w) is a ( mab,, )-extendable tree which its root has label. Let r be the root of T and degT () r = d and T has l leaves if and only if

lk1.

ka(1)1 l  dl. b

la(1) k m.

Now we construct a rooted tree T such that degT ()= r d ,for which x 12,xx ,..., d are its children. T has n vertices, l leaves and c labelled internal vertices. Then, we add the xi i i i root r and edges rx i , 1 id. The desired tree T is constructed. Software implementing this algorithm was designed and it is available at: http://bioinf.cs.ipm.ir/softwares/RAPPERhttp://bioinf.cs.ipm.ir/softwares/RAPPER

References

[1] Excoffier, L., Novembre, J., and Schneider, S. 2000. SIMCOAL: a general coalescent program for the simulation of molecular data in interconnected populations with arbitrary demography. J. Heredity. 91, 506-509.

[2] Pe're, I., Pupko, T., Shamir, R., and Sharan, R. 2004. Incomplete directed perfect phylogeny. SIAM Journal of Computing. 33, 597-607.

[3] Schaffner, S.F., Foo, C., Gabriel, S., Reich, D., Daly, M.J., and Altshuler, D. 2005. Calibrating a coalescent simulation of human genome sequence variation. Genome Res. 15, 1576-1583.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Chromatic Polynomials of Some Nanostars

Saeid Alikhani 1 and Mohammad A. Iranmanesh Department of Mathematics uept Yazd University 89195-741, Yazd, Iran ABSTRACT

Let G be a simple graph and (,)G  denotes the number of proper vertex colourings of G with at most  colours, which is for a fixed graph G , a polynomial in  , which is called the chromatic polynomial of G . Using the chromatic polynomial of some specific graphs, we obtain the chromatic polynomials of some nanostars.

Keywords: Chromatic polynomial; Nanostar

Introduction

A simple graph GVE=( , ) is a finite nonempty set VG() of objects called vertices together with a (possibly empty) set E ()G of unordered pairs of distinct vertices of G called edges. In chemical graphs, the vertices of the graph correspond to the atoms of the molecule, and the edges represent the chemical bonds.

Let (,)G  denotes the number of proper vertex colourings of G with at most  colours. G. Birkhoff [5], observed in 1912 that (,)G  is, for a fixed graph G , a polynomial in  , which is now called the chromatic polynomial of G . More precisely, let G be a simple graph and  N . A mapping f :VG ( ) {1,2, , } is called a  -colouring of G if f ()u  f ()v whenever the vertices u and v are adjacent in G . The number of distinct  - colourings of G , denoted by PG(,) is called the chromatic polynomial of G .

A topological index is a real number related to a graph. It must be a structural invariant, i.e., it is fixed by any automorphism of the graph. There are several topological

1 E-mail: [email protected]

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010. indices have been defined and many of them have found applications as means to model chemical, pharmaceutical and other properties of molecules. The Wiener index W and diameter are two examples of topological indices of graphs (or chemical model). For a detailed treatment of these indices, the reader is referred to [8].

The nanostar dendrimer is a part of a new group of macromolecules that seem photon funnels just like artificial antennas and also is a great resistant of photo bleaching. Recently some people investigated the mathematical properties of this nanostructures in (1).

In Section 2, we introduce two graphs with specific structures and compute their chromatic polynomials. Using our result in Section 2, we study the chromatic polynomials of some nanostars in Section 3.

Chromatic polynomial of certain graphs

In this section we consider some specific graphs and compute their chromatic polynomial.

Let Pm 1 be a path with vertices labeled by yy01,,, ym , for m  0 and let G be any graph. Denote by Gm() (or simply Gm(), if there is no likelihood of confusion) a graph v 0 obtained from G by identifying the vertex v 0 of G with an end vertex y 0 of Pm 1 (see

Figure 5). For example, if G is a path P2 , then Gm()=() P2 m is the path Pm 2 .

The following theorem gives the formula for computing the chromatic polynomial of graphs Gm() and GmG12() as shown in Figure 1:

Figure 1: Graphs G(m) and G1(m)G2, respectively.

Theorem 3 . Let m N. Then

the chromatic polynomial of Gm() is

PG((),)=(1)(,). m  m PG

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

the chromatic polynomial of GmG12() is

(1)  m 1 PG((),)= mG PG (,)(,). PG 12 1 2

Chromatic polynomial of some nanostars

In this section we shall compute the chromatic polynomials of some nanostars. We consider the nanostars NS1[] n , NS2[] n and NS3[] n as shown in Figure 2.

Figure 2: NS1[3], NS2[3] and NS3[2], respectively.

NS[] n NS[] n The following theorem give us the chromatic polynomial of nanostars 1 , 2 NS[] n and 3 .

Theorem 4 .

The chromatic polynomial of nanostar NS1[] n is

32nn2 5 4 3 2 32 PNSn([]1 ,)=(1) (  5  10  105 ).

The chromatic polynomial of nanostar NS2[] n is

42 nn115 4 3 2 2 PNSn([]2 ,)=(1) (  51 01 05 ).

The chromatic polynomial of nanostar NS3[] n is

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

32nn152 4 3 2 32  PNSn([]3 ,)=(1) (  51 0  105 ).

By study of the chromatic polynomials of nanostars, we obtain some properties of these nanostars.

References

1 A. R. Ashrafi and M. Mirzargar, PI, Szeged, and edge Szeged indices of an infinite family of nanostar dendrimers, Indian J. Chem., 47A (2008), 538-541.

2 P. Bharathi, U. Patel, T. Kawaguchi, D. J. Pesak, J. S. Moore, Improvements in the synthesis of Phenylacetylene monodendrons including a solid-phase convergent method, Macromolecules, 28 (1995), 5955-5963.

3 G.D. Birkhoff, A determinantal formula for the number of ways of coloring a map, Annals of Mathematics, 14 (1912) 42-46.

4 A. Heydari, and B. Taeri, Szeged index of TUC48 C() S nanotubes, European Journal of Combinatorics 30 (2009) 1134-1141.

5 M. A. Iranmanesh. and A. Adamzadeh, On Diameter of Zig-Zag Polyhex Nanotubes, Journal of Computational and Theoretical Nanoscience Vol. 5, (2008) 1-3.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

First and second Zagreb indices of generalized hierarchical product of graphs

Majid Arezoomand a, 1 , Bijan Taeri b , 2 a Islamic Azad University, Majlesi Branch, Isfahan, Iran b Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran

Abstract. In this paper first and second zagreb indices of generalized hierarchical product of graphs, which is generalization of standard hierarchical and Cartesian product of graphs, will be presented. As a consequence we compute the zagreb group indices for some famouse chemical graphs.

Keywords: graph operation, hierarchical product, zagreb indices, chemical graph.

Introduction

Let G be a simple connected graph with vertex and edge sets VG() and E ()G respectively. For a graph G , the degree of a vertex v is the number of edges incident to v and denoted by  ()v .

The zagreb indices have been introduced by Gutman and Trinajstic [4]. They defined first and second zagreb indices M 1 and M 2 as follows

2 M 1()=Gv  () vV () G

M 2 ()=Gu  ()().v uv E() G

In [5] Barriére et. al. defined a new product of graphs as follows: For N graphs

GVEiii=( , ) and non-empty subset UVii , iN=1,2, , 1, the generalized hierarchical product HGGUNNNN=(óó11)(ó GUGU 2211)ó() is the graph with vertex set

VV1  N and adjacency in H N is given by (where we write uv: if uv E() G )

1E-mail: [email protected]

2E-mail: [email protected] (Corresponding author)

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

(,,,,)xN  x321 x y ify 1: x 1 inG 1  (,,,,)xN  x321 y x ify 2: x 2 inG 2 andx 1 U 1  (,,,,)xNN x321 x x:: (,,,,) x y 321 x x ify 3 x 3 inG 3 andx i Ui ,=1,2 i   (yNN , , x321 , x , x ) ify: xN inGN andxi Ui , i = 1,2, , N  1.

If all the subsets Uuii={ } are singletons (u i is called a root vertex), then the resulting graph is the (standard) hierarchical product HGGNNN= óóóó12 GG1 (see [8]) and if

UVii= for all 11iN , then HGGNNN=  12 GG 1, is the Cartesian product of Gi . It is easy to see that

NN11N 1 iN |()|EHNN=| E ||||||| U i E1 V i (|||| E i  U j || V k). (1) ii=1 =2i =2 j =1k =i 1

By Corollary 2.2 in [5] the generalized hierarchical product is associative

GGNNóó11()=UG ó (GGNN 1 óó 2 ()U 2 óG 11 ())(UUU 1  1 ).

Thus for some of its properties, it suffices to study the two factors case.

Main Results

x =(xx , , , xx , ) The degree of a vertex NN121 in the generalized hierarchical H =(GGóó U) óG(U) product NN 22 11 is

 ()=(xx ) ( xx )( ) [  ( x )  ( x )](  x ), 11UU112 1UN 1N 1N

where  and  denote, respectively, the degree and the characteristic function of the Ui set U i .

Theorem 5 Let  =(GHUó ). Then

M 111()=|()|()||()4|()|()VG M H U M G EG u. uU

k k fi()=0 fi()=1 If for jk> , we put  ij= and  ij= then we can simplify the first Zagreb index of generalized hierarchical product of N graphs as follows:

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

GVE=( , ) UV H =(GGUóó)(ó GU) Corollary 6 Let iii, ii and NNNN11 11, N >2. Then

N iN1 MH11()=[()||Ni MG Uj || Vk] i =1 jk=1 =i 1

NN1 j 1 N  4 [ (uE )(  [|jkr | | UV | | |])]. iuU=1i ji = 1 kki=1, rj = 1

If for every x VG(), we define Nx[] as the set of all adjacent vertices of x , then we have

Theorem 7 Let  =(GHUó ). Then

MV22()=|()G|()||()2MHUM2G|()EG|   () x uUxNu[] 2 M 1(GxyEHxyU )|{ ( )| , }| | EG ( )| ( u )  MG1 ( ) ( u ) uU uU

If we put UVH=() or Uz={ } then we have the following corollary:

Corollary 8 Let GH and GHó be the Cartesian and standard hierarchical product of G and H with root vertex z , respectively. Then

MG11( H ) = | VG ( )| MH ( ) | VH ( )| MG1 ( ) 8| EG ( )|| EH ( )| MGHV( ) =| ()|GMHVHMGE ( )| ( )| ()3|(HMGE )| ()3|()|GMH ( ) 22211 MG11()ó H=| VG()|( MH)( MG1)4| EG()| ( z) 2 MG22(ó H ) = | VGMH ( )| ( ) MG2 ( ) 2| EG ( )|  ( x )  | EG ( )| ( z )  MG 1 ( )(). z xNz []

At the end of the paper we consider some chemical graphs as a generalized hierarchical product of their subgraphs and give an algorithm for computing of the first and second zagreb indices of these structures.

References

[4] I. Gutman and N. Trinajstic, Graph theory and molecular orbitals, total  electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535-538.

[5]L. Barrière, C. Dalfó, M.A. Fiol, and M. Mitjana, The generalized hierarchical product of graphs, Discrete Mathematics, 309 (2009) 3871-3881.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Application of experimental design based on Full factorial & Central Composite Design assisted dispersive liquid– liquid microextraction to the determination of Triton X-100 in aqueous samples by Reversed Phase High Performance Liquid Choramoatogeraphy

A. Asghari*, M. Ghazaghi, M. Bagheri-roochi, M. Rajabi Department of Applied Chemistry, Faculty of Science, Semnan University, P.O.Box 35195-363, Semnan, Iran (E-mail: [email protected]; [email protected]; [email protected]; [email protected])

ABSTRACT

An improved method for the determination of triton X-100 in water samples has been developed using dispersive liquid–liquid microextraction (DLLME) prior to liquid chromatographic analysis. The variables of interest, such as the volume of extraction solvent

(CCl4), disperser solvent (acetone) and sample solution, pH, ionic strength, and extraction time were optimized in a 26–3 Full factorial design. The next level, the signiﬁcant factors were optimized by using a central composite design (CCD) and the quadratic model between the dependent and the independent variables was built. The optimum experimental conditions found from this statistical evaluation were included: 9 ml volume of sample, 130 µL carbon tetrachloride, 0.5mL acetone, 4 min centrifugation with 4000 rpm, natural pH containing 2.5% (w/v) NaCl and time of extraction 6 min. At the optimized conditions, the preconcentration factor was 120. Limit of detection (LOD) of 9 ngL-1 and linear dynamic range (LDR) of 01-100 mgL−1 were obtained for Triton X-100 determinations.

Keywords: DLLME; Triton x-100; Experimental design; HPLC

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010. MATCING IN FULLERENES

Afshin Behmaram, Hasan Yousefi-Azari and Ali Reza Ashrafi School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, I. R. Iran E-mail: [email protected]

Abstract:

Ever since introduced by the Czech chemist Friedrich August Kekulé von Stradonitz in 1865, Kekulé valence structures have played a major role in organic chemistry. Fullerenes are cage polyhedral carbon molecules such that all faces are pentagons and hexagons. It follows directly from Euler’s formula that a fullerene has exactly 12 pentagonal faces. These graph theoretic fullerenes are designed to model large carbon molecules: each vertex represents a carbon atom and the edges represent chemical bonds. The icosahedral C60, known as the buckminsterfullerene, was predicted by Osawa (1970) and discovered by Kroto et al. (1985).

k-matching in a graph are k independent edge .The aim of this talk is to present our resent results on the problem of computing the number of k_matching (k=1…4) in Some family of fullerene graphs. also we find relation between the number of matching and the number of vertices and number of hexagons. Keywords: Fullerene, Kekulé structure, matching

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

The Extremal Generalized Theta Graph with Respect to the Hosoya Index

A. Dolati 1 and S. Golalizadeh 2 1 Department of Mathematics, Shahed University, Tehran, Iran. 2 Department of Mathematics, Arak University, Arak, Iran.

Keywords: generalized theta graph; Hosoya index; matching Abstract The Hosoya index or z-index z G)( of a graph G V ((= G), E(G)) is the total number of it's matchings plus one, where a matching is a non-empty subset M  E G)( with the property that no two different edges of M share a common vertex. If m G k),( denotes the

n/2 number of its k  matchings, matchings consisting of k edges, then z G =)( m G k),( , k 0= where n is the number of the vertices of G . It is convenient to set m G o 1=),( . By its definition, we deduce that m G k 0=),( when k > n/2 .

The generalized theta graph  n is an n -vertices graph that consists of a pair of s1,...,sk n end vertices joined by (kk  3) internally disjoint paths of lengths s1,...,sk >=1. By k we denote the family of n -vertices theta graph with k paths. In this paper we characterize the n extremal n -vertices generalized theta graphs in k with respect to the Hosoya index.

References [6] J.I. Brown, C. Hickman, A.D. Sokal, D.G. Wagner (2001), On the Chromatic Roots of Generalized Theta Graphs, Journal of Combinatorial Theory, Series B. 83, 272-297.

[7] H. Deng, S. Chen (2008), The extremal unicyclic graphs with respect to Hosoya index and Merrifield-Simmons index, MATCH Commun.Math.Comput.Chem. 59 171-190.

[8] R. Todeschini, V. Consonni (2000), Handbook of Molecular Descriptors (Wiley- VCH,Weinheim).

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Haplotype block partitioning and tagSNP selection under the perfect phylogeny model

Changiz Eslahchi 1,2*, Hamid Pezeshk 3, Mehdi Sadeghi4, Narjes Afzaly3, Ali Katanforoush1 1 Faculty of Mathematical Sciences, University of Shahid Beheshti, Tehran, Iran 2 School of Computer Science, Institute for Research in Fundamental Sciences 3 School of Mathematics, Statistics and Computer Sciences, Center of Excellence in Biomathematics, College of Science, University of Tehran, Tehran, Iran 4 National Institute of Genetic Engineering and Biotechnology, Tehran, Iran Email addresses: CE: [email protected] - HP: [email protected] MS: [email protected] - NA: [email protected] - AK: [email protected]

Introduction

One of the major interest of current genomics research is understanding the genomic differences in human population so as to be able to find out what makes us different rather than what we have in common. Single Nucleotide Polymorphism (SNP), i.e. single base pair difference between individuals in a population, is believed to be an important reason for variations occur in human genome. To date, millions of SNPs have been identified. An SNP haplotype is a sequence of SNP alleles in a certain region of chromosome. For simplicity, we will use haplotype rather SNP haplotype from now on. Many studies suggest that human genome can be viewed as a partitioning of haplotype blocks in which common variations of haplotypes within a certain population are distinguished by a relatively small number of SNPs called haplotype tagging SNPs (tagSNPs) [1-5].

Different methods have been suggested for defining blocks structure. Haplotype blocks in methods of Patil[3] and Zhang[6] are subjected to get a limited Haplotype Diversity. The former approach applies a greedy algorithm to find edges of the haplotype blocks and the latter performs a dynamic programming to achieve minimum number of tagSNPs. Methods based on statistical association are essentially derived from the measures of Linkage Disequilibrium (LD). The method introduced by Gabriel [2] is the most commonly used one in this category. Characterizing SNP pairs to three association categories, Gabriel’s method determines regions with large number of strongly associated pairs as blocks.

* Corresponding author

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

In this paper we present an algorithm for haplotype block partitioning based on Perfect Phylogeny. Given a constant parameter, σ (0<σ ≤1), we find the partitioning in which haplotypes are governed by the Perfect Phylogeny model after that at most (1- σ)×100% of SNPs alleles were flipped in all haplotypes of each block. In case that σ =1, our method find the partitioning in which no recombination within each block can be observed that is called pure perfect. In a partially perfect, when σ<1, at most (1- σ)×100% of SNPs have to be ignored so that others satisfies the Perfect Phylogeny constraints. The conventional approach to find haplotype blocks under the model of Perfect Phylogeny is the Four Gamete Test approach. The first and only practical implementation of this approach is attributed to Wang et al [7] in which haplotype blocks starting with a single SNP extend until δ percent of SNP pairs fail the Four Gamete Test. Gramm et al [8] have shown that problem of haplotype partitioning with minimum haplotype blocks under the Perfect Phylogeny model is NP-hard. Our approach obtains an optimal partitioning for problem assuming blocks as continuous regions on chromosome.

It is also shown that if there is no missing data in a perfect block, then the number of tagSNPs is equal to the number of mutually distinct haplotypes minus one and in this case, tagSNPs can be identified using a polynomial time algorithm. For blocks with missing data we use an approximation algorithm to find minimum set of tagSNPs.

Definitions

Let c1 and c2 be two columns of a matrix A. Let V (c1, c2) be the set of values that the pair of columns can take on over all the rows of A, so V (c1, c2) ε {(0, 0), (0, 1), (1, 0), (1, 1)}. The necessary and sufficient condition for A to have a perfect phylogeny is that, for every pair c1 and c2 of columns of A, |V (c1, c2)| ≤ 3. This criterion is known as Four Gamete Test. In this paper we assume that the major and minor alleles are represented by 0 and 1 respectively. A missing SNP is denoted by N. We also assume the Minor Allele Frequency (MAF) is less that 50%. Therefore, in every two columns of A, gamete (0, 0) exists. So in our algorithm we only consider three gametes {(0, 1), (1, 0), (1, 1)} and the condition that states |V (c1, c2)- {(0, 0)}| ≤ 2.

A haplotype h' of length n is said to be a cover of type (1) of another haplotype h (of the same length as h) if for each SNP, 1  i  n , for which ih   N then h'(i)=h(i). We denote h<< h' ih  N this by 1 . It is called the cover of type (2), if   then h'(i)=h(i) or h'(i)=N and h<< h' will be denoted by 2 .

A haplotype matrix Pm'× n is said to be a pattern matrix for another haplotype matrix

A  nm m '  m if it satisfies the following conditions: h<< h' (1) h  A h' P  1 ,

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

h<< h' (2) h' P h  A  1 , (3) h,h' P then

Theorem 1: Suppose P be a pattern matrix for A. If P is perfect then A is so.

Details of proofs are presented in Appendix I.

By the following, we introduce an algorithm which seeks the minimum number of transformations of entries 1 to 0 until perfect block will be obtained.

For every two columns i and j (i

1 A = ti, p|  p = 1 ij it jt 2 A = ti, p| = 1 and p = 0 ij it jt 3 A = ti, p| = 0 and p = 1 ij it jt

Now, the problem is to find the minimal set S   ,tk Pk t 1 in a way that for each i

1 (1) For each ti ,, ,tj  Ai j ;  , ti  S or  , tj  S

2 (2) Ai j  S  .

3 (3) Ai j  S  .

Algorithm PerfectBlock

Let ~ be a total order on the set  for which i0 , j0  is the minimal element of 3  ~,  S 0 A  . At the first step of the algorithm, set equal to ji 00 provided that 3 2 3 1 2 2 3 A  A A  A . S 0 A A  A ji 00 ji 00 and ji 00 ji 00 would be set equal to ji 00 if ji 00 ji 00 and A 2  A 1 A 1  A 2 A 1  A 3 S ji 00 ji 00 . If ji 00 ji 00 and ji 00 ji 00 then 0 would be constructed using A 1 i , j  one of the coordinates of each ordered pair of ji 00 . Let k k be the kth element of  . Define:

1 and  ti,  S k 1 and  tj,  S k 1 A1 = tj,,ti, : tj,,ti,  A , i jkk

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

A'  A 2  S 2 ji kk k 1 , A'  A 3  S 3 ji kk k 1 . ` ` ` S  S  S Similar to the first step, we obtain S 0 for A1 , A 2 , A 3 . Define k 0 k 1 . The algorithm would be terminated after   steps.

Table 1: The properties of the haplotype blocks of Chromosome 21 defined by Perfect Block algorithm

C M C c C si ommon A % ommon bl size/bloc ommon ommon ta ethod ze/block haplotyp ll blocks of all SNPs/blo ocks k haploty SNPs gSNPs (kb) e/block (%) bases ck (SNPs) pe/block (%) (%)

σ n 6 2 1 4. 9 1 4 4 3 = 1.00 >10 57 0.1 7.80 45 1.3 7.6 8.6 0.8 070

1 8. 5. 3. 9 5 4 5 6 3  n 964 28 44 80 5.1 2.6 4.4 0.2 539

n 1 2. 1. 2. 9 2 6. 9. 1

<3 116 75 50 39 9.3 9.9 9 5 608

σ n 7 2 2 4. 7 4 7 6 4 = 0.98 >10 44 9.85 4.18 36 5.8 0 4.8 8.6 077

1 9. 5. 3. 8 5 2 3 3 3  n 019 81 76 98 7.5 4.8 4.4 0.9 604

n 9 3. 1. 3. 9 5. 0. 1. 1

<3 7 51 99 27 3.1 2 8 1 92

References

[1] Daly MJ, Rioux JD, Schaffner SF, Hudson TJ, Lander ES: High-resolution haplotype structure in the human genome. Nat Genet 2001, 29:229-232.

[2] Gabriel SB, Schaffner SF, Nguyen H, Moore JM, Roy J, Blumenstiel B, Higgins J, DeFelice M, Lochner A, Faggart M, Liu-Cordero SN: The structure of haplotype blocks in the human genome. Science 2002, 296:2225-2229.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

[3] Patil N, Berno AJ, Hinds DA, Barrett WA, Doshi JM, Hacker CR, Kautzer CR, Lee DH, Marjoribanks C, McDonough DP: Blocks of limited haplotype diversity revealed by high-resolution scanning of human chromosome 21. Science 2001, 294:1719-1723.

[4] Dawson E, Abecasis GR, Bumpstead S, Chen Y, Hunt S, Beare DM, Pabial J, Dibling T, Tinsley E, Kirby S: First-generation linkage disequilibrium map of human chromosome 22. Nature 2002, 418:544-548.

[5] Johnson GC, Esposito L, Barratt BJ, Smith AN, Heward J, Di Genova G, Ueda H, Cordell HJ, Eaves IA, Dudbridge F: Haplotype tagging for the identification of common disease genes. Na. Genet 2001, 29:233-237.

[6] Zhang K, Deng M, Chen T, Waterman MS, Sun F: A dynamic programming algorithm for haplotype block partitioning. Proc Natl Acad Sci USA 2002, 99:7335-7339.

[7] Wang N, Akey JM, Zhang K, Chakraborty R, Jin L: Distribution of recombination crossovers and the origin of haplotype blocks: the interplay of population history, recombination, and mutation. Am J Hum Genet 2002, 71:1227-1234.

[8] Gramm J, Hartman T, NierHoff T, Sharan R, Tantau T: On the complexity of SNP block partitioning under the perfect phylogeny model. Discrete Mathematics 2009, 309: 5610-5617.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

On Matching in Structural Bioinformatics via Statistical Shape Analysis

Mousa Golalizadeh1and Seyed Morteza Najibi2 Department of Statistics Faculty of Mathematical Sciences Tarbiat Modares University Tehran, Iran

Abstract

An active area of research in bioinformatics is to study the structure of protein. This involves gathering information from the complex biomolecule structure. Although the roles of mathematics and computer have been highlighted, how statistics can be of any help on encountered problems is less identified. One of the new fields of statistics is statistical shape analysis which is initiated, expanded and implemented over the last three decades. It deals with the geometrical information about objects which are invariant under some mathematical operations. In this paper, we discuss how statistical shape analysis can be utilized to tackle some problems in bioinformatics. In particular, leaving technical complexity aside, we clarify matching the structural biomolecule via this new field of statistics.

Keywords: Bioinformatics, Matching molecule, Statistical shape analysis, Mathematical algorithm, Complex structure.

1 Introduction

Bioinformatics is a science consisting of mathematics, computer and biosciences. It is, precisely, introduced as a branch of sciences in which the computer, mathematics and statistics are used to analysis the biological and molecular information [4]. Analysing the information gathered from the complex structural objects falls in the domain of bioinformatics. The prediction of biological structure of a protein based upon the information available in the Protein Data Bank (PDB) are also the subject of this science. It is worth to

[email protected]

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010. note that the number of PDB links are increasing as the web knowledge and technology became popular as the result of this era.

Statistics is also a science to study gathering, summarizing and inferring the data [7]. Statistics shape analysis, as a new sub-branch of statistics, grown and developed over the last three decades, is dealing with all geometrical information about an object. Particularly, Kendall (1977) has defined shape of an object as entire geometrical information remained after filtering out location, scale and rotation effects. A comprehensive treatment of this field is appeared in some monographs (see, for example, [3] and [9]). Since structural biomolecules are, statistically, geometrical information, this new field of statistics has capability on quantifying, analysing and predicting the random feature of the relevant structures.

In the next section, we shall highlight matching structural biomolecules, the main biological problem in bioinformatics. Also, some key features of statistical shape analysis which are to match structural biomolecules, are clarified. The paper is ended by the conclusion aiming to create a collaboration link with other sciences involved in bioinformatics.

2 Matching Structural Biomolecules

2.1 Biological View

Biomolecules are the vital elements of an organism. The proteins are a great deal of importance as a main category of biomolecules which play an important structural and metabolic role. Also, they are key factors in functioning the living organism, such as color of hairs and nails [4].

The matching of proteins structure, as an unsolved problem, is a recent area of research attracting researchers concerned about the geometrical aspect of the subject. The core roots of the subject is based on similarity and differences. The main challenges are on proposing sensible inferential methods and proper prediction algorithm for structural biomolecules and the function induced by a protein [8]. The popular way to tackle the problem are biophysical, sequence based and homology approaches. Also, combination of these three approaches with the chemical physical relationship can be considered.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

2.2 Statistical View

The random geometrical objects, the random samples in statistical shape analysis, lies on space known as shape space which is a non-Euclidean space [3]. The procedure to remove the three effects leads to shape coordinate systems. The main such systems are Kendall and Bookstein and Procrustes coordinates [3]. The methodology of deriving the latter one comes from matching two objects as close as possible to obtain the minimum of sum square of the distances between corresponding Euclidean coordinate. This quantity is, in spirit, related to Root Mean Square Deviation (RMSD), a popular measure in bioinformatics, in comparing quality of alignment protein structures.

In statistical shape analysis there are different approaches to analysis the objects that either have label or are unlabeled. This is the case in bioinformatics in which sometimes matching structural protein is to be done via superimposition of the homologous atoms, while in some circumstances atoms are unlabeled.

Statisticians recently made important contributions in this subject using both traditional and new statistical tools. Invoking Markovian property for the sequences of amino acids was the initial implemented method [1]. Constructing a Bayesian framework was then considered. In this approach, constructing the hierarchical models [5] and also using random fields to tackle pairwise (multiple) alignment were the main theme[2].

As Mardia et al. (2003) proposed, the study of structural bimolecular through unlabel points makes more sense to align proteins. This is due to either non-equal landmark points for two molecules or lack of matching for part of them. For the latter case a mixture model can be considered to model simultaneously the matched and unmatched points.

3 Conclusion

Structural comparison of biomolecules is attracted the great attention in bioinformatics in order to understand biological, mathematical, physical and chemical properties of involved components. The process are usually based on similarities and differences which is one of the main subject of statistical comparison of data. Moreover, to determine the function in which a newly structure does, to match with previously available one, is the key factor to classify the complex structures.

Statistical shape analysis, with capability on matching, geometrically, random objects play an important role in bioinformatics. Recently, some modification of this field is applied in discovering some features of structural biomolecules. However, there are many challenges to overcome through close collaboration with other researchers in non-statistical fields.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

References

[1] Asai, K. and Hayamizu, S. and Handa, K. Prediction of protein secondary structure by the hidden Markov model. Computer applications in the biosciences, 9(2):141--146, 1993.

[2] Czogiel, I. and Dryden, I.L. and Brignell, C.J. Bayesian alignment of unlabelled marked point sets using random fields. Submitted, 2009.

[3] Dryden, I.L. and Mardia, K.V. Statistical shape analysis. Wiley, New York, 1998.

[4] Ewens, W.J. and Grant, G.R. Statistical Methods in Bioinformatics. Springer, New York, 2005.

[5] Green, P.J. and Mardia, K.V. Bayesian alignment using hierarchical models, with applications in protein bioinformatics. Biometrika, 93(2):235--254, 2006.

[6] Kendall, DG. The diffusion of shape. Advances in Applied Probability, :428-- 430, 1977.

[7] Koosis, D.J. Statistics: a self-teaching guide. John Wiley & Sons, New York, 1997.

[8] Mardia, K.V. and Taylor, C.C. and Westhead, D.R. Structural bioinformatics revisited. In proceedings in stochastic geometry, biological structure and images, :11--18, 2003. Edited by R.G. Aykroyd, K.V. Mardia and M.J. Langdon.

[9] Small, CG. The statistical theory of shape.. Springer, New York, 1996.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Some results on the Wiener and terminal Wiener index of balanced trees

Abbas Heydari * Department of Science, Islamic Azad University, Arak Branch,Arak, Iran

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Some polynomials and topological indices in nanocones with even or odd edges in the apex

Aleksandar Ilića, Mircea V. Diudeab Farzane Gholami-Nezhaadc and Ali Reza Ashrafic

[email protected] , [email protected] [email protected] and [email protected]

Abstract

The Wiener index; it equals the sum of distances between all unordered pairs of

vertices of G. The Szeged index is the sum of all products of non-equidistant, vertices nu(e)

and nv(e). The vertex PIv index is defined as the sum of nu(e) and nv(e) and the PIe and

Edge-Szeged indices are sefinde as PIe(G) = [mu(e)+ mv(e)] and SZe(G) = [mu(e)mv(e)], in

which mu(e) is the number of edges lying closer to vertex u than to v over all edges e=uv of G. A sixth index, called Cluj-Ilmenau, is calculated from the derivatives of Omega polynomial. In this paper, these indices are calculated in several families of nanocones, C(m,n).

Keywords: Wiener index, Szeged index, Cluj-Ilmenau index, PI index, Nanocones.

1. Introduction Conical nano-structures1 have been reported in Nanoscience since 1968, before the discovery of fullerenes. If a graphite sheet is divided into six sectors (Figure 1), each with an angle of 60º, and if k of these sectors (with k=1 to 3) are deleted sequentially, three classes of graphs, are obtained; their apex polygon will be a pentagon (m=5), a square (m=4) or a triangle (m=3), respectively. (Figure 1).

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Figure 1. the six sectors on the lattice and the nano-cones C(4,3) and C(5,3).

2. Definitions

Let G be a simple connected graph with the vertex and edge sets V(G) and E(G), respectively. The distance between the vertices u and v of a connected graph G is denoted by d(u,v) and it counts the number of edges in a shortest path joining the vertices u and v. A topological index is a numeric quantity, derived from the structure of a graph, which is invariant under automorphisms of the given graph. The Wiener index, introduced by Harold Wiener2 equals WG() d (,) uv the sum of topological distances between all (,)uv V ( G ) G

unordered pair vertices. The Szeged Sz v G)(  [nu )( ne v (e)] and Edge-Szeged uve

3,4 Sz e G)(  [mu e)( mv (e)] are other indices, where nu(e) is the number of vertices of G lying uve closer to u than to v and nv(e) is the number of vertices of G lying closer to v than to u, and mu(e) is the number of edges of G lying closer to u than to v and mv(e) is the number of edges of G lying closer to v than to u. A polynomial was defined by Ashrafi is

m e)((  m e))( u v that its first derivatives is PIe index: PI e G)(  [mu e)(  mv (e)] . A PI xG ),(   uve x e uve 4,6 related index, defined by Ashrafi et al. on vertices, is the vertex PIv:

 PIv() G PI v (1)  n uv,, n vu V E m uv ,. euv euv

Let G be a bipartite, plane graph; two edges e = (x,y) and f = (u,v) of G are called codistant (briefly: e co f ) if dxv(,)(,)1(,)1(,) dxu dy vd  y u .

Let C e  {:)( f  E(G); f co e} denote the set of edges in G, codistant to the edge e  E G)( . If relation co is an equivalence relation, then G is called a co-graph. The set C(e) is called an orthogonal cut (oc for short) of G, with respect to edge e. If G is a co-graph then its orthogonal cuts C(G)= C1,C2 ,...,Ck form a partition of E(G): EG( ) C12 C ... Ck ,

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Ci  C j  , i  j . We say the edges e and f of a plane graph G are in relation opposite, e op f, if they are opposite edges of an inner face of G. Using the relation op we can partition the edge set of G into opposite edge strips, ops5,8 .

Let G be a connected graph and SG( )  ss12, ,..., sk be the set of ops of G. Then ops form a partition of E(G). Denote by m(s) the number of ops of length s==|sk| and define the Omega s polynomial as 5 (x)  m(s)  x .Its first derivative (in x=1) provides single number  s topological descriptor as (1)  ms  eEG (). On Omega polynomial, the Cluj-  s Ilmenau5,9 index, was defined asCI() G { [(1)]2  [ (1) (1)]}.

3. Main Results The number of vertices in the cone C(m,n) is v((,))Cmn  m(1)n  2 and the number of edges in the row k (numbering starting from the external row) is as:

n eCmn((,)) mn[( 1) (1nk)](/2)( m 3nn2 52) (1) k 1

3.1. Case: m=even

In cones with m>4, m=2s, the Wiener index can be obtained by using a cutting procedure, which sums the products of the number of vertices lying to the left and to the right of each orthogonal line (Figure 2):

Figure 2. Cones with various even apex values m= 4, 6, 8; the cutting procedure.

WC((2,)) sn  (1 / 15)sn ( 1)(80sn4 76nn4  169 3 15sn23 230sn3  111n2  (2) 452s22n  20sn2 19 n 457sn2  0 sn15s2 )

which in case of s=3 reduces to:

39 The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

WC( (6, n )) (1/ 5)( n 1)(164 n432  656 n 954 n 596 n 135) (3)

A similar procedures leads to the formula for the Szeged index SZv

24 4323 3 SZve( C ( mven , n )) ( m / 4)( n 1) ( 18 n 9 mn 36 n m n 24 mn 21mn222 19 nm 3nm22  3nnm  2 6nm 2 ); (4) mss2; 2

The formulas for calculating the Wiener and Szeged indices in C(4,n) cones are:

WC( (4, n )) 8 (736 /15) n  108 n23  (340 / 3) n  58 n45  (58 / 5) n (5)

SZ( C (4, n )) 16 (538 / 5) n  (4129 /15) n23  370 n  veven (6) (1669 / 6)nn45 (557 / 5) (557 / 30) n 6

SZ( C (4, n )) (31/ 2) (523 / 5) n  (8213 / 30) n23  370 n vodd (7) (1669 / 6)nn45 (557 / 5) (557 / 30) n 6

In this case for m>4 the Omega polynomial is as:

n1 2(nn 1) k ((Cmeven ,),)( n x m /2) x mx (8) k2

2 (Cm (even , n ),1) ( m / 2)(3 n 5 n 2) . (9)

32 2 CI( C ( meven , n )) (1/12) m ( n 1)(27 mn 28 n 63 mn 48 mn 12 m 50 n  24) (10)

Cones with apex polygon m=even are clearly bipartite and a first result is the formula 6 for the PIv polynomial (Ashrafi et al. ) and index (as the first derivative, in x=1):

v 2 nm  )1( 2 PIv C(( meven ,n), x)   xe  (m 3)(2/ n  5n  )2  x . (11)

PI C((' m , n )1),  e  v  (m 3)(2/ n 2  5n  )2  m(n  )1 2 v even (12)  )2/1( m 2 3( n  )(2 n  )1 3

W Sz C P P

V| E| I Ie Iv 11 75 7 7 5 4 8 704 920 176 176 632 35 28 1 1 1 00 40 912 9864 8480 8480 4000

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33 22 1 1 1 6 32 204 0824 6572 6572 2672 10 84 4 4 3 50 10 1751 2850 2420 2420 1500 67 45 2 2 2 28 76 312 3152 9840 9840 2528 20 17 7 7 5 00 80 5168 18800 6160 6160 6000 Table 1. Topological indices in nanocones C(m,n), m=even.

3.2. Case: m=odd

For m>3, we have the following formula:

1 W C 2(( s  n)),1  (n  2)(1 s  16)1 n  n2  54n3  36n4 15s 115ns  265 2sn 30 (13)  245 3sn  80 4sn 15s2  45ns2  45 sn 22 15 sn 23 

1 SZ( Cm ( , n )) (9 m26325324 18 mn ) ( m 40 m 71 mn ) (5 m 68 m 104 mn ) vodd 4  (14) (10mmm32 52 66 )n 3  (10 mmm 32  13 13 )n 2 (5mmmnmmm32 4 3 ) ( 32 2 )

81 2 81 6 285 2 931 5 367 2 107 4 Sz C(( m n)),  ( m  )nm  ( m  )nm  ( m  )nm e odd 16 8 16 40 16 24 (15) 199 2 1337 3 2 179 2 767 1 2 ( m  )nm  2( m  )nm ( m m )n m.4 16 24 3 30 4

n1 nk ((Cmodd ,),) n x mx. (16) k 1

2 ((Cmodd ,),1)( n m /2)(3 n 5 n 2). (17)

32 2 CI( C ( modd , n )) (1/12) m ( n 1)(27 mn 28 n 63 mn 48 mn 12 m 38 n  12) (18)

In cones with apex polygon when m>3 is odd, the formulas for the PI indices are as follows .

322 PIe( C ( m odd , n )) ( m /12)( n 1)(27 mn 63 mn 46 n 56 n 48 mn 12 m  12) . (19)

2423222 2. PICmv(( odd ,))(1/2)3 n mn (11 m 2) mn (15 m 6) mn (9 m 6) mn 2 m 2 m (20)

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W S C P P S

V| E| Zv I Ie Iv Ze 5 2 3 3 2 6 8 6 421 9469 978 906 976 0408 1 1 1 1 7 2 5 05 6701 15608 0260 0110 500 32461 2 1 1 1 8 2 0 10 0790 26740 1470 1110 480 02980 6 4 2 2 2 8 25 75 3855 93000 9350 8600 1250 44185 4 3 2 2 1 4 12 54 8370 01980 2834 2330 6800 27392 1 1 5 5 4 1 75 45 48022 167075 8240 7190 2000 821309 Table 2. Topological indices in nanocones C(m,n), m=odd.

References

1. A. Krishnan, E. Dujardin, M. M. J. Treacy, J. Hugdahl, S. Lynum, T. W. Ebbesen, Graphitic cones and the nucleation of curved carbon surfaces, Nature 388 (1997) 451-454. 2. H. Wiener, Structural determination of the paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 1720. 3. I. Gutman, A formula for the Wiener number of trees and its extension to graphs containing cycles, Graph Theory Notes N.Y. 27 (1994) 915. 4. A. R. Ashrafi, F. Gholaminezhaad, The edge Szeged Index of One-Pentagonal Carbon Nanocones, Int. J. Nanoscience & Nanotechnology (IJNN), 4 (2008), 135–138. 5. M. V. Diudea, Omega polynomial, Carpath. J. Math., 22 (2006) 43-47. 6. M. H. Khalifeh, H. Yousefi-Azari, A. R. Ashrafi, Vertex and edge PI indices of Cartesian product graphs, Discrete Appl. Math. 156 (2008) 1780-1789. 7. S. Klavžar, A bird’s eye view of the cut method and a survey of its applications in chemical graph theory, MATCH Commun. Math. Comput. Chem, 60 (2008) 255274. 8. M. V. Diudea, Counting polynomials and related indices by edge cutting procedures, J. Chem. Inf. Model. (2009), submitted. P. E. John, A. E. Vizitiu, S. Cigher, M. V. Diudea, CI Index in Tubular Nanostructures, MATCH Commun. Math. Comput. Chem. 57 (2007) 479484.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Computing the Cluj Index of TUC4C8(S)

S. Zohreh Imani Golafshani 1 , Ali Iranmanesh *1 , Mircea V. Diudea 2 1 Department of Mathematics, Tarbiat Modares University P.O. Box 14115-175, Tehran, IRAN. [email protected] , [email protected] 2 Faculty of Chemistry and Chemical Engineering, Babes-Bolyai University, 400084 Cluj, ROMANIA [email protected]

Mathematical calculations are absolutely necessary to explore the important concepts in chemistry. Mathematical chemistry is a branch of theoretical chemistry for discussion and prediction of the molecular structures using mathematical methods without necessarily referring to the quantum mechanics. Chemical graph theory is an important tool for studying the molecular structures. This theory had an important effect on the development of chemical sciences. A topological index is a real number related to a molecular graph. It must be a structural invariant. It does not depend on either the labeling or the pictorial representation of a graph. Several indices have been so far defined and many of them have found applications as means to model chemical, pharmaceutical and other properties of molecular structures. The Wiener Index is the first topological index to be used in chemistry.

It was introduced in 1974 by Harlod Wiener as" the path number for characterization of alkanes". The unsymmetric Cluj matrix([4]), UCJ, has been proposed by Diudea ([5]). It is defined by using either the distance or the detour concept. The non-diagonal entries; []UM ij ; M = CJD (Cluj-Distance) or CJ  (Cluj-Detour), are defined as:

[UM ] = max | v | k = 1,2,... ij i,, j pk

vv= { |vV (G );d

where, ||v is the cardinality of the set v ; ijp,,k ijp,,k

In this paper, we obtain an exact formula for the Cluj Index of TUC48() C S .

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

References

[9]I.Gutman, M.V.Duidea, Defining Cluj Matrices and Cluj Matrix Invariants, J. Chem. Inf. Compute. Sci. 34 (1994) 899-902.

[10]M.V.Diudea, Ivan Gutman, Jantschi Lorentz, Molecular Topology

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Eccentric connectivity Index of HAC5C7[p,q] and HAC5C6C7[p,q] Nanotubes

Ali Iranmanesh and Yaser Alizadeh Department of Mathematics, Tarbiat Modares University, P.O. Box 14115-137, Tehran, Iran [email protected]

Abstract For a simple connected graph G, Eccentric connectivity index denoted by  c ()G is defined as: c ()GE (()id ())i, where E(i) is the eccentricity of vertex i ,d(i) is the iVG () degree of vertex i and V(G) is the set of vertices of G. Eccentricity of vertex i in a graph G is the path length from verte i to vertex j that is farthest from i.

In this paper, an algorithm is presented for computing the Eccentric connectivity index of any simple connected graph. Furthermore the Eccentric connectivity index is computed for

HAC5C7[p,q] and HAC5C6C7[p,q] Nanotubes by GAP program.

Keywords: Eccentric connectivity index, HAC5C7[p,q] Nanotubes, HAC5C6C7[p,q] Nanotubes.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Application of chemometric Method and spectrophotometric technique in study thermodynamic of the dimerization equilibrium of toluidine blue in different ionic strengths

S. Lotfei1, M. Mazloum-Ardakani2, G.Ghasemi3, S. Ghahri Saremi1 1Department of Chemistry, Payame Noor University(PNU), Gilane Gharb ceter, Kermansh

2Department of Chemistry, Faculty of Science, Yazd University, Yazd

3Department of Chemistry, Faculty of Science, KNT University, Tehran

Abstract

The monomer–dimer equilibrium of toluidine blue (TB) has been investigated by means of UV-Vis spectroscopy. The dimerization constants of TB have been determined by studying the dependence of their absorption spectra on the temperature in the range of 25–80 ºC at different total concentrations of TB (from 8.45×10- 5 to 2.50×10-4 M) and different concentrations of NaCl as supporting electrolyte. The equilibrium parameters of the dimerization of TB have been determined by chemometrics refinement of the absorption spectra obtained from thermometric titrations performed at different ionic strengths. The quantitative analysis of the data of undefined mixtueres, was carried out by simultaneous resolution of the overlapping spectral bands in the whole set of absorption spectra. The dimerization constants are varied by changing the ionic strength and the degree of dimerization are increased by increasing of the ionic strengths of the medium. The enthalpy and entropy of the dimerization reactions were determined from the dependence of the equilibrium constants on the temperature (van’t Hoff equation).

Introduction

The UV-Vis absorption spectroscopy is one of the most suitable methods for quantitative studying the aggregation properties of dyes as function of concentration. In the

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010. commonly used concentration range (10-3 to 10-6 M) the main equilibrium is monomer-dimer reaction [1]. It is well known that the ionic dyes tend to aggregate in diluted solutions, leading to dimer formation, and sometimes even higher order aggregates. In such a case the molecular nature of dye is strongly affected and related to parameters such as dye concentration, structure, ionic strengths, temperature and presence of organic solvents [2].

In this work, we used some physical constraints to determine the dimeric constants of TB in pure water and at different ionic strengths. Data analysis was carried out by DATAN package that developed by Kubista groups [3,4].

Spectra recorded at different temperature are arranged as rows in an n×m matrix A, where n is the number of temperature intervals and m is the number of data points in each spectrum. A is decomposed into an orthogonal basis set by NIPALS or any equivalent method:

r T T T A  TP  E  TP   i pt i (1) i 1

where ti (n×1) is orthogonal target vectors and piT (1×m) are orthogonal projection vectors. These are mathematical constructs and do not correspond to any physical property of the system. r is the number of spectroscopically distinguishable components, and E is the error matrix containing experimental noise, if the right value of r is selected. For a well- designed experiment, E is small compared to TPT and can be discarded. Assuming linear response the recorded spectra are also linear combinations of the spectral responses, vi (1×m), of the components:

r A  CV  E  CV   vc ii (2) i1

where ci(n ×1) are vectors containing the component concentrations at the different temperatures. The two equations are related by a rotation:

1 C  TR (3)

T V  RP (4)

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

where R is an r×r rotation matrix. For a two-component system:

r11 r12  1 1  r22  r12  R    and R    (5) r21 r22  11rr 22  12 rr 21  r21 r11 

Since a single sample is studied, the total concentration must be constant, constraining matrix R. For a monomer–dimer equilibrium, the total concentration of monomers is constant:

c T  2c T  c c  2c  c x x  tot or x x2 tot (6)

kD 2X  X 2

These can be determined, for example, by fitting the target vectors to a vector with all elements equal to ctot.

Experimental section All the chemicals used were of analytical reagent grade. Double distilled water was used throughout. Toluidine blue was purchased from Fluka and was used without additional purification. A stock solution (5×10-2 M) was prepared by dissolving solid TB in water. In all experiments the ionic strengths were maintained constant by NaCl (Fluka) at 0.5 to 4.0 mol l− 1 with interval 0.5 solutions of this salt.

Results and discussion The absorption spectra TB, at different total dye concentrations and ionic strengths, were recorded in the wavelength 475-800 nm and temperature 20–75 ºC at 5 °C intervals. As it is expected, by increasing the temperature and decreasing the concentration, the monomer form would be predominant over the dimmer form. So it is wise to choose the spectrum of the dye at the highest temperature and at lowest concentration as an initial estimate for the monomer in the subsequent calculations. The KD, dimer spectrum, ΔS and ΔH, correspond to minimum value of the χ2 statistics, are selected as the final results. With increasing temperature the absorption peak 670 nm grows and the absorption shoulder around 600 nm decreased. We analyzed the temperature titrations assuming monomer–dimer, monomer– dimer–trimer and even some models including higher order aggregates, and it was found that

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010. a monomer–dimer model most adequately describes the data in these ranges of dyes concentrations. The presence of exactly two species is also evidenced by appearing of the isobestic points at 650 nm. The dimerization constants (KD) were calculated at different temperatures and dye concentration in pure water. As expected KD decreased with increasing temperature, while it is virtually independent of total dye concentration. From the dependence

◦ ◦ of log KD on 1/T (Fig. 1), ΔH and ΔS values were determined. The dimerization constants at 25 ◦C and at different concentrations and thermodynamic parameters of the dimerization reactions of the TB dye are listed in Table 1.

Table1. Dimeric constant (KD) and thermodynamic 11 parameters values of toluidine blue dye at different 10 concentrations without adjusting ionic strength in water 9 8 tolui L -ΔHº -

ln(Kd) dine blue ogK Sº 7 D Δ (mol l-1 ) (kjm 6 ( ol-1) (j -1 -1 5 25°C) mol k ) 2.7 2.9 3.1 3.3 3.5 (1/T)/1000 8.45 4. 63.32 88. ×10−5 55 14

1.74 4. 78.59 12 ×10-4 39 0.69

1.98 4. 67.29 11 6×10−4 26 6.14

2.28 4. 79.14 14 ×10−4 11 1.49

2.50 4. 69.52 11 ×10−4 05 2.34

Fig. 1. The van’t Hoff equation plot at 8.45×10−5

mol l−1concentrations of toluidine blue dye

The observed relationship between entropy and enthalpy reflects an electrostatic nature of the dimerization phenomenon of TB dye. The relative dependence of the concentrations of

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010. the monomer and dimer of methylene green on the temperature in different concentrations are shown diagrammatically in Fig. 2

1 0.8 0.6 0.4

Molar ratio Molar 0.2 0 290 310 330 350 Temperature

Fig. 2. Mol. ratio of toluidine blue dye monomer (□) and dimer (Δ), compared to mol. ratios predicted by the temperature dependence of the equilibrium constant (shown as line) at 8.45×10−5 mol l−1concentrations of toluidine blue dye

References:

[1] L. Antonov, G. Gergov, V. Petrov, M. Kubista, J. Nygren, Talanta 49 (1999) 99.

[2] J. Ghasemi, A. Niazi, G. Westman, M. Kubista, Talanta 62 (2004) 835.

[3] M. Kubista, R. Sjoback, J. Nygren, Anal. Chim. Acta 302 (1995) 121.

[4] M. Kubista, J. Nygren, A. Elbergali, R. Sjoback, Crit. Rev. Anal. Chem. 29 (1999)

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

THE 1- CONNECTIVITY INDEX OF DENDRIMER NANOSTARS

A. Madanshekaf, M. Ghaneei, M. Moradi Dept. of Mathematics, Faculty of Sciences, Semnan University, Semnan, Iran

Abstract

Among the numberous topological indicies considered in chemical graph theory, only a few have been found noteworthy in practical application, connectivity index is one of them. Dendrimer is a synthetic 3-dimentional micromolecule that is prepared in a step-wise fashion from simple branched monomer units. The nanostar dendrimer is a part of a new group of micromolecules that appear to photon funnels just like artificial antennes. . In this article, we compute the 1-connectivity index of PAMAM and Polyester dendrimers.1-2

Keywords: Connectivity index, Randic index, dendrimer nanostars.

1. Introduction

Let G be a simple graph and consider the m-connectivity index where runs over all paths of length m in G and denotes the degree of the vertex i. Randic introduce the 1-connectivity ∑ 1⁄ index (now called Randic index) as , where i-j ranging over all pairs of adjacent vertices of G. This index has been successfully correlated with physo-chemical ∑ 1⁄ properties of organic molecules. Indeed if G is the molecular graph of a saturated hydrocarbon then there is a strong correlation between and the boiling point of the substance.6-10 There is no universal valance connectivity index that would apply to all properties of dendrimers nanostars, but general topological indices are considered in our present work.

2. Results and discussion

Considered a graph G on n vertices, . The maximum possible vertex degree in such a graph is n-1. Suppose denote the number of edges of G connecting vertices of 2 51

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010. degrees i and j. Clearly, . Then 1-connectivity index can be written as

. Therefore, if the graph G consists of components then 11-12 ∑ . ,,…, We now consider two infinite classes and of dendrimer nanostars, Figure1, 2. Now we present our results in two theorems bellow:

Theorem 1. The connectivity index of is computed as follows:

√32 1 9.2 7√3 32 2√6

Figure1. PAMAM dendrimer

Theorem 2. The connectivity index of is computed as follows:

√3 7. √6 7 .2 2 2 2 √32 3√2 3 3 2

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Figure2. Polyester dendrimer

References

[1] Jevpraesesphant, R. et al., "The influence of surface modification on the cytotoxicity of PAMAM dendrimers", Int. J. Pharm., 252, 263–266, 2003.

[2] Haag, R. et al. "An approach to glycerol dendrimers and pseudo-dendritic polyglycerols". J. Am. Chem. Soc ., 122, 2954–2955, 2000 .

[3] A. R. Ashrafi, P. Nikzad, "Connectivity index of the family of dendrimer nanostars", 269-273(2009).

[4] M. Randic, J. Am. Chem. Soc., 97, 6609(1975).

[5] B. Bollobas,P. Erdos, Ars Combinatoria, 50, 225(1998).

[6] Z. Mihali, N. Trinajstic, J. Chem. Educ., 69(9), 701(1992).

[7] D. Morales, O. Araujo, J. Math. Chem. 13, 95(1993).

[8] M. Randic, P. J. Hansen, P. C. Jurs, J. Chem. Inf. Compute. Sci., 28, 60(1988).

[9] E. Estrada, Chem. Phys. Lett., 312, 556(1999).

[10] E. Estrada, J. Chem. Inf. Compute. Sci., 35, 1022(1995).

[11] L. Pavlovic, I. Gutman," Graphs with extremal connectivity index", 53-58 (2001).

[12] Hao Li, Mei Lu," The m-connectivity index of graphs", (2009).

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

A Computational Scheme for Conductor- like Screening Model by Sinc Quadrature Technique

K. Maleknejad, B. Sohrabi, M. Alizadeh, R. Mollapourasl aDepartment of Mathematics, Iran University of Science & Technology Narmak, Tehran 16846 13114, Iran bCollege of Chemistry, Iran University of Science and Technology, Tehran, 1684613114, Iran

Abstract

In this article, our aim is to study on Hammerstein type nonlinear integral equations which forms the basis for the conductor-like screening model for real solvents (COSMO-RS). So this paper presents a powerful numerical approach based on Sinc approximation to solve conductor-like screening model for real solvents (COSMO-RS). The approach is based on the recognition of the Hammerstein integral equation for the determination of the chemical potential of a surface segment as a function of screening charge density. Then convergence of this technique is discussed by preparing a theorem which shows exponential type convergence rate and grantees the applicability of that.

Keywords: Conductor-like screening model, Real solvent, Hammerstein integral equation, Sinc approximation

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Cusp at the nucleus for the Dunning's correlation consistent basis sets

Afshan Mohajeri* and Mojtaba Alipour Department of Chemistry, College of Sciences, Shiraz University, Shiraz, 71454, Iran. E-mail: [email protected].

Introduction

The development of accurate wave functions for atomic and molecular systems has been the subject of investigations over the last years.1-3 Thus selection of a basis set is one of the most important factors in the design and application of reliable quantum mechanical calculations. The quality of a given basis set can be understood in terms of its ability either to reproduce results close to experimental values for properties of interest or to reproduce properties of approximate wave functions represented in a complete basis set.

In 1957 Kato explained that exact many-electron wave functions must exhibit cusps in regions where the coordinates of any two electrons coalesce4,5, hence the wave function and its derivative at Coulomb singularities (electron-nucleus and electron-electron cusp) are shown in the following expressions;

1  ( )r0  Z (1)  r1

and

1  1 ( )  r12 0 (2)  r12 2

where Z is the nuclear charge. However, for determinantal wave functions this condition is never fully achieved. For polyatomic molecules electronic structure calculations are most frequently carried out in atom-centered Gaussian basis sets constituted by function of the form

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

l  r2 lm r  );(  r Ylm  ),( e (3)

with a real harmonic Ylm  ),( covering the angular factor and a radial part composed of a Gaussian function and a power of r which generates the type of orbital, 1s, 2p, 3d, etc.

Despite they do not satisfy the Kato's cusp condition at the nucleus due to the computational ease for evaluating multi-center integrals the choice of using gaussian functions has been well documented. One method of ensuring that this condition met is to build multiple zeta or more polarized basis set. This research is devoted to test Kato's cusp condition as an indicator of basis set quality for Dunning's correlation consistent basis sets. Computational Method

Using  r)(     d , the Eqs. 1 and 2 can be modified in terms of density

1   r)(     Z (4) 2  r)(   r0

In order to test the Kato's cusp condition for the Dunning basis sets we have computed the electron density on the atomic species (O, C, N, F) using cc-pVXZ (X=D, T, Q) basis set. Our main goal is to investigate the effect of increasing basis set size (X) on the near cusp at the nucleus. Moreover, in the second part we have selected four diatomics (NO, CO, FO and

O2) to study whether the presence of neighboring atom with different electronegativity helps to fulfilling Kato's cusp condition. Gaussian 03 has been employed to obtain wavefunctions at B3LYP level for both atomic and diatomic systems. Results and Discussion

 r)( The plot of as function of r for oxygen atom as a representative atom at  r)( B3LYP/cc-pVDZ has been shown in Figure 1.

18.0 16.0 14.0 12.0 |

ρ 10.0 /

ρ 8.0 

| 6.0 4.0 56 20

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Figure 1.  r)( as function of r for oxygen obtained at B3LYP/cc-pVDZ level.  r)(

It is found that the more electronegative is the atom, the cusp condition is satisfied at the closer distance from the nucleus. The distances where the Kato's cusp condition are satisfied for C, N, O and F are 0.009, 0.008, 0.007 and 0.006 a.u, respectively. In the case we use more d and f functions in pVTZ and pVQZ basis sets the Kato's condition occurs at the closer distance to the nucleus; however the variation is little for the species under investigation.

In diatomics the Kato's condition is a little affected due to polarization of the nuclear 1  r)( vicinity. In Figure 2 the values of at r=0.007 a.u. for atomic oxygen has been 2  r)( compared with corresponding values for oxygen atom in four diatomics with different environments. It is observed that the electron density close to nucleus remains almost unaffected when going from atom to multiply bonded molecule. Inspection of Figure 2 indicates that in CO where there is maximum electronegativity difference, the oxygen nucleus has more resemblance to atomic oxygen.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

8.0320 8.0300 8.0280 8.0260

| 8.0240 ρ /

ρ 8.0220  | 8.0200

1/2 8.0180 8.0160 8.0140 8.0120 8.0100 Aomic O CO FO NO O2

Figure 2. 1  r)( at r = 0.007 a.u. for atomic oxygen and oxygen nucleus in 2  r)( four different environment

In general, the influence of basis set of neighboring atom on the electron density near to the nucleus seems of little influence only. Hence, the cusp at the nucleus for the Dunning's basis sets is almost the same for atomic and molecular systems. References

1. E. A. Hylleras, Z. Phys. 54 (1929) 347. 2. R. A. Bonham, D. A. Kohl, Phys. Rev. 45 (1966) 2471. 3. Porras, F. J. Galvez, Phys. Rev. A 46 (1992) 105. 4. T. Kato, Commun. Pure Appl. Math. 10 (1957) 151. 5. E. Steriner, J. Chem. Phys. 39 (1963) 2365.

58 The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Eccentricity, Radius and Diameter of Tensor Product of Graph

S. Moradi Department of Mathematics, Faculty of Science, Arak University, Arak, Iran

Abstract

Let G and H be two graphs. The tensor product G  H of G and H has vertex set V (G  H )  V G)( V H )( and edge set E(G  H)  ba dc |),)(,({ acE G)( and bd  E(H)}. In this paper we obtain some various results about eccentricity of an arbitrary vertex of tensor product G  H and apply them to find the radius and Diameter of this graph.

1. Introduction

A graph G consists of a set of vertices V(G) and a set of edges E(G). For every vertex a V G)( , the edge connecting a and b is denoted by ab and degG a denotes the degree of a in G. The distance between two vertices in a connected graph G is the number of edges in a shortest path between them. This concept has been known for a very long time and recently has received considerable attention as a subject of its own. Suppose a and b are vertices of a graph G, their distance is shown by dG(a,b). For a given vertex a of V (G) its eccentricity  G a)( is the largest distance between a and any other vertex b of G. The radius r(G) is the minimum eccentricity of the vertices and the diameter d(G) is the maximum eccentricity. Now a V G)( is a central node if G a)(  r G)( , and the center C(G) is the set of all central nodes. Graph operations play an important role in the study of graph decompositions into isomorphic subgraphs. For more details see [1,2]. For any two simple graphs G and H, the tensor product G  H of G and H has vertex set V (G  H )  V G)( V H )( and edge set E(G  H)  ba dc |),)(,({ acE G &)( bd E(H)}, we refer the reader to [3] for the proof of this fact that | E(G  H  |2|) E G ||)( E(H |) . The distance between two vertices

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010. under tensor product of graphs is studied in [4,5]. Here we obtain eccentricity of an arbitrary vertex of tensor product of two graphs.

2. Main Results and Discussion

In this section we use some concepts, from [5], for obtaining eccentricity of an arbitrary vertex tensor product of graphs.

Definition 2.1. Let G be a graph. We define dG yx ),( for , yx V G)( as follows:

If dG yx ),( is odd then dG yx ),( is defined as the length of the shortest even walk joining x and y in G, and if there is no shortest even walk then dG yx ),(   .

If dG yx ),( is even then dG yx ),( is defined as the length of the shortest odd walk

joining x and y in G, and if there is no shortest odd walk then dG yx ),(   .

(i) If dG yx ),(   , then dG yx ),(   .

Example. Let K n be a complete graph with n  3. For each ,ba V (K n ) , if a  b then d ba ),(  1 and d ba ),(  2 and if a  b then d ba ),(  0 and d  ba ),(  3. Also Kn Kn Kn Kn let C n12 be a cycle of order 2n+1, then for each two vertices ,ba V (C n12 ) , it is easy to see , d  ba ),(  2n 1 d yx ),( . In each even cycle C for each ,ba V (C ) , C n12 C n12 2n 2n d  ba ),(   . C2 n

Lemma 2.2. Let G be a graph. Then G is not bipartite graph if and only if there exists

, yx V G)( such that dG yx ),(   .

Definition 2.3. Let G be a connected bipartite graph then for each a V G)( , define

 G a)(  max{dG ba |),( b V (G)}.

Lemma 2.4. Let G and H be two nontrivial connected graphs and H be bipartite. Then for each ba ),( V (G  H) ,  HG (( ,ba ))  max{ G (a), H (b)}.

Proposition 2.5. Let G and H be nontrivial connected graphs. Moreover either H or G is not bipartite . Then for each ba ),( V (G  H) ,  HG (( ,ba ))  max{ G (a), H (b)}.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Corollary 2.6. Suppose G and H be nontrivial connected graphs, such that at least one of them is not bipartite. Then for each ba ),( V (G  H) the following equalities is hold:

max{ G (a), H (b)}   HG (( ,ba ))  max{ G (a), G (a),  H b),(  H (b)}.

Corollary 2.7. Suppose G is a connected bipartite graph Kn be a complete graph of order n . Then for each ba ),( V (K  G) ,  (( ,ba ))  max ,3{  (b)}. n Kn G G

Corollary 2.8. Suppose G is a connected bipartite graph Kn be a complete graph of order n . Then r(K n  G)  max r(,3{ G)} and d G)( d G)(  3 d(K n  G)   .  3 d G)(  3

3. Application

In this section we use the pervious results for finding the center of tensor product of complete graph Kn and a graph G.

Theorem 3.1. Let G be a connected bipartite graph and Kn be a complete graph of order n. Then the following statement are hold:

i) If r G)(  3,then C(K n  G)  K n  C G)( .

ii) If r G)(  3 then C(K n  G)  K n  D , where

D {a V G r G)(|)(   G a)(  }3 .

Corollary 3.2. Let G be a connected bipartite graph. If r(G)=1 then K n  G is a self- centered graph. References

[1] J. Bosak, Decompositions of Graphs, Kluwer Academic Publishers, 1990.

[2] W. Imrich and S. Klavžar, Product Graphs, Structure and Recognition, John Wiley & Sons, New York, USA, 2000.

[3] A. Bottreou, Y. Métivier, Some remarks on the Kronecker product of graphs, Inform. Process. Lett. 68 (1998) 55–61.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

[4] D. Stevanovic , Distance regularity of compositions of graphs, Appl. Math. Lett., 17 (2004) 337–343.

[5] S. Moradi, A note on tensor product of graphs, MATCH Commun. Math. Comput.

Chem., (to appear).

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

The Omega Polynomial of a new type of Nanostructures

M. Saheli,1 A. R. Ashrafi1 and M. V. Diudea2 1Department of Mathematics, University of Kashan, Kashan 87317-51167, I. R. Iran 2Faculty of Chemistry and Chemical Engineering, “Babes-Bolyai” University, 400028 Cluj, Romania

Abstract A quasi orthogonal cut (qoc) with respect to a given edge of a graph is the smallest subset of edges closed under taking opposite edges on faces. The omega polynomial is a counting polynomial whose k-th coefficient is the number m(G,k) of qoc's containing k- edges. In this paper an exact formula for the omega polynomial of the molecular graph of a new type of nanostructures named CorCor is given.

Keywords: Omega polynomial, CorCor.

1. Introduction

Throughout this paper graph means simple connected graph. Suppose G is a graph and u, v are vertices of G. The distance d(u,v) is defined as the length of a shortest path connecting u and v in G. A graph can be described by a connection table, a sequence of numbers, a matrix, a polynomial or a derived unique number which is called a topological k index. When we describe a graph by a polynomial as P(G,x) = km(G,k)x , then we must find algorithms to compute coefficient m(G,k), for each k.

Suppose G is a connected bipartite graph, with the vertex set V(G) and edge set E(G).

Two edges e = uv and f = xy of G are called co-distant (briefly: e co f ) if d(v,x) = d(v,y) + 1 = d(u,x) + 1 = d(u,y). It is far from true that the relation "co" is equivalence relation, but it is reflexive and symmetric.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Let C(e) = { f  E(G) | f co e} denote the set of edges in G, co-distant to the edge e  E(G). If relation “co” is an equivalence relation then G is called a co-graph. Consequently, C(e) is called an orthogonal cut "oc" of G and E(G) is the union of disjoint orthogonal cuts. If two consecutive edges of an edge-cut sequence are opposite, or “topologically parallel” within the same face/ring of the covering, such a sequence is called a quasi-orthogonal cut “qoc” strip. This means that the transitivity relation of the “co” relation is not necessarily obeyed. Any oc strip is a qoc strip but the reverse is not always true.

Let m(G,c) denote the multiplicity of a qoc strip of length c (i.e., the number of cut-off edges); for the sake of simplicity, we define m = m(G,c) and e = |E(G)|. Three counting c c polynomials can be defined in simple bipartite graphs as (G,x) = emx , (G,x) = emcx ec and (G,x) = emcx .

Recently, some researchers interested to counting polynomials of nanostructures. In this talk, our recent results on the problem of computing omega polynomial of nanostructures are presented. We focus on a new type of nanostructures named CorCor and its omega polynomial will be computed, Figure 1. Our notation is standard and mainly taken from the standard books of graph theory.

Figure 1. The Molecular Graph of CorCor.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Reference

M. V. Diudea, Omega polynomial, Carpathian J. Math., 22 (2006), 43-47.

A. E. Vizitiu, S. Cigher, M. V. Diudea and M. S. Florescu, Omega polynomial in ((4,8)3) tubular nanostructures, MATCH Commun. Math. Comput. Chem. 57 (2007), 457-462.

M. V. Diudea, A. E. Vizitiu, F. Gholaminezhad and A. R. Ashrafi, Omega polynomial in twisted (4,4) tori, MATCH Commun. Math. Comput. Chem. 60 (2008), 945-953.

M. V. Diudea, Omega polynomial in twisted/chiral polyhex tori, J. Math. Chem. 45 (2009), 309-315.

M. V. Diudea, Omega polynomial in twisted ((4,8)3)R tori, MATCH Commun. Math. Comput. Chem. 60 (2008), 935-944.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

A Note on The Bipartite Edge Frustration

Z. Yarahmadia,1, T. Došlićb, A. R. Ashrafia aDepartment of Mathematics, Faculty of Science, University of Kashan, Kashan 87317-51167, I. R. Iran bFaculty of Civil Engineering, University of Zagreb, Kačićeva 26, 10000 Zagreb, Croatia

Abstract

The smallest number of edges that have to be deleted from a graph to obtain

a bipartite spanning subgraph is called the bipartite edge frustration of G and denoted by G)( . In this paper we determine some properties of the bipartite edge frustration of some classes of composite graphs.

1. Introduction

Let G be a graph with the vertex and edge sets V (G) and E(G) respectively. The bipartite edge frustration of a graph G is defined as the minimum number of edges that have to deleted from the graph to obtain a bipartite spanning subgraph and denoted by  G)( . It is  G)( clear that if G is bipartite then φ(G) = 0, see [1-5] for details. The quantity is, in general, difficult to compute; it is NP-hard for general graphs. Hence, it makes sense to search for classes of graphs that allow its efficient computation.

We recall some definitions that will be used in the paper. The Cartesian product G  H of graphs G and H has the vertex set V (G  H)  V G)( V (H) and ,( ua vb ),)( is an edge of G  H if a  b and uv  E(H) , or ab  E(G) and uv. If G1,G2 ,...,Gn are graphs then n we denote G1  G ... Gn by i1 Gi .

1 [email protected] 66

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

The graphG is obtained from G by adding a new vertex and making it adjacent to all vertices of G. The graph is called suspension of G. In fact we have G  G  K1 , see [6,7]. The corona product GoH of two graphs G and H is defined as the graph obtained by taking one copy of G and |V(G)| copies of H and joining the i-th vertex of G to every vertex in i-th copy of H . It is clear from the definition that corona product of two graphs is not commutative. Obviously, GoH is connected if and only if G is connected. Also if H contains at least one edge then GoH is not bipartite graph, see [6,8]. The join G +H of graphs G and H with disjoint vertex sets V (G) and V (H) and edge sets E(G) and E(H) is the graph union G H together with all the edges joining vertices of V(G) and V(H). In this paper the bipartite edge frustration of above operations are investigated. Finally, we calculate the bipartite edge frustration of complete graph and obtain some its results.

2. The Bipartite Edge Frustration of Some Graph Operations

n Theorem 1. Let G1, · · · ,Gn be connected graphs, G  i1 Gi . Then

n n  (G j )  G)(   V (| Gi |)  . i1 j1 V (| G j |)

Corollary 2. Suppose G is a bipartite graph of order n. Then for every graph H,

 (G  H)  n (H) . In particular

a)  (Pn  C2 m1 )  n ,

b)  (C2n  C2 m1 )  2n ,

c)  (C n12  C2 m1 )  (2 m  n  ).1 Theorem 3. Let G be a connected bipartite graph with bipartition (A, B). Then

 (G)  min{| A ||, B |}.

The following results can be easily verified by direct computation.

a)  (Sn ) 1,

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

b)  (Wn )  n 1,

n c)  (Pn )  2 ,

n d)  (Cn )  2 . Lemma 4. Let G be a connected bipartite graph on n vertices. Then

 (Sn )   (G)   (Pn ) .

 (S )   (G) G  S Notice that n if and only if n , but there exist n-vertex bipartite G  P  (G)   (P ) graphs G such that  n and n ; any even cycle is an example.

Theorem 5. Let G be a graph. Then there exists bipartite subgraph G0 of G by deleting

 (G) edges of G, such that  (G)   G)(  (G0 ) .

Corollary 6. Let G be a connected graph on n vertices. Then

 G)(   (Sn )   (G)   G)(   (Pn ) .

Theorem 7. Let G and H be two connected graphs and let H be bipartite. Then

 (GoH )   G)(  Va G)( , where a is the size of the smaller class in the bipartition

(A,B) of H.

Theorem 8. Let G1 and G2 be two connected bipartite graphs with bipartitions (A1,B1) and (A2,B2), respectively. Let us denote ai = |Ai|, bi = |Bi|, i = 1, 2, and

let ai  bi for i = 1, 2. Then

 (G  G )  min{ aa  bb , ba  ba , 1 2 21 21 21 12 a1 |V (G2 ||) E(G2 ,|) a2 |V (G1 ||) E(G1 |,|) E(G1 ||) E(G2 ) |}.

3. Calculation of Bipartite Edge Frustration of Complete Graph and Its Results

 1 n(n  )2 |2 n Theorem 9. Let K be complete graph of order n. Then  (K )  4 . n n  1 2  4 (n  )1 |2 n

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Corollary 10. Let G be graph on n vertices. Then 0   G)(   (K n ) ). Moreover, for every integer m, 0  m   (K n )there exists a graph G with n vertices, such that  G)(  m .

Corollary 11. Let G be a graph then G)(  1 if and only if all of odd cycles of G are common in one edge and this edge doesn’t lie in any even cycle.

Corollary 12. Let G be a Hamming graph. ThenG  K  K ... K for some n1 n2 ns positive integers n1 ,n2 ,...,ns . For Hamming graph G we have,

s s  G)(  (K ) n .  ni  j i1 j1 ij

Corollary 13. (K )  min{m m , m m , m m }. ,, mmm 321 21 32 31

References

[1] J. A. Bondy, S. C. Locke, Largest bipartite subgraphs in triangle-free graphs with maximum degree three, J. Graph Theory 10 (1986) 477–504.

[2] Q. Cui, J.Wang, Maximum bipartite subgraphs of cubic triangle-free planar graphs, Discrete Math. 309 (2009) 1091–1111.

[3] T. Došlić, D. Vukičević, Computing the bipartite edge frustration of fullerene graphs, Discrete Appl. Math. 155 (2007) 1294–1301.

[4] P. Erdös, On some extremal problems in graph theory, Israel J. Math. 3 (1965) 113–116.

[5] M. Ghojavand, A. R. Ashrafi, Computing the bipartite edge frustration of some

nanotubes, Digest J. Nanomaterials and Biostructures 3 (2008) 209–214.

[6] W. Imrich, S. Klavžar, Product Graphs, Structure and Recognition, John Wiley and Sons, New York, USA, 2000.

[7] D. B. West, Introduction to Graph Theory, Prentice-Hall, Upper Saddle River, NJ,1996.

[8] F. Buckley and F. Harary, Distances in Graphs, Addison-Wesley, Redwood City, CA, 1990.

Posters

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Third-order Connectivity Index of an Infinite Class of Dendrimers

M.B.AHMADI, M. SADEGHIMEHR Department of Mathematics, College of Sciences, Shiraz University Email: [email protected]

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Computing frustration index using a mathematical programming model

M.B.AHMADI, Z. SEIF Department of Mathematics, College of Sciences, Shiraz University Email: [email protected]

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

The Neural Network Approach for Computing the Topological Index Am1 of C4C8[p,q] Nanotours

M.B.AHMADI, M. SALAMI Department of Mathematics, College of Sciences, Shiraz University Email: [email protected]

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Proposing an Upper and a Lower Bound for the Energy of Spider Graph

A. Afzal Shahidi, S. Rahimi Sharebaf Department of Mathematics Shahrood University of Technology , Shahrood , Iran [email protected] [email protected] Abstract

The energy of a graph , denoted by , is defined as the sum of the absolute values of all eigenvalues of . In this paper we show that if is a complete Spider graph with vertices and edges, then

. (1) 212 4 , Also, several classes of graphs3 are 1known2 that satisfy 9 the condition . In this paper we denote . Keywords: , Energy of graph, Triregular graph, Upper and a Lower Bound complete spider graph.

Introduction

Let be a graph. If is the set of vertices of , then The adjacency matrix of , , is an matrix, where if and are adjacent and otherwise. The ,,…, spectrum of graph , Spec , is the set of the eigenvalues of A, denoted by 1 0 [1]. For a graph , the choromatic number of , , is the minimum number of colors needed to color the vertices of such that no two adjacent vertices have the same color[1]. The energy of is defined and denoted by [2]. Gutman specified several classes of graphs without quadrangle that their energies exceeded the number of their vertices[3],[4]. In this paper we have introduced a class ∑of graph that they have quadrangle and this condition holds.

A spider is a tree with at most one vertex of degree more than two, called the center of Spider (if no vertex of degree more than two, then any vertex can be the center). A leg of a Spider is a path from the center to a vertex of degree one. Thus, a star with legs is a Spider of legs, each of length 1.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Let be an integer. The Spider graph is a Spider with legs each of length at least two, such that two vertices of two different legs and are adjacent if 3 ,,…, , and where O is the center of T. , ||A subclass1, of 1 Spider graphs, , is complete Spider graph . A complete Spider graph is a Spider graph with legs . If any leg has length for , and for any two vertices of two different legs and with , we have is adjacent to that ,,…, 1,2, … , , and denote by [5]. , || 1, 1 , In 1,2, this … ,paper we consider the, complete Spider graph that any legs has a vertex of degree one.

1. Main result

Lemma 1. [1] For any graph , , where denotes the largest eigenvalue of . 1 Lemma 2. [1] Let be a graph with vertices and edges, and suppose the eigenvalues of are . Then , . Applying the Cauchy-Schwartz inequality to and leads to the bound ∑ 0 ∑ 2 . (2) ,…, / 1, … ,1 Thus, inequality (2) implies .

Theorem 1. [5] if 2 is even, and if is odd.

Proof. Since has, triangles,3 we have ,4. Also if is odd, then the nearest cycle to the center is an odd cycle and therefore has chromatic number 3. But the center is adjacent to all of , ,3 those vertices. So . Let . Let C be the nearest cycle to the center.

If is even, then,4 we consider a 2-coloring,,…, , where . We color the center by 3. For any vertex in leg if is even we color by the color of , and if : 1,2 is odd we color of . So . , , If is odd, then we consider a 3-coloring,3 , where . We color the center by 4. For any vertex in leg if is even we color by the color of , and if : 1,2,3 is odd we color of . So . , , In this paper, we have a pendent vertex,4 at the end of leg, so we color the pendent vertex with the color of central vertex, therefore there was not be any problem in proof.

The Third Conference and Workshop on Mathematical Chemistryr (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Fige 1

Theorem 2. If is a Spider graph with vertices and edges, then we have

, . 212 4 , Proof. Suppose that 31 are 2the eigenvalues9 of . The n with Lemma1, we have

For , we have 1

, . 2 Moreover, since 3

Using this together with the Cauchy-Schwartz inequalit2y, applied to the vectors and (1, … ,1) with n-1 entries, we obtain the inequality ||,…,||

|| 12 Thus, we must have

(2) ,|| 12

85 The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Now, since the function decreases on the intervals and , we see that or must hold. Since 12 √2 Lemma 2, we have, belongs to interval or , and hence √2 √3 √2 √2 2 √3 3 must hold as well. From this √fact,2, √ 2and Inequality√3, √ (2),2 it immediately follows F2 if c is even that inequality (1) holds. F3 if c is odd A graph is said to be triregular of degrees , , and , ( ), if at least one vertex of has degree , at least one degree and at least one degree , and if no vertex of has degree different from or or . The set of all n-vertex triregular graphs of degrees , , and will be denoted by . If , then the number of vertices of of degree and will be denoted by and , respectively. We have, Θ, , Θ1, , ,

And is number of dependent1. . vertex. . 2

Let denote the th spectral moment of the graph ,

And note ∑that for even ,

. Recall that ∑| |

and (3) Where 2 and are, respectively, the 2 number∑ of 28edges avd quadrangles in . Our starting point is the inequality (4) 2 ∑ So, Complete Spider graph is triregular of for and . if then we can consider triregular of . If then Gutman’s proof is satisfied and we ignore it. 1,4, 4,… 3,4,… 3 Theorem 3. Let is 1,3,4a Spider graph2 with vertices and edges. If then . , 8 30 31 Proof: 31 Let 0 . according, to structure of complete Spider graph, we have,

4 , .

21 1 ∑ 1 . . . 1686 1

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

The number of quadrangle of complete Spider graph is equal to . So, using Eq. (3) and (4), we obtain . 31 3031

But

1 If and only if and this inequality always holds. References 8 30 31 31 0

[1] N.Biggs, Algebriac Graph Theory, Cambridge University Press, Cambridge, 1974.

[2] D. M. Cvetkovic, M. Doob, H. Sachs, spectra of graphs theory and applications, Johann Ambrosius Barth, Heidelberg, (1995).

[3] I. Gutman,on graphs whose energy exceeds the number of vertices, Linear algebra and its applications (2002)

[4] I. Gutman. , more graphs whose energy exceeds the number of vertices, IJMSI, 2 (2007) 57-62.

[5] S.Rahimi Sharebaf, Vertex, Edge and Total coloring in Spider graph, Applied Mathematical Sciences,Vol.3,2009,no.18,877-881.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

An Analytical Approximation to the Solution of Chemical Kinetics System

Hossein Aminikhah  Department of Mathematics, School of Mathematical Sciences, Shahrood University of Technology, P. O. Box 316, P.C. 3619995161, Shahrood, Iran Abstract

In this paper, homotopy perturbation method is applied to solve chemical kinetics problem. Theoretical considerations are discussed. Numerical results are presented, to illustrating the efficiently and simplicity of the method.

Keywords: Homotopy perturbation method, Chemical kinetics model.

1. Introduction

In every phenomenon in real life, there are many parameters and variables related to each other under the law imperious on that phenomenon. When the relations between the parameters and variables are presented in mathematical language we usually derive a mathematical model of the problem, which may be an equation, a differential equation, an integral equation, a system of integral equations and etc. Consider a model of a chemical process [1] consisting of three species, which are denoted by A,B and C . The three reactions are

A ®,B (1)

B +®+,CAC (2)

B +®.BC (3)

Let, y12,y and y3 denote the concentrations of A,B and C , respectively. We assume these are scaled so that the total of the three concentrations is 1, and that each of three constituent reactions will add to the concentration of any of the species exactly at the expense of corresponding amounts of the reactants. The reaction rate of (1) will be denoted by k1 This means that the rateat which y1 decreases, and at which y2 increases, because of this reaction, will be equal to ky11. In the second reaction (2), C acts as a catalyst in the production of A from B and the reaction rate will be written as k2 , meaning that the increase of y1 , and the decrease of y3 , in this reaction will have a rate equal

Corresponding author: Tel & Fax: (+98)2733392012.

E-mail address: [email protected], [email protected] (H. Aminikhah).

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

to kyy223. Finally, the production of C from B will have a rate constant equal to k3 , meaning that 2 the rate at which this reaction takes place will be ky32. Putting all these elements of the process together, we find the system of differential equations for the variation with time of the three concentrations to be

dy 1 =-ky + kyy , dx 11 223

dy 2 =-ky kyy - ky2, dx 11 223 32

dy 3 = ky2. dx 32

If, the three reaction rates are moderately small numbers are not greatly different in magnitude, then this is a straightforward problem.

The homotopy perturbation method introduced by He for solving functional equations at the year 1998 [2]. In this method the solution is considered as the summation of an infinite series which usually converges rapidly to the exact solutions. Considerable research works have been conducted recently in applying this method to a class of linear and non-linear functional equations [3-9].

The rest of this paper is organized as follows:

In section two, the uses of homotopy perturbation method for solving system of chemical kinetics is presented. And conclusion will be appeared in section 3.

2. HPM for system of chemical kinetics

Consider the following non-linear system of ordinary differential equations (chemical kinetics model) with initial conditions

dy 1 =-ky + kyy,( y 0), =a dx 11 223 1 1 dy 2 =-ky kyy - ky2,(0), y =a (4) dx 11 223 32 2 2 dy 3 ==ky2,( y 0)a . dx 32 3 3

For solving system (4), by homotopy perturbation method, we first construct a homotopy as follows

dYdy dY (1--++-=pp )(111,0 ) (kYkYY ) 0, dx dx dx 11 223 dYdy dY (1--+-++=pk )(222,0 ) (YkYYkY2 ) 0, (5) dx dx dx 11 223 32 dYdy dY (1--+-=pk )(333,0 ) (Y2 ) 0. dx dx dx 32

And consider the solution of Eq. (5) as a series say:

2 Yxii()=+ Y,0 () x pYx i ,1 () + pYx i ,2 () +L , i = 1,2,3, (6)

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

where Yxiij, ( )==K 1, 2, 3, j 0, 1, 2, are functions which will be determined.

Substituting (6) into (5) and comparing the coefficients of identical degrees of p , for i =1,2,3, we have

ì dY dy ï 1, 0= 1, 0 , ï dx dx ï ï dY2,0 dy 2,0 p0 :,íï = ï dx dx ï ï dY3,0 dy 3,0 ï = , ï dx dx îï (7) ì dY dy ï 1,1=- 1, 0 -kY + kY Y , ï dx dx 1 1,0 2 2,0 3,0 ï ï dY2,1 dy 2,0 pk12:,íï =- +Y -kYY -kY ï dx dx 1 1,0 2 2,0 3,0 3 2,0 ï ï dY3,1 dy 3,0 2 ï =- +kY32,0. îï dx dx

j- 1 ïì dY 1,1 ï =-kY + k Y Y , ï dx 11,1jk-- 2å 2,3,jk- 1 ï k= 0 ï jj--11 ï dY 2,1 pkj :,íï =-YkYYk -YY j =K1, 2, ï 11,122jk--åå,3,132jkk--,2,1jk- ï dx kk==00 ï ï dY j- 1 ï 3,1 = kYY . ï dx 32å ,2kjk,1-- îï k= 0

The initial approximation Yxi,0() or yi,0()x can be chosen freely, here we set

Yxii,0()=== yx ,0 ()a i , i 1,2,3. (8)

Therefore, the solution of Eqs. (7) can be readily obtained

j- 1 xx Yx1,jj( )=- k 1 Y 1,-- 1 () tdtk + 2 YtY 2, k () 3,jk- 1 () tdt , òò00å k = 0 jj--11 xx x Yx2,11jj( )=- k Y,--1 () tdtk 22 YtY,3 k (),132jk-- () tdtk - YtY,2 k (),1jk- () tdt , j =K1, 2, (9) òò00åå ò 0 kk==00 j- 1 x Yx3,jk( )= k 3 YtY 2, () 2,j--k 1 () tdt . ò0 å k= 0

Therefore, we will have

¥ yxiii()== lim() Yx Y,j (), x i = 1,2,3. (10) p® 1 å j = 0

If we set, kk12==0.04, 0.003, ky 31 = 0.05, (0) = 10, yy 23 (0) == 5, (0) 20 , the numerical results are presented in Table 1 and Figure 1.

Table 1: Numerical values of solutions of chemical kinetic model by HPM for kk12==0.04, 0.003, k 3 = 0.05 .

The Third Conference and Workshop on Mathematical Chemistryr (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

xi * * * yx1( ) y2(x) y3(x) 0 10 5 20 0.5 9.944651132 4.494033438 20.56131543 1 9.880161150 4.098646466 21.02119238 1.5 9.808548358 3.786293183 21.40515846 2 9.731598340 3.550846463 21.71755520 2.5 9.651195643 3.422280269 21.92652409 3 9.569769303 3.494212621 21.93601808 3.5 9.490852205 3.964309224 21.54483857 4 9.419754293 5.187547743 20.39269796 4.5 9.364349610 7.742342740 7.742342740 5 9.335977191 12.50953127 13.15449154

Fig. 1. The plots of approximations of chemical kinetics model for k12==0.04,k 0.003,k3 = 0.05 .

If we set, k1 = 0.1,kk23= 0.02, = 0.009,y1 (0) ==10,y2 (0) 5,(0)20y3 = , the numerical results are presented in Table 2 and Figure 2.

Table 2: Numerical values of solutions of chemical kinetic model by HPM for kk12==0.1, 0.02,k3 = 0.009 .

xi * * * yx1( ) y2(x) y3(x) 0 10 5 20 0.5 10.43503165 4.464605890 20.10036245 1 10.76109494 4.057152920 21.18175214 1.5 11.00402787 3.746196924 20.24977521 2 11.18359480 3.509324189 20.30708101 2.5 11.31487799 3.332706833 20.35241517 3 11.40922038 3.212502512 20.37827711 3.5 11.47471966 3.158098469 20.36718187 4 11.51627359 3.197199938 20.28652648 4.5 11.53517660 3.382762863 20.08206054 5 11.52826765 3.801770988 19.66996136

91 The Third Conference and Workshop on Mathematical Chemistryr (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Fig. 2. The plots of approximations of chemical kinetics model for k1 = 0.1,kk23= 0.02, = 0.009 .

3. Conclusions

Homotopy perturbation method has been known as a powerful device for solving many functional equations such as ordinary, partial differential equations, integral equations and so many other equations. In this article, we have applied homotopy perturbation method for solving the nonlinear system of chemical kinetics. Numerical methods such as Runge-Kutta and Euler methods commonly used for solving these equations, either need a lot of computations and have less convergence speed and accuracy or solve only certain types of problems. In many cases one or two terms approximations have enough accuracy, i.e. fast convergence which can be mentioned as an advantage of the homotopy perturbation method. The main advantage of the HPM over ADM. is that this method provides the solution without a need foor calculating Adomian's polynomials [9]. It can be concluded that homotopy perturbation method is very powerful and efficient technique in finding exact solutions for wide classes of problems. The computations associated with the examples in this paper were performed using Maple 10.

92 The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

References

[1] J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons (2003). [2] J.H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering 178 (1999) 257–262. [3] J.H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, International Journal of Non-Linear Mechanics 35 (1) (2000) 37–43. [4] J.H. He, Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation 135 (2003) 73-79. [5] J.H. He, Comparison of homotopy perturbation method and homotopy analysis method, Applied Mathematics and Computation 156 (2004) 527-539. [6] J. Biazar, H. Ghazvini, Exact solutions for nonlinear Burgers equations by homotopy perturbation method Numerical Methods for Partial Differential Equations [In Press]. [7] L. Cveticanin, Homotopy–perturbation method for pure nonlinear differential equation, Chaos, Solitons and Fractals 30 (2006) 1221–1230. [8] A. Beléndez, T. Beléndez, A. Márquez, C. Neipp, Application of He’s homotopy perturbation method to conservative truly nonlinear oscillators, Chaos, Solitons and Fractals, 37 (3) 2008 770-780. J. Biazar, H. Ghazvini, Homotopy Perturbation Method for Systems of Ordinary Differential Equations, Journal of Nature Science and Sustainable Technology, [In press].

Reverse Wiener Index of Cartesian Product of Graphs

Majid Arezoomand a, 1 , Akbar Zare Chavoushi b , a Islamic Azad University, Majlesi Branch, Isfahan, Iran

b Department of Mathematics, Isfahan University of Technology, Isfahan, Iran

Abstract. Let G be a graph, dG() the diameter of G and duv(,) the topological distance between vertices u and v . Starting from the distance matrix of graph G and subtracting from dG() each duv(,) value, one obtains a new symmetrical matrix which, like the distance matrix,

1E-mail: [email protected] (Corresponding author)

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010. has zeroes on the main diagonal and, in addition, at least a pair of zeroes off the main diagonal corresponding to the diameter in the distance matrix denoted by RW() G . The reverse-Wiener index of graph G is defined as ()=GR [W ()]G. In this work we give an exact {,}uv V ( G ) uv, formula for the reverse-Wiener index of Cartesian product of graphs. As a consequence we compute this topological index for C 4 -nanotube and C 4 -nanotori.

Keywords: Cartesian product, reverse-Wiener index, C 4 -nanotube, C 4 -nanotori.

Introduction

A topological index (TI) is a number associated with a chemical structure represented by a connected graph (usually a hydrogen-depleted graph) wherein atoms are represented by vertices (points) and covalent bonds by edges (lines) connecting adjacent vertices. The first TIs were introduced in the late 40s by Wiener and by Platt. Since then, many new TIs have been added for quantitative structure-property relationship (QSPR) and especially quantitative structure-activity relationship (QSAR) studies. Several hundreds of mathematical descriptors derived from molecular graphs were proposed in the literature, but only a few of them were found useful in QSPR models.

Wiener’s index, denoted by WG(), is defined as the sum of all topological distances in the hydrogen-depleted graph G . The topological distance duv(,) between two graph vertices u and v is the number of edges along the shortest path between these two vertices. The matrix which has as entries duv(,) (topological distances) is called the distance matrix D of the graph; it is symmetrical and has zeroes only on its main diagonal. Then WG()= duv (,). {,}uv V ( G )

By subtracting from the graph diameter dG() all topological distances one obtains a new symmetrical matrix, reverse Wiener RW() G , with zeroes on the main diagonal, whose sums over rows or columns give rise to new integer-number graph invariants whose halfsum is a novel topological index (TI), the reverse Wiene index ()G . One can see that |()|(|()|1)VG VG  ()=Gd ()GW ()G. 2

Theorem 9 Let GH be the Cartesian product of G and H . Then

|VG ( )|| VH ( )| (GH )= (()(|()|1)()(|()|1))|()|()|()|() dGVH  dHVG  VG22  H  VH  G 2

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Corollary 10 Let GG1  n be the Cartesian product of n graphs GVEiii=( , ). Then

nn()1G 1 1 ()GGV =||2 ((i dG)()) 1 ni2   ii=1||VVii 2 =1 ||||V

n where ||=VV | |. i =1 i

n 1 n It is easy to see that dC()= when n is odd and dC()= otherwise. Also n 2 n 2 nn(1)(3) n nn2 (2 ) ()=C when n is odd and ()=C otherwise. Also dP()=1 n and n 8 n 8 n nn(1)(5) n ()=P . n 6

Example 11 Let T and S be the C 4 nanotube and nanotori, respectively. Then

TPC= kn and SC= mn C for some mn,3 and k  2 . By theorem [4] we have

kn(4 k22 n 3 kn 12 kn 12 k 12 2 n  if2| n  24 kn(4 k22 n 3 kn 18 kn 9 k 18 2 n ()=Ti f2Œn .  24   

mn() m n(2 mn ) if2| n , 2| m  8  mn(2 m22 n n m mn2 n m 2)  if2| n , 2Œ m  8 ()=S  mn(44 m22 n n m mn m n )  if2,ŒŒ n 2 m  8  mn(2 m22 n n m mn2 m n 2)  if2,Œ n 2| m  8

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

References

[11]A. T. Balaban, D. Mills, O. Ivanciuc and S. C. Basak, Reverse Wiener Indices,Croatica Chemica Acta,73 (4) (2000) 923-941.

[12] Z. Du. B. Zhou, On the Reverse Wiener Indices of Unicyclic Graphs, Acta Appl Math,106 (2009) 293-306.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Computation of the edge-Wiener index of the rooted product of graphs

Esmaeil Babaei and Ali Iranmanesh Department of Mathematics, Tarbiat Modares University P.O.Box: 14115-137, Tehran, Iran

Abstract

Let be a connected graph with the vertex and edge sets and , respectively. Throughout this paper, we suppose that is connected. The Wiener index, is defined by

where ∑, denotes, the | distance between vertices and .

This index, |was introduced by the chemist Harold Wiener within the study of relations between the structure of organic compounds and their properties.

The edge-Wiener index was defined as:

For , ∑, , | ,0 where, 4. is the line graph of , i.e., a graph that the vertices of are the edges of , with two vertices in adjacent whenever the corresponding edges of 0 are. , | ,| Also , | 1 , | where 0 , such that and .

, | , ,,,, ,. and , | , | where, 0 , such that and .

, are not, distances| max and , ,,,, , for. all .

, | 97, | ,

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Thus for the first edge-Wiener index we have:

And for the second edge-Wiener ∑, index:, | ∑, , | .

The rooted product of graph and rooted graph ,∑ ,, is obtained, |by taking one copy of and copies of , and by joining the root vertex of the th copy of to the th vertex of for . || 1,Let 2, … , ||be a labeled graph on vertices, be a sequence of rooted graphs , then denotes the graph obtained by identifying the root of with the th vertex of , called the rooted product of by . Thus, . , ,…, , … , In this paper, we compute the edge-Wiener|| index of the rooted product of graphs and also obtain this index for some dendrimers.

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

A computational study of ammonia adsorption on boron nitride nanotube

Javad Beheshtian, Farzaneh Zanjanchi, Hadi Behzadi, Nasser L. Hadipour* Department of Chemistry, Tarbiat Modares University, P.O. Box: 14115-175, Tehran, Iran

Abstract

The ammonia adsorption on the surface of (5,0) zigzag BNNT was studied by density functional theory (DFT) calculations. Geometrical optimizations were carried out at the B3LYP/6- 31+G* level of theory. Particular attention is paid to knowing the role of increasing the ammonia cluster size in the process. Four configurations of ammonia adsorption process including monomer (1N), dimer (2N), trimer (3N) and tetramer (4N) clusters were obtained. The strengths of interactions were analyzed by the equilibrium geometries, binding energies, and charge transfer. The natural bonding analyses (NBO) were also performed to investigate electronic properties. The results show that the adsorption of ammonia is more favorable as the cluster size increases.

Key words: Boron nitride nanotube, Density functional theory, ammonia, adsorption

* To whom correspondence should be addressed. Phone: (+98) 218288-3495. Fax: (+98)

218288-9730. E-mail: [email protected]

Introduction

In recent years, a main attention has been put on to the research of boron nitride nanotubes (BNNTs) because of their potential applications in future mechanical and electronic devices. In line with theoretical calculations, BNNT is a wide band gap semiconductor independent of tube radius, chirality, and other geometrical factors.1,2 These have exceptional mechanical properties,3 high thermal conductivity,4,5 resistance to oxidation, and chemical stability,6 which makes them most useful in devices working in hazardous and high-temperature environments.

Ammonia is normally encountered as a gas with a characteristic pungent odour. Ammonia contributes significantly to the nutritional needs of terrestrial organisms by serving as a precursor to foodstuffs and fertilizers. Ammonia, either directly or indirectly, is also a building block for the synthesis of many pharmaceuticals. Although in wide use, ammonia is both caustic and hazardous. Ammonia is a major pollutant in animal agricultural operation and industrial production. Dramatic

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010. increases in atmospheric ammonia emission have been reported in recent years in areas of intensive animal agriculture 7.

Within this work, we performed a computational study on the interactions of a single ammonia molecule and also ammonia cluster, dimer, trimer and tetramer with the outer layer of the single walled BNNTs. The bond strengths, binding energies, and electronic properties in the adsorption process water models on the (5,0) BNNT are investigated. The results of this study reveal new insights to better understand the ammonia molecular adsorption on the surface of tube and the role of co-operativity effects in this process.

Figure 1. Lowest energy configurations of zigzag (5,0) BNNT (a) 1N, (b) 2N, (c) 3N and (d) 4N

(a)

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(d)

Table 1. Interaction energies, Eb (kcal/mol), BSSE correction energy EbBSSE (kcal/mol), distance of tube-ammino molecule of pristine and amino cluster adsorbed zigzag (5,0) BNNT.

E E d Str tu ucture b bBSSE be-cluster 1N - - 1.6 H3 19.18 16.06 7 2N - - 1.6 H3 17.88 24.56 4 3N - - 1.6 H3 34.38 30.65 2 4N - - 1.6 H3 40.30 35.96 1 Table 2. B hybrid and N hybrid in bond, HOMO, LUMO energies (eV) and HOMO-LUMO energy gap (eV) calculated for pristine and water adsorbed on zigzag (5,0) BNNTs.

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Str B N L H g ucture hybrid hybrid UMO OMO ap B - - 3 - - N 6.74 2.93 .81 1 S S - - 3 N p4.60 p1.80 6.43 2.51 .92 2 S S - - 3 N p4.16 p1.85 6.27 2.42 .85 3 S S - - 3 N p3.84 p1.94 6.21 2.36 .85 4 S S - - 3 N p3.66 p1.93 6.15 2.30 .85

References

(1) Blase, X.; Rubio, A.; Louie, S. G.; Cohen, M. L. Europhys. Lett. 1994, 28, 335.

(2) Rubio, A.; Corkill, J. L.; Cohen, M. L. Phys. ReV. B 1994, 49, 5081.

(3) Chopra, N. G.; Zettl, A. Solid State Commun. 1998, 105, 297.

(4) Xiao, Y.; Yan, X. H.; Cao, J. X.; Ding, J. W.; Mao, Y. L.; Xiang, J. Phys. ReV. B 2004, 69, 205415.

(5) Han, W. Q.; Mickelson, W.; Cumings, J.; Zettl, A. Appl. Phys. Lett. 2002, 81, 1110.

(6) Golberg, D.; Bando, Y.; Kurashima, K.; Sato, T. Scripta Materialia 2001, 44, 1561.

(7) Phillips, J.;Control and pollution prevention options forammonia emissions. Research Triangle Park, NC: Control Technology Center Report 1995,#EPA-456/R-95-002;.

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MLR modeling for non-nucleoside HIV-1 reverse transcriptase inhibitors

Reza Behjatmanesh-Ardakani, Fariba Dehghan-Chenari Department of Chemistry, School of Science, Payame Noor University (PNU), Ardakan, Yazd, Iran.

Introduction:

From 1981, acqucred immuno deficiency syndrome (AIDS) has affected 75 million people. This disease caused by Human Immuno deficiency virus type 1 (HIV-1). Almost more than 90% of these people live in developing countries. HIV is a kind of retrovirus that belongs to the lentivirus family. Lentiviruses are slow viruses that cause neurological illness and immune disorders. Retroviruses contain genomic single-stranded RNA in conjunction with RNA-dependent DNA polymerase (reverse-transcriptase enzyme). HIV-medicines have been categorized to four groups containing non-nucleoside HIV-1 reverse transcriptase (NNRTIs). Nucleoside HIV-1 reverse transcriptase inhibitors (NRTIs). Protease inhibitors (pIs), and fusion inhibitors (FIs). In this work we have used chemometric methods to build a model for prediction of PEC50 (-log EC50) of NNRTIs. Table 1 shows the molecules structures of inhibitors,experimental and predicted values used for this study. Quantitative structure activity relationship (QSAR) approach is used to model the behavior of the drugs. The experimental data were taken from reference 1.

Multiple linear regression (MLR) technique is used to relate PEC50 to the descriptors. In this work, topological and quantum mechanical descriptors are used to build this kind of relationship. Table 2 shows the variety of descriptors.

Gaussian 98 has been used to calculate quantum mechanical descriptors presented in table 2 [2] . HF/6-3IG level of theory has been utilized for this purpose. topological descriptors have been calculated by Dragon software [3]. Results show the statistical prameters calculated by SPSS program [4] for the training and predication set. The best unstandardized and standardized MLRs are as follows:

Unstandardized: PEC50=5.431+0.081 VAR-0.097 CV + 1.006 qzzN1 (1)

Standardized: PEC50 = 1.035 VAR- 0.380 CV + 0.204 qzzN1 (2)

2 R  0.898 Radj= 0.887 S.E= 0.39452 R=0.948

2 RCV  0.812 F=76.617 D.W=1.654

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Table 3 slows that there is not any colinearity between data. The colinearity is shown by Varience inflation factor (VIF). Figure 1 shows the residual for training set. The data shows that the residual are distributed normally. Figure 2 and 3 shows the predicted data versus experimental data for the training and test sets, respectively.

References:

1- E. chichero, S. cesarini, P. Fossa, A. spallarossa, L. Mosti, J. Europ. Med. Chem.

2009, 44, 2059.

2- M. J. Frisch. et al., Gaussian, Inc, Pittsburgh, PA, 1998.

3- Dragon 5.0 http://www.disat.unimib.it/chm.

4- SPSS is a statistical software of spss inc.,IL, USA.

Table1: NNRTIs structures,their experimental and predicted values used for this study

Com R1 R PEC5 PEC50 pound 0(exp.) (pred.) 1* H 2,6- 4.12 4.41 * 2 H (CH3)2 4.52 4.78 * 3 H 2,3-Cl2 4.68 4.62 * 4 H 2,6-Cl2 5.4 4.99 * 5 H 3,5-Cl2 4.64 4.54 6* H 2-Cl-6- 6.10 6.12 * 7 H CH3 4.32 4.45 8* H 3-Cl-4- 4.96 5.61 * 9 H CH3 5 4.71 10* CH 5-Cl-2- 7.7 6.94 * 11 3 OCH3 7.7 7.73 * 12 CH 2,3,4-Cl3 7.15 7.06 * 13 3 2,4,6-Cl3 8 8.21 * 14 CH 4-CH3 7.52 7.88 * 15 3 4-C2H5 6.7 6.43 16* CH 4- 6.4 6.24 * 17 3 COCH3 6.3 6.18 18* CH 4-CN 5.92 6.18 * 19 3 4-OCH3 7.10 7.54 * 20 CH 3,4-Cl2 7.40 6.98 * 21 3 3-Cl-4- 7.52 7.67 * 22 CH CH3 6.22 6.55 * 23 3 4-Cl-3- 5.75 5.48

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* 24 CH CF3 5.70 5.7 * 6.52 7.40 ـــ 3 25 * 26 H 4-C2H5 6.7 6.32 27* H 4-Cl 5.10 5.23 * 28 H 4-OCH3 5.22 5.96 29* H 4- 4.87 5.23 * 30 H N(CH3)2 5.43 5.29 * 31 H 3-CH3 4.92 4.60 32* H 3-Cl 5.25 4.76 33* H 4- 6.30 6.50 * 34 H CH(CH3)2 5 4.27 * 35 H 3,4-Cl2 7.52 7.22 * 36 H 2,5-F2 6.85 6.86 H 2,4- H (CH3)2 H 3-CF3 H 2-CH3 H 3,5- H (CH3)2 CH 2-OCH3 3 2,4-F2 H 2,3,5,6- F4 4-F 4- COCH3 37** H Cyclope 4.95 5.01 ntyl 38** CH Cyclohe 6.52 5.65 3 xyl * Structure 1 in Fig.1 has been used for the substitution.

** Structure 2 in Fig.1 has been used for the substitution.

Table 2: The name of descriptors used for MLR equation.

Name Formula

Homo EHomo Lumo ELomo Dipole moment DM Gibbs Free energy G Polarizability MP

Mulliken charges qi Total Energy Etot

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Constant volume molar heat capacity CV Entropy S Chemical hardness η Softness S Electronegativity x Electrophilicity ω  The largest positive atomic charg Q max

Sum of ralues of atomic charg on Oxygen QO Electric field gradient on Nitrogen1 qzzN1 Electric field gradient on Nitrogen2 qzzN2 Wiener Index W Balaban distance connectivity index J Kier flexibility index PHI Variation VAR Surface area approx SAA Surface area grid SAG Volume V Lipophilicity logp Refractivity R Mass M Table3: VIF For the training set

Variable VIF

VAR 1.757

CV 1.604 qzzN1 1.123

(1) (2)

Fig. 1: Two main drugs have been chosen for the investigation.

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Fig. 2: Residuals for the training set.

8 y = 1.0431x - 0.5592 9 2 R2 = 0.8984 7 R = 0.8867 8 7 6 6 5 5 4 4 ypred 3 ycalculated 3 2 2 1 1 0 0 02468 0246810 yexpected yobs

Fig. 3: Predicted pEC50 versus Fig. 4: Predicted pEC50 versus experimental ones for the training set experimental ones for the test set

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The Hosoya and Merri_eld-Simmons indices of a denderimer nanostars

A. Dolati1, M. Haghighat2, and M. Safari2 1 Department of Mathematics, Shahed University, Tehran, Iran. 2 Department of Mathematics, Arak University, Arak, Iran.

Abstract The Hosoya index and Merrifield-Simmons index of a graph G is defined as the total number of independent edge subsets and total number of independent vertex subsets of a graph, respectively. In this paper we compute the Hosoya index and Merrifield-Simmons index of dendrimer nanostar. Keywords: Hosoya index, Merrifield-Simmons index, dendrimer nanostars. Introduction Dendrimer synthesis is a relatively new field of polymer chemistry defined by regular, highly branched monomers leading to monodisperse, tree-like or generational structure. Synthesizing monodisperse polymers demands a high level of synthetic control which is achieved through stepwise reactions, building the dendrimer up one monomer layer, or ''generation'' at a time. Each dendrimer consists of a multifunctional core molecule with a dendritic wedge attached to each functional site. The core molecule is referred to as ''generation 0''. Each successive repeat untile along all branches forms the next generation, generation 1, generation 2, and so on until the terminating generation. The Hosoya index or z-index z(G) of a graph G is the total number of it's matchings plus one, where a matching is a subset M of the edge-set of G with the property that no two different edges of M share a common vertex. If z G)( denotes the Hosoya index of G and m G k),( the number of its n/2 k-matchings, matchings consisting of k edges each, then z G =)( m G k),( , where n is the order k 0= of G, the number of its vertices, and n/2 is the integer part of n/2 . It is convenient to set m G o 1=),( and m(G,1) = m , the number of the edges of graph G. By its definition, we deduce that m G k 0=),( when k > n/2 . The Hosoya index is a prominent example of topological indices which are of interest in combinatorial chemistry. The z-index was introduced by Hosoya[4] in 1971, and it turned out to be applications with physico-chemical properties such as boiling point, entropy or heat of vaporization are well studied. The Merrifield-Simmons index or i- index i(G) of a graph G is the total number of independent vertex subsets. If Gi k),( denote the number of its independent n vertex subsets, consisting of k vertex, then Gi =)( Gi k),( , where n is the number of the k 0= vertices of G. If e=uv is an edge of G, then z(G)=z(G-e)+z(G-u,v). If v is a vertex of G, then z(G)=z(G-v)+z(G-v,x).

If G a graph with components G1,G2 ,G3,...,Gk

then z G =)( z(Gi ) . [5]

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(Sz n =) n ; (Cz n =) (nf 1)  (nf 1) . z(P0 0=) , (Pz 1 1=) , and z(Pn =) (nf 1) n  2 , where f (0) 0= , f (1) 1= , and nf =)( (nf 1)  (nf  2) n  2

If v is a vertex of G, then Gi (=)( Gi {v})  (Gi  N vG ][ ) .

If v and u be adjacent vertices of G, then Gi (=)( Gi  ,{ vu })  (Gi  N uG ][ )  (Gi  N vG ][ ) . If v and u are not adjacent vertices of G, then

Gi (=)( Gi  ,{ vu })  (Gi  v}{  N uG ][ )  (Gi  u}{  N vG ][ )  (Gi  N uG ][  N vG ][ ) .

If G a graph with components G1,G2 ,G3,...,Gk

then Gi =)(  (Gi i ) .

(Cz n =) (nf 1)  (nf 1) . z(Pn =) (nf  2) n  2 , where f (0) 0= , f (1) 1= , and nf =)( (nf 1)  (nf  2) n  2

main result and discussion

The number of vertices nanostar dendrimer. At first we compute the number of vertices nanostar dendrimer. With regard to the process of geometric growing hexagonal and edges between them, we can show that V (| NS[n]) |= 8.2n2 . theorem 2.1.The Hosoya index of nanostar dendrimer z(NS[n]) for n 1 is equal to z(NS[n]) = z [( nA ])2  [( nBz 2.])

where z [( nA ]) = z [( nA 1])(z [( nA 1]) az )(  [( nBz 1]) (bz )),  [( nBz 1])(z [( nA 1]) cz )(  [( nBz 1]) (dz )).

Z [( nB ]) = z [( nA 1])(z [( nA 1]) az )(  [( nBz 1]) (bz )),  [( nBz 1])(z [( nA 1]) cz )(  [( nBz 1]) (dz )). such that z(A[0]) = 26 and z(B[0]) =18. Theorem 2.2.The Merrifield-Simmons index of nanostar dendrimer NS n][ for n 1 is equal to i(NS[n]) [(= nBi ])2  [(2 nBi (]) Ai [n]) where [( nBi ]) = [( nBi 1])( [( nBi 1])i )(  (Ai [n 1])i( )),  (Ai [n 1])( [( nBi 1])i  )(  (Ai [n 1])i( )),

(Ai [n]) = [( nBi 1])( [( nBi 1])i()  (Ai [n 1])i())  (Ai [n 1])( [( nBi 1])i( )  (Ai [n 1])i()).

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such that (Ai [0]) =13 and (Bi [0]) =18 .

References [13] H.Hosoya,Bull.Chem.soc.Jpn 44(1971)2332-2329. [14] I.Gutman,O.E.Polansky(1986), Mathematical concepts in organic chemistry,springer,Berlin.

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COMPUTING THE ZAGREB INDICES OF A TYPE OF DENDRIMER NANOSTARS

A. Dolati and S. TAROMI Department of Mathematics, Shahed University, Tehran, Iran.

INTRODUCTION

Figure 1: A nanostar with n stages Graph theory has provided chemist whit a variety of useful tools, such as topological indices. Molecules and molecular compounds are often modeled by molecular graph. A molecular graph is a representation of the structural formula of a chemical compound in terms of graph theory, whose vertices correspond to the atoms of the compound and edges correspond to chemical bonds.

A topological index is a graph invariant applicable in chemistry. The wiener index is the first topological index introduced by chemist Harold wiener.There are some topological indices based on degrees such as the first and second zagreb indices of molecular graphs. the first zagreb index Z = Z G)( and the second Zagreb index Z = Z G)( of a graph G are defined as: g1 g1 g2 g2

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Z =  ud )([  (vd )] g1 =  GEuve )(

Z =  ()([ vdud )] g2 =  GEuve )(

where ud )( denotes the degree of a vertex u in G. In this paper we compute the Zagreb indices of the dendrimer nanostar depicted in Figure 1.

References 1.M.H.Khalifeh, H. Yousefi-Azari, A R. Ashrafi,The first and second Zagreb indices of some graph operations, Discrete Applied Mathematics,157(2009)804-811.

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Harary index of Armchair polyhex nanotubes

Mehdi Eliasi1 and Ali Iranmanesh 2 Department of Pure Mathematics, Faculty of Mathematical Science, Tarbiat Modares University ,P.O.Box: 14115-137 Tehran, Iran

The Harary index is a novel topological index for the characterization of chemical graphs, derived from the reciprocal distance matrix([4]). In modern chemistry and biochemistry topological indices are use to study the quantitative structure-property relationship between the structure of a molecule and chemical, physical and biological properties.

Let G be a connected simple graph(i.e., G has no loops, multiple or directed edges) with the set of vertices VG()={, v1  , vn }. The reciprocal distance matrix Rd of G , also called the

Harary matrix, is an nn matrix Rd ij such that

 1  if i= j d ij  Rdij =  0=if i j ,  

where d ij is the distance between the vertices v i and v j in G .

In this paper we find an algorithm and write a MAPLE program for computing the Harary index of Armchair polyhex nanotubes (TUVC6[2 p , q ]).

References

[15]D. Plavsi c , S. Nikol, N. Trinajst and Z. Mihali, On the Harary Index for the Characterization of Chemical Graphs. J. Math. Chem., 12:335-350, 1993.

1E-mail: [email protected]

2E-mail: [email protected]

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Topological Symmetry of Fullerenes

M. Faghani1 and A. R. Ashrafi2 1Department of Mathematics,Payam-e-Noor University,Tehran,Iran. 2Department of Mathematics, Faculty of Science, University of Kashan, Kashan 87317-51167, Iran

Abstract

The aim of this paper is to introduce a new algorithm for computing topological symmetry of fullerenes.

Keywords.Topological symmetry, fullerene

Introduction

Fullerenes are carbon-cage molecules in which a large number of carbon atoms are bonded in a nearly spherically symmetric configuration. Let F be a fullerene molecule with exactly p pentagons, h hexagons, n carbon atoms and m bonds. Since each atom lies in exactly 3 faces and each edge lies in 2 faces, the number of atoms is n = (5p+6h)/3, the number of edges is m = (5p+6h)/2 = 3/2n and the number of faces is f = p + h. By the Euler’s formula n − m + f = 2, one can deduce that (5p+6h)/3 – (5p+6h)/2 + p + h = 2, and therefore p = 12, v = 2h + 20 and e = 3h + 30. This implies that such molecules made up entirely of n carbon atoms and having 12 pentagonal and (n/2  10) hexagonal faces, where n  22 is a natural number equal or greater than 20.

In mathematics, groups are often used to describe symmetries of objects. This is formalized by the notion of a group action: every element of the group "acts" like a bijective map (or "symmetry") on some set. To clarify this notion, we assume that G is a group and X is a set. G is said to act on X when there is a map  : G  X X such that all elements x  X, (i) (e,x) = x where e is the identity element of G, and, (ii) (g, (h,x)) = (gh,x) for all g,h  G. In this case, G is called a transformation group, X is called a G-set, and  is called the group action. For simplicity we define gx = (g,x).

Main Results

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Detecting topological symmetry of molecules is a well-studied problem with applications in a large number of areas. The Euclidean matrix of a molecular graph G is a matrix D(G) = [dij], where for i  j, dij is the Euclidean distance between the nuclei i and j. In this matrix dii can be taken as zero if all the nuclei are equivalent. Otherwise, one may introduce different weights for different nuclei.

In this paper a new algorithm for computing topological symmetry of molecules, specially fullerene molecules, is presented.

References

1. Kroto H.W., Heath J.R., O’Brien S.C., Curl R.F., Smalley R.E.: Nature 1985, 318, 162.

2. Fowler P.W., Manolopoulos D.E.: An Atlas of Fullerenes, Oxford Univ. Press, Oxford, 1995.

3. Randić M.: Chem. Phys. Lett. 1976, 42, 283.

4. Randić M.: J. Chem. Phys. 1974, 60, 3920.

5. Balasubramanian K.: J. Chem. Phys. 1980, 72, 665.

6. Balasubramanian K.: Int. J. Quantum Chem. 1982, 21, 411

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Some Remarks on DNA Sequences

Parisa Farhami and Ali Rza Ashrafi Department of Mathematics, University of Kashan, Kashan 87317-51167, I. R. Iran

Keywords: DNA sequence, DNA sequence matrix.

The extremely long DNA molecule is actually made of a long string of chemical building blocks called “nucleotides.” There are four different nucleotides, which are labeled adenine (A), thymine (T), guanine (G), and cytosine (C). A DNA sequence is a succession of letters A, C, G and T representing these four nucleotide bases of a DNA strand.

DNA computing is a form of computing which uses DNA and biochemistry and molecular biology, instead of the traditional silicon-based computer technologies. DNA computing is fundamentally similar to parallel computing in that it takes advantage of the many different molecules of DNA to try many different possibilities at once. Generally, in DNA computing, the DNA sequences used for the computation should be critically designed in order to reduce error that could occur during computation.

Since 2000, Randic’s research group had proposed several visualization schemes for DNA sequences [1-8]. In one of his method, the four vertices associated with a regular tetrahedron are assigned to four nucleotides. The mapping between four nucleotides and corresponding 3-D coordinates is shown below:

(1,-1,-1)  A, (-1,1,-1)  G, (-1,-1,1)  C, (1,1,1)  T.

For a positive integer n, n denotes the set of all DNA sequences of length n and = n1

n. Suppose  = {A, C, G, T}, w = x1x2...xn is a DNA sequence of length n and Li(w) = |{j | 1 ≤ j ≤ i & xj = L}|, where L  . Our other notations are standard and taken mainly from [9]. He and J. Wang [10] considered the coordinates of a graphical representation of DNA sequence introduced by

Zhang [11]. They presented an action of the symmetric group S4 on Z-curves and the DNA matrices and proved that the information entropy is invariant under the action of S4. In this paper we consider another model presented by Randić and his group [8] and extend the results given by He and Wang mentioned above.

 Corresponding author (e-mail: [email protected]) 116

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References

[1] M. Randić, Chem. Phys. Lett. 317, 29 (2000).

[2] M. Randić, Chem. Phys. Lett. 386, 468 (2004).

[3] M. Randić and A. T. Balaban, J. Chem. Inf. Comput. Sci. 43, 532 (2003).

[4] M. Randić and S. C. Basak, J. Chem. Inf. Comput. Sci. 41, 561 (2001).

[5] M. Randić G. Krilov, Chem. Phys. Lett. 272, 115 (1997).

[6] M. Randić and M. Vracko, J. Chem. Inf. Comput. Sci. 40, 599 (2000).

[7] M. Randić, A. F. Kleiner and L. M. De Alba, J. Chem. Inf. Comput. Sci. 34, 277 (1994).

[8] M. Randić, M. Vračko, A. Nandy and S. C. Basak, J. Chem. Inf. Comput. Sci. 40, 1235 (2000).

[9] H.-H. Hsu, Advanced Data Mining Technologies in Informatics, Idea Group Inc., London, 2006.

[10] P. He and J. Wang, J. Phys. A: Math. Gen. 37, 7135 (2004).

[11] R. Zhang and C. T. Zhang, J. Biomol. Struct. Dyn. 11, 767 (1994).

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The Hyper-Wiener Polynomial of Graphs

G. H. Fath-Tabar and A. R. Ashrafi1 Department of Mathematics, Faculty of Science, University of Kashan, Kashan 87317-51167, I. R. Iran

Abstract Let G be a graph. The distance d vu ),( between the vertices u and v of the graph G is equal to the length of a shortest path that connects u and v . Define Gd )( 2 WW G x =),( 1/2 x ),(  yxdyxd ),( , where d G)( is the greatest distance between any two vertices. In i 1= this paper the hyper-Wiener polynomials of the Cartesian product, composition, join and disjunction of graphs are computed. We apply some of our results to compute the hyper-Wiener polynomials of a C4 nanotube and nanotorus, respectively.

Keywords and phrases: Hyper-Wiener polynomial, graph operation, C4 nano-structure.

AMS Subject Classification Number: 05C12, 05A15, 05A20, 05C05.

Introduction

All graphs we consider are assumed to be finite, connected, and to have no loops or multiple edges. The vertex and edge sets of a graph G are denoted by V G)( and E G)( , respectively. The distance between the vertices u and v of V G)( is denoted by d vu ),( and it is defined as the number of edges in a minimal path connecting the vertices u and v . The Wiener index is one of the most studied topological indices defined as the sum of distances between all pairs of vertices of the respective graph. In 1993, Milan Randic proposed a generalization of the Wiener index[5] for trees. Then Klein et al. [1,2], generalized the Randic's definition for all connected graphs. It is defined as 1 1 WW G =)( W G)( + d 2 vu ),( , where d 2 vu =),( vud ),( 2 . 2 2   GVvu )(},{ The Cartesian product[3,4] G  H of graphs G and H has the vertex set V (G H =) V G)( V (H) and ,( xa yb ),)( is an edge of G  H if a = b and xy E(H) , or ab E G)( and x = y . If G ,G ,,G are graphs then we denote G G by n G . Let G 1 2 n 1 n i 1= i and H be two graphs with disjoint vertex sets V G)( and V (H) and edge sets E G)( and E(H) . The join G  H is the graph with vertex set V (G  H =) V G)( V (H) together with all the edges joining vertices V G)( and V (H) . The composition G[H] is the graph with vertex set V G)( V (H) and u vu ),(= is adjacent with v ba ),(= whenever (u is adjacent with a ) or (u = a

1Corresponding author. ([email protected]) 118

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010. and v is adjacent with b ). Consider two arbitrary graph G and H . The disjunction G  H is the graph with vertex set

V G)( V (H) and u v11 ),( is adjacent with (u2 ,v2 ) whenever uu 21  E G)( or vv 21  E(H ) . The symmetric difference G  H is the graph with vertex set V G)( V (H) and

E(G  H =) {(u1,u2 )(v1,v2 |) vu 11  E G)( or vu 22  E(H ) but not both} . Definition. Let G be a graph. The hyper-Wiener polynomial of G is defined as 1 ),( dbad 2 ba ),( WW G x =),( x . 2   GVba )(},{

Main Results

In this section, exact expressions for the hyper-Wiener polynomials of composition, Cartesian product, join, disjunction symmetric difference and power of graphs are computed.

Theorem 2. Suppose G1 and G2 are graphs with V (| G1 |=) n1 , V (| G2 |=) n2 , E(| G1 |=) m1 and 1 1 E(| G |=) m . If G is connected then WW (G [G ], x =) n2WW (G x),  n m x2  n (n  m )x6 2 2 1 1 2 2 1 2 22 2 1 2 2 2 . The kth power graph, denoted by Gk has the same vertex as G . Two vertices are adjacent in Gk if their distance in G is at most k . For further information on the power graph, we encourage the reader to consult [4] and references therein.

Theorem 3. Let G be a graph then kn 1]/[ k k 1)(( ii 2) WW (G ) =  nG ( j  ik)x i 0= j 1= kn  kn 2)]/1)([]/([  (nG (1 kkn )]/[  nG (n))x ,

where n  k, and nG (n 1) = nG (n  2) =  0= . If k | n then the hyper Wiener polynomial of kn ]/[ Gk becomes n )( xj ii 1)( . i 1=  1)( ki 1  ikj G Proof. By definition of the power graph Gk , V (G k =) V G)( and for every vertex ,ba V G)( a and b are adjacent if and only if dG ba ),(  k . There are nG (1) pair of vertices at distance 1

(edges), nG (2) vertices at distance 2,  and, nG k)( vertices that are at distance k . These vertices k become at distance one in Gk . Hence the coefficient of x is n j)( in Gk . One can generalize  j 1= G this idea by taking the distinct pairs of vertices in G whose distance lies in the set

Ai {= ik  , jj = 1,2,...,k}, where 0  i  kn ]/[ 1. There are nG (ik 1)  nG (ik  k) distinct pairs of vertices in G whose distance lies in Ai . These distinct pairs of vertices become at distance i 1 k k in G . Hence we have nG (ik 1)  nG (ik  k) distinct pairs of vertices in G that are at distance i 1 . This gives the hyper Wiener polynomial of Gk . ฀

Theorem 4. Let G and H be graphs with n1 =|V G |,)( n2 =|V (H ) , m1 =| E G)( and

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m2 =| E(H ) . Then

1 2 2 2 1 2 2 6 WW (G  H x =), (n1 m2  n2 m1  2m m21 )x   nn 21  n1 m2  n2 m1  2m m21 x . 2 2    2 

Theorem 5. Let G and H be graphs with n1 =|V G |,)( n2 =|V (H ) , m1 =| E G)( and m2 =| E(H ) . Then

1 2 2 2 1 2 2 6 WW (G  H x =), (n1 m2  n2 m1  4m m21 )x   nn 21  n1 m2  n2 m1  4m m21 x . 2 2    2 

Theorem 6. Let G1,G2 ,,Gk be graphs with ni =|V (Gi |) and mi =| E(Gi |) , 1 i  k . Then k k 1   2 1 6 WW (G1  G2  Gn =) mi   nn ji x  ni  mi x . 2  i 1=  ji  2 i 1=    2  In particular, if G is a graph with n vertices and m edges then 1 1 WW (kG x =), [km  k n2 ]x2  n  mx6 . 2 2   2  2  Theorem 7. Suppose G and H are graphs and d = d G)(  d(H) . Then d k1 1   kk 1)( WW(G H x =), 2nG )( nj H (k  j  V G |)((|) nH k  V (|)( H |) nG k)( x . 2 k 1=  j 1= 

Proof. By Lemma 1(a), we have d HG (( , xa ,(), yb )) = dG ba ),(  dH yx ),( . Thus,

n HG k)( = |{{( , xa ,(), yb |)} d HG (( , xa ,(), yb )) = k |}

= |{{( , xa ,(), yb |)} dG ba ),(  d H xa =),( k |}

= |{{( , xa ,(), yb |)} dG ba =),( ,dj H ,( yx )) = k  , jj = 0,1,k |} k = 2nG )( nj H (k  j) j 0= k 1 = V G ||)(| nH k  V (||)( H ||) nG k |)( 2nG )( nj H (k  j), j 1= which completes the proof. ฀

References

[16] G. G. Cash, Relationship Between the Hosoya Polynomial and the Hyper-Wiener Index, Appl. Math. Letters, 15 (2002), 893-895. [17] I. Gutman, Relation between hyper-Wiener and Wiener index, Chem. Phys. Letters, 364 (2002), 352-356.

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[18] M. H. Khalifeh, H. Yousefi-Azari and A. R. Ashrafi, Vertex and Edge PI Indices of Cartesian Product Graphs, Disc. Appl. Math. (2007), doi: 10.1016/j.dam.2007.08.041. [19] M. H. Khalifeh, H. Yousefi-Azari and A. R. Ashrafi, The hyper-Wiener index of graph operations, Computers and Mathematics with Applications (2008), doi:10.1016/j.camwa.2008.03.003 [20] H. Wiener, Structural determination of the paraffin boiling points, J. Am. Chem. Soc., 69(1947), 17-20.

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Theoretical study of the ternary oil-water- asymmetric mixed surfactants systems by lattice Monte Carlo simulation

S.Garmroudi-Sabet1, B.Sohrabi1, S.M.Mousavi-khoshdel2, S. M. Hashemianzadeh 1 1. Molecular Simulation Research Laboratory Department of Chemistry, Iran University of Science & Technology, Tehran, Iran. 2. Shahed University

Abstract

Monte Carlo simulation provides some insight into the self-assembly of surfactants in aqueous environment. . In the presence of an appropriate solvent, they have the ability to aggregate into structures called micelles. In this work, we investigated some physical properties of micellar aggregates such as the CMC, aggregation number, and premicellar concentration for asymmetric mixed surfactants-oil-water systems using lattice Monte Carlo simulation. Keywords: Lattice Lattice Monte Carlo simulation, self-assembly, critical micelle concentration (CMC), premicellar concentration, surfactant.

Introduction Nowadays computer experiments play an important role in industry. They are complement and alternative to lab experiments. By computer methods, we can investigate unusual temperature/pressure regions, simulate dangerous experiments, find alternative for hazardous chemicals, gain an atomistic description of a reaction and save lab costs. Also, they can help us to characterize and understand reaction mechanisms. We can use these experiments in environmental sciences, biology, chemistry and physics with the language of mathematics. They need models and theories to describe the laws of nature.

Computer simulations are progressively being used to study the structure and thermodynamics of micelles and properties of self-assembling amphiphilic systems. Simulation results are shown to be useful for microscopic understanding of thermodynamic properties and for further improvement of theoretical models and their limitations [1].

Surfactant molecules have two structural features: one hydrophobic and the other hydrophilic. In the presence of an appropriate solvent, they aggregate into structures called micelles [2].

 [email protected] 122

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Surfactant mixtures are very useful products in industry. They frequently show synergistic interactions in solution that can be used to reduce the total amount of surfactant needed in applications, thus reducing costs [3].

Moreover, for many applications additives are needed for better stability and control of the physical properties. In this work, oil additives are used. Micellar aggregates can solubilize oil molecules, thus bringing them into aqueous solution [4].

This study is based on standard Metropolis algorithm [5]. From all possible configurations for the system, a starting configuration is randomly selected and the total internal energy of this configuration, Eold, is calculated. Then, another configuration is randomly generated and its energy, Enew, is calculated. The only type of move used for modification of the configuration is the reptation move. The probability of acceptance of this move is calculated according to the standard Metropolis algorithm [5],

Pacc .  Min ,1{ exp(   E )} (1)

where  E is the difference in the total internal energy between the trial and old configurations [6,7]. If a move was not accepted, the old configuration was maintained and a new move was tried [6].

This attempted move is called a Monte Carlo step, and is repeated until equilibrium is reached for a particular temperature. In this work, more than 108 moves were done in order to reach a region where the total energy was almost constant.

To analyze mixtures of surfactants, the regular solution theory is the most widely used [8]. In the regular solution theory, the micelles are considered as a macroscopic phase in equilibrium with a bulk solution phase containing only monomers. Nonideality is introduced via the activity coefficients γi [9] .To evaluate these activity coefficients, the excess free energy of mixing gE is expressed as a series expansion. This leads to the following expressions for the activity coefficients:

2  1  exp  1(  x1 ) (2)

2  2  exp (x1 ) (3)

β is the interaction parameter for micellization and xi is the mole fraction of surfactant i in the mixed micellar phase. β has been interpreted as the enthalpic contribution to nonideality:

w  (w  w 2/)   12 11 22 k T B (4) 123

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

where w11 and w22 are the interaction energies between the surfactant species in pure micelles and w12 is the interaction energy between the surfactant species in mixed micelles. k is the Boltzmann constant, and T is the absolute temperature. β serves as an index of the magnitude of the nonideality of the interactions between the surfactants in the micellar phase. When β is negative, there is a negative deviation from ideality or a synergism between surfactants. On the other hand, when β is positive there is a positive deviation from ideality or a repulsive interaction between the surfactant species.

In this paper we have presented Monte Carlo simulations of asymmetric mixed surfactants, H4T4-

H2T4, and one type of oil, T4, on a cubic lattice to predict micellar system properties such as the Critical Micelle Concentration and the aggregate size distribution. We chose a box size of 60*60*60. By having the attraction interaction between the headgroups of the surfactants species, the model was able to demonstrate the synergistic behavior that characterizes surfactant mixtures. Our results showed in the higher concentrations of our mixture, T4 molecules were in aggregated with mixed surfactants. In addition, the simulation results indicated reasonable agreement with the regular solution theory.

Figures and Tables

7.20E-03 MIX H2T4&H4T4&T4=0.005,ratio=0.1

6.20E-03 MIX H2T4&H4T4&T4=0.005,ratio=0.2

MIX H2T4&H4T4&T4=0.005,ratio=0.3 5.20E-03 MIX H2T4&H4T4&T4=0.005,ratio=0.4 4.20E-03 MIX H2T4&H4T4&T4=0.005,ratio=0.5 3.20E-03 MIX H2T4&H4T4&T4=0.005,ratio=0.6

2.20E-03 MIX H2T4&H4T4&T4=0.005,ratio=0.7

MIX H2T4&H4T4&T4=0.005,ratio=0.8 1.20E-03 MIX H2T4&H4T4&T4=0.005,ratio=0.9 2.00E-04 MIX H2T4&H4T4&T4=0.005,ratio=1 Free Monomer Concentration Monomer Free 0 0.02 0.04 0.06 0.08 0.1 0.12 -8.00E-04 Total Concentration

Figure 1. Free monomer concentration vs. total concentration of mixed H4T4-H2T4 and T4

concentration of 0.005 at different ratios of H2T4; EH1H2= - 0.7.

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0.033

0.031

0.029

0.027

0.025

0.023

0.021

0.019

0.017

Mixed Critical Micelle Concentration,CMC 0.015 0 0.2 0.4 0.6 0.8 1 Fraction of surfactant H2T4

Figure 2. Mixed CMC vs. fraction of surfactant H2T4 for mixed H4T4-H2T4 and T4 concentration of 0.005;

EH1H2= - 0.7.

Head of H4T4: Red

Tail of H4T4: Cyan

Head of H2T4: Blue

Tail of H2T4: White

Figure 3. Snapshots for micellar aggregations in T4 constant concentration of 0.005 and a) mixed H4T4-H2T4 concentration of

0.0006, b) mixed H4T4-H2T4 concentration of 0.02. ) b)

0 0.2 0.4 0.6 0.8 1 0

-0.2

-0.4

-0.6

-0.8

-1 2 ln 2 x1 Figure 4. vs. for mixed H4T4-H2T4 and T4 concentration of 0.005; EH1H2= - 0.7.

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Table 1. Our results for the simulated system of mixed H4T4-H2T4 and concentration of T4 = 0.005;

EH1H2= - 0.7 System of Mixed H2T4, H4T4; Slope of premicellar Interaction CMC* T4= 0. 005; E= -0. 7 region parameter(β) ratio= 0.1 0.0284 0.267606 -0.7346 ratio= 0.2 0.026 0.234615 -0.8327 ratio= 0.3 0.024 0.2 -0.9654 ratio= 0.4 0.022 0.177273 -1.1822 ratio= 0.5 0.021 0.171429 -1.2945 ratio= 0.6 0.02 0.165 -1.4861 ratio= 0.7 0.197 0.167513 -1.6229 ratio= 0.8 0.021 0.171429 -1.5205 ratio= 0.9 0.023 0.165217 -1.4188

References [1] R.G. Larson, L.E. Scriven and H.T. Davis, J. Chem. Phys. 83, 2411 (1985). [2] S.K.Talsania, L. A. Rodriguez-Guadarrama, K.K. Mohanty, R.Rajagopalan; “Phase Behavior and Solubilization in Surfactant-Solute-Solvent Systems by Monte Carlo Simulations”; Langmuir; 14; 2684-2692(1998). [3] L. A. Rodriguez-Guadarrama, S.Ramanathan, K. K. Mohanty,V.Vasquez; “Molecular Modeling of binary mixtures of amphiphiles in a lattice solution”; Fluid Phase Equilibria; 226; 27-36(2004). [4] S.Y. Kim, A.Z.Panagiotopoulos; “Ternary Oil-Water-Amphiphile Systems: Self-Assembly and Phase Equilibria” ;( 2001). [5] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller, J. Chem. Phys. 21;1087 (1953). [6] L. A. Rodriguez-Guadarrama, S.K.Talsania, K. K. Mohanty,R.Rajagopalan; “Thermodynamics of Aggregation of Amphiphiles in Solution from Lattice Monte Carlo Simulations”; Langmuir; 15; 437- 446(1999). [7] M.Zaldivar, R.G. Larson; “Lattice Monte Carlo Simulations of Dilute Mixed Micelles”; Langmuir; 19; 10434-10442(2003). [8] Rubingh, D. N. In Solution Chemistry of Surfactants; Mittal, K.L., Ed.; Plenum Press: New York; Vol. 1; 337(1979). [9] V. Peyre, Langmuir 18; 1014(2002).

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Estimating kinetic parameters of the Oxidative Coupling of Methane over (4%Mn+0.1%Na+3.13%W)/SiO2 catalyst by using genetic algorithm

Mahnaz Ghiasi , Azim Malekzadeh School of Chemistry, Damghan University of Basic Sciences (DUBS), Damghan, I. R. Iran. E-mail: [email protected]

The oxidative coupling of methane (OCM) has been extensively studied over the last two decades owning to the large reserves of natural gas. Since of the early work of the Keller and Bhasin on the direct conversion of natural gas to ethane and ethylene, intensive researches have been carried out to develop more active and selective catalysts [1]. However, further improvements of the catalytic performance by reaction engineering are necessary in order to make this process commercially viable [2].

The OCM reaction involves chemical species of CH4, O2, C2H4, C2H6, CO and CO2, H2O and possibly H2. The reaction kinetic is complicated and consisted of parallel and consecutive, heterogeneous and homogeneous reactions. Kinetic models help to be inform about rates, behavior of catalyst sites and to simulate OCM reactors. In this study the kinetic of

(4%Mn+0.1%Na+3.13%W)/SiO2 catalyst, as an effective catalyst for OCM, is considered. Rate equations are determined on the basis of the experimental data. Kinetic parameters, involving the reaction rate constant, activation energy and reaction orders, however, are estimated by means of genetic algorithm (GA). One of the considered reaction networks is Stansch model with 10 steps, an step of this model is shown below.

C2H4 + 2H2O 2CO + 4H2 (1) r  k e Ea /RTPm pn Equ 0 C2H4 H2O ation 1 Genetic algorithm is a technique to find the approximate solutions for optimization. The conventional optimization techniques are started by a single point whereas GAs operate on a whole population of points. Method improves the chance of reaching to the global optimum and also helps in avoiding local stationary point. A flowchart of genetic algorithm is shown in the scheme 1. GA have been successfully applied to non-linear optimization problems in many dimensions, where more traditional methods are often found to fail [3]. Genetic algorithm handles a population of

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010. possible solutions. Each solution is represented through a chromosome, which is just an abstract representation [4]. The chromosomes evolve through successive iterations, called generation. During each generation, the chromosomes are compared against each other according to a measure called fitness. The optimum values of kinetic parameters are obtained by a fitness function which is average square relative deviation of calculated data (Eq. 2) [5]. Here “C” and “D” are the measured responses in each experiment and the number of experiments, respectively.

 1  1  N  N D C  ji, cal ji, Exp Fitness = 1=i  1j Equ  D C  N ji, Exp ation 2 

For creating next generation new chromosomes called offspring are formed through mating and mutation, now chromosomes with poor fitness are replaced by the obtained offsprings. A Decimal GA, which has been used by many researchers for solving optimization problems in both scientific and engineering problems, is chose [6]. In Decimal GA crossover operator is arithmetic crossover and offsprings are obtained through following equations.

Offspring 1 = ω × Parent1 +(1 – ω) × Equ Parent2 ation 3 Offspring 2 = ω × Parent2 +(1 – ω) × Equ Parent1 ation 4

Start

Creat initial random

Evaluate fitness for each

Store best

Creating

Create next generation by applying crossover

Y Optimal or good

solution found? N Reproduce and ignore few

Perform

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Scheme 1: Flowchart of genetic algorithm.

The calculated parameters were estimated by our generated program in Matlab R2008b and also using optimization Genetic Algorithm Toolbox of the Matlab software. The values of the parameters used in the implemented GA are listed in table 1.

Populatio 10 n size 0 Mutation 0. rate 15 Crossover 1 rate Table 1: Selected GA parameters.

Precision of the models and obtained kinetic parameters from the genetic algorithm are measured with calculating average absolute relative deviation (AARD) (Eq. 5).

1 m N - N AARD = jexp,  1j Equ m N jexp, ation 5

Here “m” is the total number of experimental data.

Calculations on Carbon Monoxide Selectivity is presented in table 2.

CH4/Air = 1 CH4/O2 = 5 Argon Air CM CM CM CM T CMS CM (ºC) S*exp(%) Scal(%) (ºC) Sexp(%) Scal(%) (ºC) exp(%) Scal(%) 7.0 7.0 9.40 9.4 7 7.54 7.5 75 5034 61 75 559 82 75 3504 63 7.0 7.0 9.46 9.5 8 7.35 7.3 00 4686 58 00 299 19 00 692 52 6.9 7.0 9.93 9.7 8 7.44 7.5 25 5935 41 25 3906 3 25 993 29 *Carbon Monoxide Selectivity(%) Table 2: experimental and calculated Carbon Monoxide Selectivity in different conditions.

According to experimental and calculated Carbon Monoxide Selectivity the value of AARD% is 0.23% which is low and shows the precision of the model.

References

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[1] G. E. Keller, M. M. Bhasin; Synthesis of ethylene via oxidative coupling of methane, J.Catal, 73 (1982) 9-19.

[2] Z. Stansch, L. Mleczko, M. Baerns; Comprehensive kinetics of oxidative coupling of methane over the La2O3/CaO catalyst, Ind. Eng. Chem. Res. 36 (1997) 2568-2579.

[3] L. Elliott, D .B. Ingham, A. G. Kyne, N. S. Mera, M. Pourkashanian, C. W. Wilson; Genetic algorithms for optimisation of chemical kinetics reaction mechanisms, Progress in Energy and Combustion Science 30 (2004) 297-328

[4] Randy L. Haupt, Sue Ellen Haupt; Practical genetic algorithms, Second Edition, 2004.

[5] Mahdi Daneshpayeh, Abbasali Khodadadi, Navid Mostoufi, Yadolah Mortazavi, Rahmate Sotudeh-Gharebagh, Alireza Talebizadeh; Kinetic modeling of oxidative coupling of methane over Mn/Na2WO4/SiO2 catalyst, Fuel Processing Technology 90 (2009) 403-410.

[6] Ramin B. Boozarjomehry, Mohammad Masoori; Which method is better for the kinetic modeling: Decimal encoded or Binary Genetic Algorithm?, Chemical Engineering Journal 130 (2007) 29-37

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Different Time of sputtering Effect on the Synthesis of Cu nano particles on AISI 304 by cylindrical magnetron sputtering system

M. Ghoranneviss, M. Eshghabadi, *A. Mahmoodi Plasma Physic Research Center, Science and research branch center I.A.U, Tehran, Iran E-mail address: [email protected]

Abstract

In this paper, Cu nano particles have been synthesized on AISI 304 by cylindrical magnetron sputtering technique. We used Cu cylindrical target for a 45 m Torr Argon sputtering for deposition. Various time of sputtering were studied in order to obtain satisfactory condition for cu nano particles. After deposition, the films were annealed using oven at 500 °C for 10 min to induce the nucleation and growth of Cu nanoparticles. By the analysis of XRD the resultant particles were confirmed to be Cu. AFM and SEM was both applied to characterize the nanostructures of Cu nanoparticles in this study. Details of results will be discussed in the full paper.

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Counting Numbers of Permutational Isomers of some Hetero Fullerenes

Modjtaba Ghorbani, Maryam Jalali and A. R. Ashrafi Department of Mathematics,Faculty of Science, Shahid Rajaei Teacher Training University, Tehran, 16785 - 136, I R. Iran

Abstract

Hetero fullerenes are fullerenes where some of the carbon atoms are replaced by other atoms. Fripertinger applied SYMMETRICA to write some codes for computing

the number of C60-kBk molecules, where B is a hetero-atom such as Si. (see H. Fripertinger, MATCH Commun. Math. Comput. Chem. 33, 121 (1996)) In this paper,

the numbers of all Cn-kBk hetero-fullerenes are computed, where Cn is a fullerene. We apply the computer algebra system GAP to compute the number of permutational

isomers of hetero fullerenes of the C60 fullerene with Ih point group symmetry.

Keywords: Fullerene, Hetero Fullerene, Cycle Index, Permutation Group.

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

On Omega and Sadhana Polynomials of Some Fullerenes

Modjtaba Ghorbani and Maryam Jalali Department of Mathematics,Faculty of Science, Shahid Rajaei Teacher Training University, Tehran, 16785 - 136, I R. Iran

Abstract. The omega and the Sadhana polynomials are defined as

(G, x) m(G,c)  xc Sd(G, x) m(G,c) x|E| c  c ,  c , where m(G, c) is the number of strips of length c. The sadhana polynomial has been defined to evaluate the sadhana index of a molecular graph. The relation between this new polynomial and omega polynomial is investigated. In particular, a method of computing Sadhana polynomial and then sadhana index for for an infinite family of fullerene has been described.

Keywords: Omega polynomial, Sadhana polynomial, Fullerene.

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

On the Geometric Arithmetic GA5 Index

M. Ghorbani, A. R. Ashrafi and M. Saheli Corresponding Author: [email protected] Department of Mathematics, Faculty of Science, University of Kashan, Kashan, 87317 51167, I. R. Iran Department of Mathematics,Faculty of Science, Shahid Rajaei Teacher Training University, Tehran, 16785 - 136, I R. Iran January 14, 2010

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

A New Type of Geometric Arithmetic Index

M. Ghorbani and H. Maimani Department of Mathematics,Faculty of Science, Shahid Rajaei Teacher Training University, Tehran, 16785 - 136, I R. Iran January 14, 2010

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

COUNTING THE NUMBER OF (3,6)-FULLERENES

A. Hamzeh, S. Hossein-Zadeh and A. R. Ashrafi Department of Mathematics, University of Kashan, Kashan 87317-51167, I. R. Iran

Abstract

A (3,6)-fullerene is a cubic plane graph whose faces have sizes 3 and 6. The Euler’s formula implies that there are exactly four faces of size 3. These graphs have received recent attention from chemists due to their similarity to ordinary fullerenes.

During last decade a Japanese chemist, Shinsaku Fujita applied an extension of Polya’s theorem to compute the number of isomers of some molecules. In this work, we apply these method to compute the number of (3,6)-fullerenes

Keywords: (3,6)-Fullerene, Isomer.

INTRODUCTION

Detecting symmetry of molecules is a well-studied problem with applications in a large number of areas. Randic and then Balasubramanian considered the Euclidean matrix of a chemical graph to find its symmetry. Here the Euclidean matrix of a molecular graph G is a matrix D(G) =

[dij], where for i  j, dij is the Euclidean distance between the nuclei i and j. In this matrix dii can be taken as zero if all the nuclei are equivalent. Otherwise, one may introduce different weights for different nuclei.

Suppose  is a permutation on n atoms of the molecule under consideration. Then the permutation matrix P is defines as P = [xij], where xij = 1 if i = (j) and 0 otherwise. It is easy to see that PP = P, for any two permutations  and  on n objects, and so the set of all n  n permutation matrices is a group isomorphic to the symmetric group Sn on n symbols. It is a well- known fact that a permutation  of the vertices of a graph G belongs to its automorphism group if it t satisfies P AP = A, where A is the adjacency matrix of G. So, for computing the symmetry of a molecule, it is sufficient to solve the matrix equation PtEP = E, where E is the Euclidean matrix of

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010. the molecule under consideration and P varies on the set of all permutation matrices with the same dimension as E.

We now present a method is described how to construct a fullerene C3n from a fullerene Cn having the same or even a bigger symmetry group as Cn. This method is called the Leapfrog principle. If we are starting with a Cn cluster with icosahedral symmetry all the new clusters will be of the same symmetry, since this is the biggest symmetry group in 3-dimensional space. In the first step you have to put an extra vertex into the centre of each face of Cn. Then connect these new vertices with all the vertices surrounding the corresponding face. Then the dual polyhedron is again a fullerene having 3n vertices 12 pentagonal and (3n/2)-10 hexagonal faces. Knowing the 3- dimensional cycle index of S(Cn) acting on the sets of vertices, edges and faces it is very easy to compute the cycle index for the induced action of S(Cn) on the set of vertices of C3n. We just have to identify the vertices of Cn with the n new hexagonal faces of C3n. From Fig. 1, one can see that Le(C20) = C60.

Groups are often used to describe symmetries of objects. This is formalized by the notion of a group action. Let G be a group and X a nonempty set. An action of G on X is denoted by GX and X is called a G-set. It induces a group homomorphism  from G into the symmetric group SX on X, where (g)x = gx for all x  X. The orbit of x will be denoted by Gx and defines as the set of all (g)x, g  G. The set of all G-orbits will be denoted by G\\X : = { Gx | x  X}. Suppose g is a permutation of n symbols with exactly 1 orbits of size 1, 2 orbits of size 2, …, and n orbits of size n. Then the cycle type of g is defined as 112 2 ...n n .

We now introduce the notion of cycle index. Let G be a permutation group. The cycle index of G acting on X is the polynomial Z(G, X) over Q in terms of indeterminates x1, x2, …, xt, t = |X|,

t 1 c(p)i defined by Z(G, X) = x, in which (c1(p), ···, ct(p)) is the cycle type of the |G| pG i1 i permutation p  G. The generalized character cycle index is defined as 1 )p(c P x,x( ,..., )x   t  x)p( i , where χ(g) is the linear character of the irreducible G 1 2 t |G| Gp 1i i representation of G. In this paper we use two special cases: One is the anti-symmetric representation, that is

1,if g is a proper rotation  ()g   1,if g is an improper rotation

and the other when χ is 1 for all g. Since, all elements of a conjugacy class of a permutation group have the same cycle type, so the cycle index and the generalized character cycle index can be rephrased in the following way:

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The Third Conference and Workshop on Mathematical Chemistryr (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

1 t Z(G,x , ,x )  |C| x,c(g)iC 1t|G|CConj(G) i1 i

1 t P (x , ,x )  |C| (g )xc(giC). G 1t|G|CC onj(G) i1 ci

Enumeration of chemical compounds has been accomplished by various methods. The Polya- Redfield theorem has been a standard method for combinatorial enumerations of graphs, polyhedra, chemical compounds, and so forth. Combinatorial enumerations have found a wide-ranging application in chemistry, since chemical structural formulas can be regarded as graphs or three- dimensional objects.

Denote by Cm,n the set of all functions f: {1, 2, …, m} {x1, x2, ..., xn}. The action of p  Sm -1 induced on Cm,n is defined by pˆ (f) = fop , f  Cm,n. Treating the colors x1, x2, …, xn that comprise m the range of f  C as, independent variables the weight of f is W(f) = f (i) . Evidently, W(f) m,n i1 is a monomial of (total) degree m. Suppose G is a permutation group of degree m, Gˆ ={ pˆ :pG}, ˆ pˆ is as defined above. Let p1, p2, …, pt be representatives of the distinct orbits of G . The weight of pi is the common value of W(f), f  pi. The sum of the weights of the orbits is the pattern inventory t W (x ,x ,…,x )= W(p ). G 1 2 n i1 i

Theorem.1 (Pólya's Theorem) If G is a subgroup of Sm, the symmetry group on m symbols, ˆ then the pattern inventory for the orbits of Cm,n modula G is

1 1 pC )( 2 pC )( m pC )( WG(x1,x2,…,xn)= M M ...M , | G |  Gp 1 2 m

k k k th where Mk = x1 +x2 +…+xn is the k power sum of the x’s.

Theorem.2 (Generalization of Pólya's Theorem) Substituting Mi for xi and in the generalized  character cycle index, i = 1, 2, , t, we get the chiral generating function CGF  PMG (,,1  M k ) .

To enumerate all possibilities of the hetero-fullerene structures, we have to consider the rotation group of the fullerene, and its whole automorphism group to enumerate the number of chiral isomers. Fripertinger computed the symmetry of some fullerenes and then applied

SYMMETRICA to calculate the number of C60HkCl60-k molecules and Balasubramanian computed the number of C60H36 isomers. F. Zhang et al. for calculating the possibilities of different positional isomers used the Pólya's counting theorem. He also applied the generalization of the Pólya's theorem to compute the number of chiral isomers.

In this paper the number of (3,6)-fullerene is computed by using the Polya's method.

REFERENCES

138 The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

1. H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl, R. E. Smalley, Nature 318 (1985) 162

2. P. W. Fowler, D. E. Manolopoulos, An Atlas of Fullerenes, Oxford Univ. Press, Oxford, 1995

3. M. Randić, Chem. Phys. Letters 42 (1976) 283

4. M. Randić, J. Chem. Phys. 60 (1974) 3920

5. K. Balasubramanian, J. Chem. Phys. 72 (1980) 665

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Computation of Co-PI index of TUC4C8(R) nanotubes

F. Hasania, O. Khormalib, A. Iranmaneshc aPayame Noor University, PNU Central Branch, Nakhl Street, Lashkar Road, Tehran, Iran Email: [email protected] b Mathematics and Informatics Research Group, ACECR, Tarbiat Modares University, P.O.Box : 14115-343, Tehran, Iran Email: [email protected] cDepartment of Mathematics, Tarbiat Modares University, P.O.Box : 14115-137, Tehran, Iran Email: [email protected]

Abstract

In this paper, at first we introduce a new index with the name Co-PI index and obtain some properties related this new index. Then we compute this new index for TUC4C8(R) nanotubes.

Keywords: Vertex-PI index, Co-PI index, TUC C84 (R ) Nanotube.

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

The Study of Splices and Links of Graphs by Topological Index

S. Hossein-Zadeh, A. Hamzeh and A. R. Ashrafi Department of Mathematics, University of Kashan, Kashan 87317-51167, I. R. Iran

Abstract

Explicit formulas are given for the first and second Zagreb indices, degree-distance and Wiener-type invariants of splice and link of graphs. As a consequence, the first and second Zagreb coindices of these classes of composite graphs are also computed.

Keywords: Wiener-type invariant, degree-distance, Zagreb index, Zagreb coindex, splice, link.

INTRODUCTION

Throughout this paper graph means simple connected graphs. The distance dG(u,v) between the vertices u and v of a graph G is defined as the length of a shortest path connecting u and v. Let d(G,k) be the number of pairs of vertices of G that are at distance k,  a real  number, and W (G) = k  1d(G,k)k . W (G) is called the Wiener-type invariant of G associated to real number . Note that d(G,0) and d(G,1) represent the number of vertices and edges, respectively. The case of = 1 is called the classical Wiener index. The quantities WW = 1 1 1 1 [W1 + W2] and TSZ = W3+W2 + W1 are the so-called hyper-Wiener index and 2 6 2 3 Tratch-Stankevich-Zefirov index. Suppose G and H are graphs with disjoint vertex sets. Following Došlić, for given vertices y  V(G) and z V(H) a splice of G and H by vertices y and z, (G .H)(y,z), is defined by identifying the vertices y and z in the union of G and H. Similarly, a link of G and H by vertices y and z is defined as the graph (G ~ H)(y,z) obtained by joining y and z by an edge in the union of these graphs. The Zagreb indices have been

introduced more than thirty years ago b Gutman and Trinajestic, . They are defined as M1(G) = 141

The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

2 u  V(G)degG(u) and M2(G) = uv  E(G)degG(u)degG(v). As the sums involved run over the edges of the complement of G, such quantities were called Zagreb coindices. More formally,

the first Zagreb coindex of a graph G is defined as M 1(G)= uv E(G) [degG(u) + degG(v)], and

the second Zagreb coindex of a graph G is given by M 2(G)= uv E(G) degG(u)degG(v). The reader should note that Zagreb coindices of G are not Zagreb indices of G ; the defining sums run over E( G ), but the degrees are with respect to G. In some recent papers Dobrynin and Kochetova and Gutman introduced a new graph invariant defined as:

' ' 1 D(G) = x  V(G)D (x) = x V(G)D(x)degG(x) = x,y V(G)d(x,y)[degG(x) + degG(y)] 2

' where D (x) = D(x)degG(x), degG(x) is the degree of x and D(x) = yV(G)dG(x,y).

In this paper some graph operations under degree distance of graphs are computed. Some examples and applications are also presented.

REFERENCES

1. Došlić, T. Splices, links, and their degree-weighted Wiener polynomials, Graph Theory Notes N. Y. 48, 47-55, 2005.

2. Dobrynin, A. A. and Kochetova, A. A. Degree distance of a graph: A degree analogue of the Wiener index, J. Chem. Inf. Comput. Sci. 34, 1082-1086, 1994.

3. Gutman, I. A property of the Wiener number and its modifications, Indian J. Chem. 36A, 128-132, 1997.

4. Graovac, A. and Pisanski, T. On the Wiener index of a graph, J. Math. Chem. 8, 53-62, 1991.

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Computing Schultz and modified Schultz polynomials of C100Fullerene by GAP program

Ali. Iranmanesh, Samaneh. Mirzaie Department of Mathematics, Tarbiat Modares University P.O.Box; 14115-137, Tehran, Iran

Abstract

For a simple connected graph, the Schultz polynomial is defined as:

duv(,) HGx1(,) (uv ) x {,}uv V ( G ) And the modified Schultz polynomial is defined as:

duv(,) HGx2(,)  (uv ) x {,}uv V ( G )

In this paper, we compute the Schultz polynomial and the modified Schultz polynomial of

C100 fullerene by a GAP program.

Keywords: Schultz polynomial, Modified Schultz polynomial, C100 Fullerene, GAP programming.

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Investigation and Study of Chemical Adsorption of Biatomic Gases on the (8, 0) Nanotube Surface Using (DFT) Calculations

Author: Morteza. Keshavarza,* ,Ali kazemi Babaheydarib, Khadijeh Tavakolia Address: a) Department of Chemistry, Islamic Azad University, Shahreza Branch, P. O. Box 311-86145, Shahreza, Isfahan, Iran b) Department of Chemistry, Sharekord Branch, Islamic Azad Univercity , Sharekord, Iran

* Corresponding author Tel: +98-9133093346 Fax: +98-3125829097 Email: [email protected]

ABSTRACT

In this research the chemical adsorption of biatomic gases such as CO2 and O2 on the monolayer nanotube surfaces of carbon ((SWCNT) and (8, 0)) was investigated using quantum chemistry calculations and density function theory methods (DFT).

In this regard the gas molecules were approached to the carbon nanotube surface in three state of Top, Bridge and Center and in two horizontal and vertical shape and were calculated by B3lyp and basis sets of 6-31G and 6-311G for nanotube and gas molecule and gas molecule and nanotube in different distances in order to obtain the most stable adsorption state and in order to determine whether the nanotube of carbon (SWCNT) can be used as a proper catalytic surface for gas adsorption so in this case these nanotubes can be used in different reactions.

Keywords: Carbon nanotube (SWCNT), Chemical adsorption

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The Third Conference and Workshop on Mathematical Chemistryr (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Activity Calculations Based on A New Theory For Binary Mixtures Of Polymer/Solvent And Comparison With Experimental Data

KESHMIRIZADEH.ELHAM Applied chemistry department,Islamic Azad University –Karaj Branch P.O.BOX:31485-313Karaj-Iran Email:[email protected] Phone no:09125613268

ABSTRACT

Phase equilibrium plays an important role in the processing and application of polymer. In our previous works[1,2], a new thory based on hard – sphere limit has been developed for calculating activity of solvents(organic) in polymer/ solvent mixtures.

In this work, other seven systems, both of organic and aqueous solvent/polymer mixtures have been evaluated and compared with experimental data at various temperature and varying range of molecular weights (Mw) of polymers. The results show this new theory properly fits the experimental data and it can be concluded that this theory has only one adjustable parameter  22 0 0 named (polymer diameter) .2 97 A   60 A , to proceed with the activity calculations, an objective  22 error function root mean square difference (r.m.s.d) was definned, the was calculated by acal numerical method( non linear regression) and optimization(minimization the error function).The activities of polymer are in agreement with the experimental data and thus the consistency in the results is confirmed calculations.

CALCULATIONS

From statistical mechanics it is demonstrated that the following expression exists for the partial derivative of Helmholtz function (A) with respect to σ11, the molecular size diameter of component one, [1]:

(1)

145 The Third Conference and Workshop on Mathematical Chemistryr (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

where R is the gas constant, T is temperature, x is mole fraction, n is number density and

σ12=(σ11+σ22/2).

In Eq. (1) g11 and g12 are the pair correlation function. It is generally accepted that for liquids the following assumption is valid [1].

E E (G (at constant T and P)=A (ًat constant T and V) (2

E This result may be substituted into Eq. (1) and equation G hs/(RT) .

In a binary mixtures the excess Gibbs free energy is expressed by the following well-known equation [1]:

(3)

where γ represents the activity coefficient. The activity coefficient of solvent γ1 is [1]:

(4)

Therefore on combining Eqs. ((2) and (3)) and after rearrangement we get

(5)

And therefore the activity of solvent can be obtained from Eq. (5) and the definition; a1=γ1x1.The activity model represented by Eq. (5) was applied to seven binary polymer mixtures. The experimental data for activities of solvents in these solutions have been obtained in [3,4]. To calculate activity coefficient of solvent by Eq. (5) the value of σ11 was calculated by 1/3 σ11=2(3M1/4πNρ1) , the density of solvent ρ1 was obtained from reference [3,4], the molar density

146 The Third Conference and Workshop on Mathematical Chemistryr (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

ρ* at various temperatures was calculated from the molar densities of pure polymer and pure solvent using the following mixing rule.

(6)

Fortunately the application of this simple mixing rule was very successful since it resulted in * very small root mean square differences (r.m.s.d) in the activity calculations. The molar densities ρi * were calculated by ρi =ρi/Mi (i=1, 2) and the value of ρ2 the density of polymer was obtained from reference [3,4]. To proceed with the activity calculations an objective function (r.m.s.d) was defined in the following form:

(7)

where Np is the number of experimental data point, aexp and acal are respectively the experimental and calculated activities. The acal, was calculated by numerical method (nonlinear regression) and optimization (minimization). The polymer diameter value σ22 was calculated as a adjustable parameter. The optimum value of σ22 for each polymer solution is reported in Table 1.

RESULTS AND DISCUSSIONS

Comparison of figures 1 to 3 shows that the trend of variation of activity is the same as in experimental data, which lends support to the ideas used in presenting the new theory. The results show this new theory properly fits the experimental data and it can be concluded that this theory has only one adjustable parameter named (polymer diameter) and the error(r.m.s.d) is very low.  22

Table 1

147 The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Comparison between the results obtained from new theory and experimental data at various temperature for binary polymer/ solvent mixtures.

r.m. Temp No of Polymer(Mw)/solv s.d erature  2 experimental Ref. ent (Err ( 0C ) data point or) 2 0.0 PEG 3000+Water 55 6 [3] 9.7 04 3 0.0 PEG 5000+Water 60 5 [3] 0 2 PA-66, 3 0.0 23 9 [4] 20700+Water 2.8 01 4 0.0 P-BS 10000+Tolu 20 6 [4] 5.6 09 7 0.0 PEG 300+Water 30 8 [3] 0.6 07 Dextran 8 0.0 20 5 [4] 46300+Water 7.4 0003 6 0.0 PPG 400+Water 30 7 [3] 00 12 Mw: molecular weight, PPG:poly propylene Glycol , PEG: poly ethylene glycol,

PA-66:poly amide(nylon66), Poly(p-bromostyrene)

0.9995 T=200C 0.999 Experimental data- 0.9985 DEXTRAN46300-WATER 1 a 0.998 Calculated by this w ork 0.9975 0.997 0.9965 0.0579 0.0923 0.1616 0.2029 0.2441

FIG1- Activity of water(a1) versus weight fraction of polymer(w2)for dextran(46300)/water mixture at 20 0C .experimental data(▪), calculated by new theory().

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

1 T=600C 0.8 Experimental data-

1 0.6 PEG5000-WATER a 0.4 Calculat ed by this work 0.2 0 0 0.735 0.835 0.902 0.95

w 2

FIG2- Activity of water(a1) versus weight fraction of polymer(w2)for PEG(5000)/water mixture at 60 0C .experimental data(▪), calculated by new theory().

1 T=550C 0.8 Experimental data- PEG3000-WATER 1 0.6 a Calculated by this work 0.4 0.2 0 0 0.499 0.702 0.844 0.95 0.97

w 2

FIG3- Activity of water(a1) versus weight fraction of polymer(w2)for PEG(3000)/water mixture at 55 0C .experimental data(▪), calculated by new theory(). REFERENCES:

[1]E.Keshmirizadeh, H.Modarress, A.Eliassi, G.A.Mansoori, European polymer journal39 (2003)1141-1150.

[2] H.Modarress, A.Eliassi, E.Keshmirizadeh,Fluid Phase Equilibria235(2005)26-29.

[3]G.N.Malcolm, J.S.Rowlinson, Trans Faraday Society, Journal of Polymer Science(1957),921-931

[4]C.Wohlfarth, vapour-liquid equilibrium data of binary polymer solutions, (1994), Elsevier Science.

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

THE GROWTH OF DIAMOND NANO STRUCTURE ON SILICON SUBSTRATE USING DIFFERENT ETCHING GASSES BY HFCVD

Z.Khalaj , M.Ghoranneviss*, S.Nasiri lahegi 1-Plasma Physics Research Center, science and Research Branches, Islamic Azad University,Tehran, Iran *Correspondending Author. Tel.:=+98 21 44869624 Fax: +98 21 44869626 E-mail address: [email protected]

Abstract

In this paper, we invistigate the effects of etching gasses on the synthesization of Nano Crystalline Diamond (NCD) on silicon substrate. NCD's are synthesized at the substrate temperature of 550˚ C and the reaction pressure of 30 Torr by Hot Filament Chemical Vapor

Deposition (HFCVD), in a mixture of CH4 and H2 as a source and diluting gas, respectively. In this expriments, we use N2, H2, NH3, as the etching gasses. However, the results show that the optimum conditions can be obtained for the case of H2. Finally, we examine the crystal morphology and crystallinity of the samples using Scanning Electron Microscopy and X-Ray Diffraction systems, respectively.

Keywords: Nano Crystalline Diamond; Etching Gasses; HFCVD.

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

On Zagreb Polynomials and indices

O. Khormalia, A. Iranmaneshb,* A. Ahmadia and Ivan Gutmanc aMathematics and Informatics Research Group, ACECR, Tarbiat Modares University P.O.Box: 14115-343, Tehran, Iran bDepartment of Mathematics, Tarbiat Modares University, P.O.BOX: 14115-137, Tehran, Iran cFaculty of Science, University of Kragujevac, Kragujevac, Serbia

Abstract

For a (molecular) graph, the first Zagreb index M 1 is equal to the sum of squares of the vertex degrees, and the second Zagreb index M 2 is equal to the sum of products of degrees of pairs of adjacent vertices.

Also, let G be a graph, euvEG  (), du() be degree of vertex u . Then the Zagreb du() dv () polynomials of the graph G are defined as ZG1(;) G x  x and uv e E() G dudv()() ZG2 (;) G x  x respectively. In this paper, we generalize these indices and uv e E() G polynomials and conclude some results for them. Finally, we compute these new indices and polynomials for some graphs and particularly for some nanotubes.

Keywords: Zagreb index, Molecular graph, Zagreb Polynomials.

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Optimization of the NBR blends for hardness property by Mixture Design

Vahid Kiarostami,Mehran Davallo, Maryam vahidpur, Aida kianmehr Chemistry Department,Islamic Azad University , North Tehran Branch-Tehran-Iran

Abstract

The extreme vertices mixture design was applied for optimization of the NBR blends for hardness response. The15 treatment combinations out of 29 candidate points investigated (as proposed by Minitab 14 software) contained varying proportions of carbon black, sulfur, and MBTS (Dibenzothiazyl di sulfide) and antioxidant as four variable factors and other factors were kept constant. The significant model coefficients are interpreted in term of interacting linear and quadratic effects of the NBR variable blend constituent.

Based on the three dimensional surface, contour and response trace plot, MBTS exhibits an adverse negative effect, while sulfur exhibits a strong positive effect on the hardness. carbon black and antioxidant have, weak and moderate negative effect on the response , respectively. The model was used to predict the treatment combination of the NBR blend in the optimum hardness response. (Hardness response was54.07, A: 3.52, B: 1.5467, C: 91.9733, D: 2.96). There is no significant difference between the result of the model and that of obtained in the laboratory.

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Using algorithm at potentiometeric study of binary and mixed complex formation of some amino acids with Zn (II), Co(II) and Ni (II) in Aqueous and Non- aqueous Solution

S. Lotfei1, M. Mazloum-Ardakani2, G. Ghasemi3 1Department of Chemistry, Payame Noor University(PNU), Gilane Gharb ceter 2Department of Chemistry, Faculty of Science, Yazd University, Yazd 3Department of Chemistry, Faculty of Science, KNT University, Tehran Abstract The stability constants and complexation reaction between L-Glutamine, L-Arginine and glycin with Zn (II) , Co (II) and Ni (II) were studied potentiometrically in aqueous and ´ nonaqueous solution at 25° C and µ=0.1 M KNO3. The overall stability constants log β s of all species are obtained by computer refinement of pH- Volume data using BEST computer program.

Several model were tested and the lowest δFIT, the best one is accepted. The main species in binary complexes MHL, ML, ML2, MLOH, ML(OH)2 and for ternary complexes are ML1L2, ML1L2OH, ML1L2(OH)2. The order found for the resulting stability constants vary as Co (II) < Ni (II) < Zn(II). The concentration distribution diagram was obtained with the program SPEPLOT.

Keywords: L-Glutamine, L-Arginine, Glycin, BEST, Potentiometrically, Nonaqueous solvents

Introduction Potentiometry is one of the most convenient and successful techniques employd for metal complex equilibrium measurements. While some workers measure metal ion concentration with specific ion electrodes, or with metal electrodes, it is usually sufficient to use the highly accurate glass electrode for measuring the hydrogen ion concentration in a procedure termed potentiometric titration, whereby, for example, standard base is added in increments to a well characterized acid solution of the ligand in the absence of and in the presence of known total metal ion concentration. In most typical case, the p[H] is varied between 2-12 during which time some 50-100 equilibrium readings are obtained constituting the potentiometric equilibrium curve [1]. pH-metry has been applied most extensively for determination of the stability constants of the metal complexes of aliphatic amino acid.

In this paper, we applied pH-metry method for determine the protonation constants of L- Glutamine, L-Arginine, glycin and the stability constants of its complexes in binary and ternary systems with Zn (II) , Co (II) and Ni (II) ions in aqueous and nonaqueous solution at 25° C and

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µ=0.1 M KNO3 . The analysis is readily performed with the computer program BEST and concentration distribution diagram were obtained with the program SPEPLOT.

The basic algorithm in BEST[2-4] can be ststed in terms of equation (1):

(1) Which is a statement of the mass balance (at a given titration point) of the i-th component in

terms of j-th species summed over all species present or NS, (NS=number of species, Ti=total concentration of i-th component). Each species concentration consists of a product of the overall

stability constant and individual component concentration [Ck] raised to the power of the stoichiometric coefficient eij. The value of [Ck] is special when it represents the calculated concentration of H+, which then is compared with the measured hydrogen ion concentration. The calculation process is repeated at all measured equilibrium points. In any calculation based on a pH profile there will be some known, previously calculated, β values as well as the unknown values to be determined. The first pass of the calculation procedure uses both the known and the estimated values of the unknown constants.

Thus the use of the algorithm for computing equilibrium constant in BEST involves the following sequence:

1. Start with a set of known and estimated overall stability constants (β´s) and compute

[H+] at all equilibrium point.

2. Compute the weighted sum of the squares of the deviation in p[H] as in equation (2):

2 U  w(p[H]obs  p[H]calcd) (2)

Where W  1 is a weighting factor which serves to lesson the influence p[( H ]i1  p[H ]i1 ) of the less accurate p[H] profile.

3. Adjust the unknown stability constants and repeat the calculation until no further minimization of a U can be obtained.

Experimental section

Potentiometry The apparatus consisted of a custom-designed, thermostated 80-ml capacity all-glass vessel. In general, an experimental run involves collecting equilibrium data points throughout the entire pH range, between 2.0 and 11.0 as a function of millimoles standard KOH, added using the

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010. piston buret through a fine capillary tip immersed in the solution. The hydrogen concentration was measured using a Metrohm model CH-9101 titroprocessor connected to a confident 286 PC for data transfer and computation. The protonation and formation constants of all species were obtained through the least-squares refinement of its p[H+] profiles. Throughout this investigation the function minimized was the weighted average of the sums of squares of deviations between calculated and + observed p[H ] value (δFIT).

Reagents The nitrate salts of nickle, cobalt and zinc (all from Merck) were used as supplied and the stock solutions were standardaized by EDTA in the presence of suitable indicator. HCl, KOH,

KNO3, L-glutamine, L-arginine and glycin(all from Merck) were used without any further purification. Carbonate free KOH solution was standardized with potassium hydrogen phthalate (KHP). The HCl solution was standardized with KOH. All solutions were prepared in triply distilled deionized water.

Results and Discussion In aqueous, solution aliphatic amino acids exist as zwitter ions (HL), the amino group being protonate (-NH3+), while the carboxyl group is deprotonated (-COO-). In the case of the α- amino acids, deprotonation of the ammonium group occurs in slightly basic solution (pH=9-10) to give the species (L-). The carboxylate group undergoes protonation in acidic media (pH=2-3), and therefore + the two dissociation processes of the fully protonated cation form (H2L ) are completely separated. The overall protonation constants of ligands studied were calculated from computer refinement of the pH-volume data. The obtained values are shown in Table 1.

Table 1.Protonation constants of vari ous aminoacids in aqueous solution

Amino acids Log K1(-NH2)Log K2(-COOH)

Glycine 9.78 2.35

L-glutamine 9.21 2.25

L-arginine 9.14 2.12

It is seen from distribution curves for M2+- amino acid system mono-protonated form, MLH forms at wide range of pH (2-8). The monoprotonated form, MLH is converted to the deprotonated form ML at around pH 8.5, which in turn is converted at about pH 8.2 to ML2 form. It is also can be seen from table 2 and distribution diagrams that all binary system show that same behavior as

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010. explained. It is seem that the glycin complexes have higher stabilities. This is due to absence of alkyl group in glycine. The order of stability for chelates of the divalent transition metal ions is Co2+< Ni2+ < Zn2+.

The sample distribution curve of the glycine and glycine-Ni2+ for example are shown in Fig 1.

Fig. 1. Distribution of major species as a function of -log[H+] for glycine and glycine to Ni+2.

References [1] Z.B. Maksimovic, A. Miksa- Spiric, S.V.Ribnikar, J.Inorg. Nucl.Chem, 35, 1239(1973).

[2] M. Maeder, Y.M. Neuhold, G. Puxty, 70, 193 (2004).

[3] G.Ghasemi, A.Shokrollahi, Iran. J. Chem. &Chem. Eng. 20, 1, 22 (2001).

[4] A.Braibanti, C. Bruschi, E.Fisicaro and M.Pasquali, Talanta, 33,471(1986).

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A Kinetic-Potentiometric Method with Partial least squares (PLS) and principle component regression

(PCR) used for Determination of Paracetamol

(Acetaminophen) and P-Aminophenol

MARYAM MALEKZADEH*,a , MOHAMMAD ALI KARIMI b aDepartment of Chemistry, Payame Noor Tabas University, Tabas, Iran bDepartment of Chemistry, Payame Noor Sirjan University, Sirjan, Iran * e-mail: [email protected]

ABSTRACT

Partial least squares (PLS) and principle component regression (PCR) multivariate calibration methods were applied to the simultaneous determination of Acetaminophen and P- Aminophenol using kinetic data from novel potentiometry methods(1-3). These methods were based on the rate of chloride ion production in the reaction of Acetaminophen and P-Aminophenol with N-chlorosuccinimide (NCS) which was monitored by a chloride ion-selective electrode. These methods are based on the differences observed in the production rate of chloride ions. The results show that simultaneous determination of Acetaminophen and P-Aminophenol can be performed in their concentration ranges of 0.5 - 65.0 and 0.1 - 75.0 μg mL-1, respectively. The total relative standard errors for applying PLS and PCR methods to 8 synthetic samples in the concentration range of 8.0-55.0 µgmL-1 for Acetaminophen and 0.5-40.0 µg mL-1 for P-Aminophenol were 4.069,4.743 and 3.931,4.797 respectively.

Key word: simultaneous determination, kinetic-potentiometric , Acetaminophen; P- Aminophenol, PLS, PCR

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Introduction

In recent years the usage of chemometrics methods including principal component regression (PCR), PLS, artificial neural network (ANN) and HPSAM in electroanalytical chemistry has received considerable attention for the extraction of more information from experimental data as in other areas of analytical chemistry. In the field of potentiometry, several methods have been reported based on flow injection system and titration using PLS and ANN as modeling methods(4-7). The methods were based on the differences observed in the production rate of chloride ions in the reaction of Paracetamol and P-Aminophenol with N-chlorosuccinimide (NCS). The reaction rate of production of chloride ion was monitored by a chloride ion-selective electrode.

Multivariate calibration consists of establishment of a relationship between matrices of chemical data. These methods are based on a first calibration step in which a mathematical model is built using a chemical data set (e.g., potential values) and a concentration matrix data set. The calibration is followed by a prediction set in which this model is used to estimate unknown concentrations of a mixture from kinetic profile. Multivariate calibration methods are being successfully applied to the multi-component kinetic determination to overcome some of the drawbacks of classical methods. Soft algorithms such as PCR, PLS and ANN, which avoid colinearity problems, have been used for simultaneous determination of the analytes having the same chemical properties that can not be resolved with common methods.

RESULTS AND DISCUSSION

I. Study of electrode characteristics

The characteristics of the chloride-selective electrode in the acetate buffer were studied. The fast response of chloride ISE and its nernstian behavior with respect to chloride ions in medium acidic solutions indicated that this electrode might be employed effectively in kinetic studies of reaction involving changes in the chloride ion concentration.

II. Effect of NCS concentration

The effect of NCS concentration on the reaction rate of Paracetamol and P- Aminophenol and mixture of them was investigated. The reaction rates of both species increase

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010. with the concentration of oxidant (NCS) in the range of 5 × 10-5 - 5 × 10-2 M. Further increase in oxidizing agent concentration was avoided due to the limited solubility of NCS in water. Therefore, a concentration of 5 × 10-2 M NCS was selected as the optimum concentration for further studies.

. Effect of pH

The effect of pH on the reaction rates of Paracetamol and P-Aminophenol with NCS over the pH range of 2.0 to 9.0 was examined. Since, maximum differences in kinetic behavior Paracetamol and P-Aminophenol were observed at pH 4.5. IV. Potential-times behavior

Figure1shows the potential-time behavior of reactions of NCS with Paracetamol and P-Aminophenol and mixtures of them at the optimized conditions.

Under the optimum conditions, the amount of (10∆E/S-1) of the produced at the reaction of NCS with Paracetamol and P-Aminophenol is linear with their concentrations.Characteristics of calibration graphs for determination of Paracetamol and P-Aminophenol are given in Table 1. .

Table 1. Characteristics of calibration graphs for the determination of Paracetamol and P- Aminophenol.

Analyte Slope Intercept Correlation coefficient Range Limits of detection (DL)a ( μg ml -1) (n = 12) (μg ml-1) (μg ml-1) P-aminophenol 0.1-75.0 0.1603 -0.2089 0.9990 0.021 Paracetamol 0.5-65.0 0.0042 0.4984 0.9993 0.20 a Defined as DL = 3Sb/ m, where DL, Sb and m are limit of detection, standard de

Figure 1. Potential-time curves for the reaction of NCS with 5 μg mL -1 of Paracetamol(a), 10 μg mL -1 of P-Aminophenol (b) and mixture of them (c).

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b E(mv)

c 10mv

020406080100120 Time/s

REFERENCES

1. B.R. Kowalski, Anal. Chem., 50 (1978) 1309A. 2. S. Brown, “Chemometrics”, in Encyclopedia of analytical chemistry, R. A. Myers, Ed., 9671- 678, Wiley, Chichester, 2000. 3. E.R. Malinowski, “Factor analysis in chemistry” 3rd ed., Wiley, New York, 2002.

4. Haaland, D. M. and Thomas, E. V. 1988. Partial least-squares methods for spectral analyses. 1. Relation to other quantitative calibration methods and the extraction of qualitative information. Anal. Chem. 60: 1193-1202. 5. Forina, M., Casolino, M. C. and de la Pezuela Martinez, C. 1998. Multivariate calibration: applications to pharmaceutical analysis. J. Pharm. Biomed. Anal. 18: 6- 20. Goicoechea, H. C. and Olivieri, A. C. 2002. Chemometric assisted

spectrophotometric determination of four-component nasal solutions with a reduced number of calibration samples Anal. Chim. Acta 453: 289-300. 7. Karimi, M. A., Mazloum Ardakani, M., Behjatmanesh Ardakani, R., Moradlou, O.and Banifatemeh,F.2007.Kinetic-spectrophotometric determination of hydrazine and its derivatives by partial least squares and principle component regression methods. J. Chin. Chem. Soc. 54: 15-21.

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Application of factorial design in digestion of ore samples by microwave and determination of some metals by AAS

M. Mazloum-Ardakania, P. Pourhakakb

aDepartment of Chemistry, Faculty of Science, Yazd University, Yazd, IRAN bDepartment of Chemistry, Mashhad Payamnoor University, Mashhad, IRAN

Abstract In statistics, a full factorial experiment is an experiment whose design consists of two or more factors, each with discrete possible values or "levels", and whose experimental units take on all possible combinations of these levels across all such factors. A full factorial design may also be called a fully-crossed design. Such an experiment allows studying the effect of each factor on the response variable, as well as the effects of interactions between factors on the response variable. An important property of a fractional design is its resolution or ability to separate main effects and low- order interactions from one another. Formally, the resolution of the design is the minimum word length in the defining relation excluding [1].

For the vast majority of factorial experiments, each factor has only two levels. For example, with two factors each taking two levels, a factorial experiment would have four treatment combinations in total, and is usually called a 2×2 factorial design.

If the number of combinations in a full factorial design is too high to be logistically feasible, a fractional factorial design may be done, in which some of the possible combinations (usually at least half) are omitted. A method is described for determining trace amounts of some metals in ores by atomic absorption spectrometry. The ores was dissolved by two methods, one a microwave digestion procedure and the other a hot-plate dissolution method for two methods digestion a combination of

HCl, HNO3 and CH3COOH acids were used. Comparable results for amounts of Fe, Ni, Cd, Cu, Pb, Zn are achieved by the two methods of dissolution. The microwave digestion improves the absorbance signal and the precision of the measurement. Time of digestion by microwave method decreased (t<5min). The amounts of solvents by it decreased. The statistical models obtained make it possible to evaluate the weight of any experimental value and obtain an idea for possible analytical application of the systems studied.

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Reference: Box, G.E.; Hunter, J.S., Hunter,W.G. (2005). Statistics for Experimenters: Design, Innovation, and Discovery, 2nd Edition. Wiley.

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Automorphism Group Symmetry of Nanostars

M. Mirzargar Institute of Nanoscience and Nanotechnology, University of Kashan, Kashan 87317-51167, Iran

Abstract

A dendrimer is an artificially manufactured or synthesized molecule built up from branched units called monomers. In this paper, the mathematical tools of group theory have been used extensively for the analysis of the symmetry properties of these macromolecules. We prove that it is possible to write the symmetry of a dendrimer, as wreath product of some finite groups. To prove, we consider two infinite classes of dendrimers and compute their topological symmetry groups.

Keywords: Symmetry group, dendrimer, wreath product.

1. Introduction

We first recall some algebraic definitions that will be used in the paper. The symmetry of a physical object can be formalized by the notion of a group action: every element of the group "acts" like a bijective map on some set. To clarify this notion, we assume that G is a group and X is a set. G is said to act on X when there is a map  : G  X X such that for all elements x  X, (i) (e,x) = x where e is the identity element of G, and, (ii) (g, (h,x)) = (gh,x) for all g,h  G. In this case, G is called a transformation group; X is called a G-set, and  is called the group action. For simplicity we define gx = (g,x). In a group action, a group permutes the elements of X. The identity does nothing, while a composition of actions corresponds to the action of the composition. For a given X, the set {gx | g  G}, where the group action moves x, is called the group orbit of x. The subgroup which fixes is the isotropy group of x.

Let G be a group and N be a subgroup of G. N is called a normal subgroup of G, if for any gG and xN, g-1xgN. Moreover, if H is another subgroup of G such that HN = {e} and G = HN = {xy | xH, yN}, then we say that G is a semidirect product of H by N denoted by HN.

Suppose X is a set. The set of all permutations on X, denoted by SX, is a group which is called the symmetric group on X. In the case that, X = {1, 2, …, n}, we denote SX by Sn or Sym(n).

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Let H be a permutation group on X, a subgroup of SX, and let G be a group. The set of all mappings X  G is denoted by GX, i.e. GX = {f | f: X  G}. It is clear that |GX| = |G||X|. We X X X X - put G∿H = G × H = {(f; π) | f  G , π  H}. For f  G and π  H, we define fπ  G by fπ = foπ 1, where “o” denotes the composition of functions. It is easy to check that the composition law (f ;

π) (f ; π) = (ffπ ; π π), makes G∿H into a group. This group is called the wreath product of G by H [1].

In some leading papers, Balasubramanian [2-9] introduced the wreath product formalism for computing symmetry of molecules. Then the present authors continued the mentioned works [10- 13] to present a computational approach which is valuable in practical problems. Our calculation given the paper was done by the computer algebra system GAP [14], which is freely accessible from internet.

34 33 32 41 40 42 39 35 31 43 38 16 15 18 36 19 37 44 7 36 30 45 20 17 17 6 14 35 37 8 7 29 46 16 34 2 21 38 2 15 2 47 9 13 6 3 22 3 1 39 18 48 14 3 1 5 49 12 8 3 23 10 4 5 3 40 50 19 24 13 30 25 4 11 51 11 12 9 41 52 28 29 60 21 53 20 10 24 25 27 59 45 26 42 23 54 22 44 58 43 57 55 56

Figure 1. The Forth Generation Figure 2. The Forth Generation of

of Dendrimer Molecule D1[4]. Dendrimer Molecule D2[4].

2. Main Results and Discussion

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At first, we consider the dendrimer molecule D1[n], Figure 1. In order to characterize the symmetry of this molecule we note that each dynamic symmetry operation of D1[1], considering the rotations of XY2 groups in different generations of the whole molecule D1[n], is composed of n sequential physical operations. We first have a physical symmetry of the framework (as we have to map the XY2 groups on XY2 groups which are on vertices of the framework). Such operations form the group G of order 6, which as is well known to be isomorphic to S3 or Sym(3). After accomplishing the first framework symmetry operation we have to map each of the three XY2 group on itself in the first generation and so on. This is a group isomorphic to H = ((…(Z2 ∿ Z2) ∿ Z2) ∿

… )∿ Z2) ∿ Z2 with n – 1 components. Therefore, the whole symmetry group is isomorphic to H ∿ G. This is a group of order . 62 We now consider the dendrimer molecule D2[n], Figure 2. The topological symmetry group of the core of this dendrimer is isomorphic to S4. This group can be generated by a1 = (1,2), a2 = (1,3) and a3 = (1,4). In order to characterize the symmetry of this molecule we note that each dynamic symmetry operation of D2[n], considering the rotations of XY2 groups in different generations of the whole molecule D2[n], is composed of n sequential physical operations. We first have a physical symmetry of the framework (as we have to map the XY2 groups on XY2 groups which are on vertices of the framework). Such operations form the group G of order 24, which as is well known to be isomorphic to S4 or Sym(4). After accomplishing the first framework symmetry operation we have to map each of the four XY2 group on itself in the first generation and so on. This is a group isomorphic to H = ((…(Z2 ∿ Z2) ∿ Z2) ∿ … )∿ Z2) ∿ Z2 with n – 1 components. Therefore, the whole symmetry group is isomorphic to H ∿ G. This is a group of order . 24 2

References

1. D. J. S. Robinson, A course in the theory of groups, Second edition. Springer-Verlag, New York, 1996. 2. 6. K. Balasubramanian, J. Chem. Phys. 72 (1980) 665–677. 3. 7. K. Balasubramanian, J. Chem. Phys. 75 (1981) 4572–4585. 4. 8. K. Balasubramanian, Studies Phys. Theor. Chem. 23 (1983) 149–168. 5. 9. X. Y. Liu and K. Balasubramanian, J. Comput. Chem. 11 (1990) 589–602. 6. 10. K. Balasubramanian, Theoretica Chimica Acta 78 (1990) 31–43. 7. 11. K. Balasubramanian, J. Phys. Chem. 108 (2004) 5527–5536. 8. 12. K. Balasubramanian, Chem. Phys. Letters 391 (2004) 64–68. 9. 13. K. Balasubramanian, Chem. Phys. Letters 398 (2004) 15–21. 10. 14. A.R. Ashrafi and M. Hamadanian, Croat. Chem. Acta 76 (2003) 299–303. 11. 15. M. Hamadanian and A.R. Ashrafi, Croat. Chem. Acta 76 (2003) 305–312.

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12. 16. A. R. Ashrafi, MATCH Commun. Math. Comput. Chem. 53 (2005) 161–174. 13. 17. A. R. Ashrafi, Chem. Phys. Letters 403 (2005), 75–80. 14. The GAP Team, GAP, Groups, Algorithms and Programming, Lehrstuhl De für Mathematik, RWTH, Aachen, 1995.

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The Relationship between Szeged and Wiener Indices of Graphs

M. J. Najafi-Arani, H. Khodashenas and A. R. Ashrafi Department of Mathematics, Faculty of Science, University of Kashan, Kashan 87317-51167, I. R. Iran

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Theoretical Study of Free Energy of Electrontransfer Properties of Metal Nitride Cluster Fullerene Y3N@C80 Based Dyads and cis-Geometry, Thiocrown Ethers Supramolecular Complexes

Avat (Arman) Taherpour1 and Amir Mohammad Hashemi Chemistry Department, Faculty of Science, Islamic Azad University,P. O. Box 38135-567, Arak, Iran E-mail: avatarman.taherpour@ gmail.com

Abstract:

The first pyrrolidine and cyclopropanederivatives of the trimetallic nitride templated (TNT) endohedral metallofullerenes Y3N@C80 connected to an electrondonor unit (i.e., tetrathiafulvalene, phthalocyanine or ferrocene) were successfully prepared by 1,3-dipolar cycloaddition reactions of azomethine ylides and Bingel–Hirsch-type reactions. The unsaturated, cis-geometry, thiocrown ethers, (1-9), (described as [X-UT-Y], where X and Y indicate the numbers of carbon and sulfur atoms, respectively), are a group of crown ethers that display interesting physiochemical properties in light of their conformational restriction compared to a corresponding saturated system, as well as the sizes of their cavities. Topological indices have been successfully used to construct mathematical methods that relate structural data to various chemical and physical properties. To establish a good relationship between the structures of 1-9 with 10, a new index is introduced, cs. ox In this study, the relationships between this index and oxidation potential ( E1) of 1-9, as well as the first and second free energies of electron transfer (Get(n), for n=1,2) between 1-9 and C18H14O6-

Y3N@C80 (10) as [X-UT-Y]@[C18H14O6-Y3N@C80] supramolecular complexes are presented and investigated.

Keywords: Fullerenes; [X-UT-Y]@[C18H14O6-Y3N@C80]; Oxidation potential; Electrochemical behaviors; Unsaturated thiocrown ethers; Molecular Modeling; Molecular topology; Reduction potential.

Introduction:

1 Corresponding author. 171

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Carbon nanostructures such as fullerenes and carbon nanotubes (CNTs) represent a fascinating class of materials which are attracting considerable attention due to their potential technological applications, including those in electronic and optoelectronic devices. More recently, a wide variety of new carbon nanostructures, including endohedral fullerenes, have emerged as new and exciting carbon-containing materials whose chemical and physical properties are currently being explored. Among the endohedral fullerenes, the trimetallic nitride templated (TNT) endohedral metallofullerenes—carbon cages that encapsulate trimetallic nitride clusters—have been the focus of great interest since their first synthesis in 1999 by the process developed by Dorn and co-workers. This method has allowed the preparation of TNT endohedral metallofullerenes in highyields. The stabilization of a large variety of endohedral carbon cages, including different isomeric structures or compounds that violate the isolated pentagon rule has been realized, simply by the judicious choice of the trapped metal atoms. In addition, due to their interesting physical properties TNT endohedral metallofullerenes have been investigated for potential applications in the fields of biomedicine and nanomaterials sciences.[1-4]

O O O

O O n O S

S S Y3N@C80

[X-UT-Y] H H

n 0 1 2 3 4 5 6 7 8 H H SS S No 1 2 3 4 5 6 7 8 9 H SS S H H H H H

Figure 1: The conjectured structures of [X-UT-Y]@[C18H14O6-Y3N@C80].

The unsaturated, cis-geometry thiocrown ethers, (1-9), comprise a group with interesting physiochemical properties, in light of their conformational restrictions compared to corresponding saturated systems and the sizes of their cavities. The presence of sulfur atoms gives rise to the unique properties of thiocrown ethers. The cis-unsaturated thiocrown ethers (described as [X-UT- Y], where X and Y indicate the numbers of carbon and sulfur atoms, respectively), 1-9, were synthesized and their structures were confirmed.[25-34] 1,4-dithiin is the smallest member of compounds 1-9 that has been widely studied.[35-45] In 2001, the structures of [X-UT-Y] (X = 6, 9, 12, 15, 18, 21, 24 and 27 and Y = 2-9) 1-9 were reported by Tsuchiya et al. [25] In that report, the

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1H and 13C-NMR spectra, x-ray crystallographic data, ORTEP drawings, cavity size, and UV spectra of [X-UT-Y] 1-9 were carefully considered.[25] The x-ray crystal structures and ORTEP drawings for some members of 1-9 [X-UT-Y], namely X = 15, 18, 21, 24 and 27 and Y = 5-9, show the presence of cavities and a nearly coplanar arrangement of sulfur atoms.[25] This study elaborates upon the relationship between the cs index and the electron transfer (Get(1) and Get(2)) Red. Red. of [C18H14O6-Y3N@C80], on the basis of the four reduction potentials ( E1 and E2), as assessed by applying the eleectron transfer equation to create [X-UT-Y]@[C18H14O6-Y3N@C80]. [5-13]

Graphing and Mathematical Method:

All graphs were generated using the Microsoft Office Excel 2003 program. Using the cs index, several valuable properties of the fullerenes derivatives can be calculated. The values were used to calculate the four free energies of electron transfer (Get(1) and Get(2)), for [X-UT-

Y]@[C18H14O6-Y3N@C80] compounds. See figure-1. Both linear (MLR:Multiple Linear Regressions) and nonlinear (ANN:Artificial Neural Network) models were used in this study.

Discussion:

By electrochemical means it was shown that the Y3N@C80-based dyads contain adducts attached to a [5,6]-double bond ring junction, while the cycloaddition reactions with Ih- Y3N@C80 occur at a double bond at a [6,6]-ring junction. Theoretical calculations confirmed the differences between the redox potentials obtained for the different dyads and provided evidence of the importance of these structures as electron transfer model systems. To confirm this prediction, photophysical investigations were carried out with dyad, where the existence of a photoinduced electron-transfer process was easily detected. Interestingly, for the charged-separated radical ion pair generated after photoexcitation, a lifetime three times longer than that of an equivalent C60 dyad was observed.[1-7]

The reduction potentials of 1-9, shown in Table 1, demonstrate that the cs index decreases with increasing molecular size. In Table 1, related values for the supramolecular complexes of [X-

UT-Y] 1-9 with [C18H14O6-Y3N@C80] (1) are also shown. Table 1 shows the calculated values of ox oxidation potential ( E1), as well as the first, second and third free energies of electron transfer

(Get(n), n=1-2) between some of the [X-UT-Y] and the complexes 1 for supramolecular [X-UT-

Y]@[C18H14O6-Y3N@C80] complexes. The group of supramolecular complexes were neither synthesized nor previously reported.[8-13]

Table-1: The data values on the 4 free energies of electron transfer (Get), in

–1 kcal mol [X-UT-Y]@[C18H14O6-Y3N@C80]. 173

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ox E1

Formula of * cs (Volt)

[X-UT-Y] o

1-9 Get(1) Get(2)

6-UT-2(1,4- 1.02 53.73 62.50 dithiin) 0.7500

9-UT-3 0.5000 0.97 52.57 61.34

12-UT-4 0.3750 0.89 50.73 59.49

15-UT-5 0.3000 0.82 49.12 57.88

18-UT-6 0.2500 0.79 48.43 57.19

21-UT-7 0.2143 0.73 47.04 55.80

24-UT-8 0.1875 0.69 46.12 54.88

27-UT-9 0.1667 0.66 45.43 54.19

30-UT-10 0.1500 0.63 44.74 53.50

DG 56

40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

References:

[1] N. Mart_n, Chem. Commun. 2006, 2093 –2104.

[3] A. Hirsch, Angew. Chem. 2002, 114, 1933 –1939; Angew. Chem. Int. Ed. 2002, 41.

[4] J. L. Barh, J. M. Tour, J. Mater. Chem. 2002, 12, 1952 –1958.

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[5] S. Niyogi, M. A. Hamon, H. Hu, B. Zhao, P. Bhomwik, R. Sen, M. E. Itkis, R. C. Haddon, Acc. Chem. Res. 2002, 35, 1105 –113.

[6] Y.-P. Sun, K. Fu, Y. Lin, W. Huang, Acc. Chem. Res. 2002, 35, 1096 – 1104; e) S. Banerjee, M. G. C. Kahn, S. S. Wong, Chem. Eur. J. 2003, 9, 1898 – 1908;

[7] D. Tasis, N. Tagmatarchis, V. Georgakilas, M. Prato, Chem. Eur. J. 2003, 9, 4000 –4008

[11] A.A.Taherpour, Chem. Phys. Lett., 469 (2009) 135-139.

[12] A.A.Taherpour, J. Phys. Chem. C 113 (2009) 5402.

[13] A.A.Taherpour, Chem. Phys. Lett., 483 (2009) 233-240.

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Theoretical Study of Free Electrontransfer Energy Properties of [Sn(Tpp)(Fc-COO)2]@Cn Supramolecular Comlexes

Avat (Arman) Taherpour*a and Rouhollah Jalajerdib aChemistry Department, Faculty of Science, Islamic Azad University, P. O. Box 38135-567, Arak, Iran bChemistry Department, Payamenoor University,Zanjan Branch, P. O. Box 45138-69987, Zanjan, Iran E-mail: avatarman.taherpour@ gmail.com

Abstract:

Photophysical properties of a newly-synthesized porphyrin derivative, trans-bis(ferrocene carboxylato)-(5,10,15,20-tetraphenylporphyrinato)tin(IV) [Sn(TPP)(FcCOO)2] were investigated by means of steady-state and fs-time resolved laser spectroscopic techniques, and compared with those of a standard molecule, transdichloro(5,10,15,20-tetraphenyl-porphrinato)tin(IV) [Sn(TPP)Cl2].

Since the discovery of fullerenes (Cn), one of the main classes of carbon compounds, the unusual structures and physiochemical properties of these molecules have been discovered, and many potential applications and physicochemical properties have been introduced. Up to now, various empty carbon fullerenes with different numbers “n,” such as C60, C70, C76, C82 and C86, have been obtained. Topological indices are digital values that are assigned based on chemical composition. These values are purported to correlate chemical structures with various chemical and physical properties. They have been successfully used to construct effective and useful mathematical methods to establish clear relationships between structural data and the physical properties of these materials. In this study, the number of carbon atoms in the fullerenes was used as an index to establish a relationship between the structures of [Sn(TPP)(FcCOO)2] (1) and fullerenes Cn (n=60, 70, 76, 82 and 86), which create [Sn(Tpp)(Fc-COO)2]@Cn. The relationship between the number of carbon atoms and the free energies of electron transfer (Get(1) to Get(4)) are assessed using the electron transfer equation for [Sn(Tpp)(Fc-COO)2]@Cn complexes 2-6. Calculations are presented Red. Red. for the four reduction potentials ( E1 to E4) of fullerenes Cn.

Keywords: Fullerenes; [Sn(Tpp)(Fc-COO)2]@Cn; Free energy of electron transfer; Electrochemical properties; Reduction potential.

Introduction:

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The chemical, physical and mechanical properties of empty and endohedral fullerenes have been the subject of many studies. [1–8] The compressive mechanical properties of fullerene molecules Cn (n = 20, 60, 80, and 180) were investigated and discussed in detail using a quantum molecular dynamics (QMD) technique by Shen. [2,16] The unique stability of molecular allotropes such as C60 and C70, was demonstrated in 1985. [1,2] This event led to the discovery of a whole new set of carbon-based substances, known as fullerenes.

After the discovery of C60 peapods [3-8], the aligned structure of encapsulated molecules, due to the molecule-molecule and/or molecule-SWNT interactions, has been studied as a new type of hybrid material [5,6]. Zhang et al. [3] reported evidence for the latter interaction measuring the thermal stability of C60 peapods. [3-7]

[Sn(Tpp)(Fc-COO) ]@C + 2 n n=60, 70, 76, 82 and 86

C60 C70 C76 C82 C86

Figure 1: The conjectured structures of [Sn(Tpp)(Fc-COO)2] 1 and fullerenes Cn (n=60, 70, 76, 82 and 86)

which create [Sn(Tpp)(Fc-COO)2]@Cn .

Tin(IV) porphyrins have been attracting much attention because of many advantages in various applications of the particular properties conferred by the highly charged main group metal center. The large Sn(IV) ion can be accommodated in the porphyrin core without distorting the planarity of the macrocyclic ligand. tin(IV) porphyrins are diamagnetic and readily form stable six- coordinate complexes with the trans-diaxial anionic ligands. The preferential coordination of the Tin(IV) porphyrins to oxyanionic ligands has been used to develop the elaborate multiporphyrin arrays and porous structures with uniform channels. It has been also applied to develop nano materials as the promising photoelectronic materials including photocatalysts and phototherapeutical agents.[9,10]]

This study elaborates upon the relationship between the number of carbon atoms and the four free energies of electron transfer (Get(1) to Get(4)) of fullerenes Cn (n=60, 70, 76, 82 and 86), on Red. Red. the basis of the four reduction potentials ( E1 to E4), as assessed by applying the eleectron transfer equation to create [Sn(Tpp)(Fc-COO)2]@Cn.

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Graphing and Mathematical Method:

All graphs were generated using the Microsoft Office Excel 2003 program. Using the number of carbon atoms contained within the Cn fullerenes, several valuable properties of the fullerenes can be calculated. The values were used to calculate the four free energies of electron transfer (Get(1) to Get(4)), for [Sn(Tpp)(Fc-COO)2]@Cn compounds. See figure-1.

Both linear (MLR:Multiple Linear Regressions) and nonlinear (ANN:Artificial Neural Network) models were used in this study.

Discussion:

The photophysical properties of the tin(IV) porphyrins have not been systematically investigated with regard to the excited-state electron transfer as compared to those of other metal- centered porphyrins even though the electrochemical properties have been reported frequently. In the course of the systematic photophysical studies of tin(IV) porphyrins, we have synthesized a new tin(IV) porphyrin complex, trans-bis (ferrocenecarboxylato) (5,10,15,20-tetraphenylporphyrinato) tin(IV) [Sn(TPP)(FcCOO)2], and characterized its molecular structure by X-ray crystallography as shown in Scheme 1. The cyclic voltammetry for Sn(TPP)(FcCOO)2 exhibits three distinctive redox couples consisting of one oxidative wave and two reductive waves due to the ferrocenecarboxylato ligands and the porphyrin ring, respectively, implying electron transfer between ferrocene and porphyin. Thus, it is expected that Sn(TPP)-(FcCOO)2 should exhibit the photoelectronic properties, and in this work we examined photophysical properties of the newly synthesized [Sn(TPP)(FcCOO)2] in conjunction with the electron transfer dynamics by using fs-time resolved laser spectroscopic techniques as well as steady-state spectral measurements.[9,10]

Here, we calculated four free energies of electron transfer (Get(1) to Get(4)) of [Sn(Tpp)(Fc-

COO)2]@Cn. The supramolecular complex structures which were discussed here, and the calculated values of Get(n) (n=1-4) corresponding to these supramolecular complexes were neither synthesized nor reported before.[11-13]

–1 Table-1: The data values on the 4 free energies of electron transfer (Get), in kcal mol [Sn(Tpp)(Fc-

COO)2]@Cn.

–1 (Get) in kcal mol

*Formula [Sn(Tpp)(Fc-COO)2]@Cn o. of Get( Ge Get G Cn 1)* t(2)* (3)* et(4)*

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C60 29.56( 36.71 48.95 60.6 27.67) (36.43) (46.81) 8(57.42)

C70 28.86( 35.29 46.47 57.8 26.98) (35.97) (44.97) 4(54.88)

26.23( 31.75 44.79 55.8 C76 23.52) (30.90) (44.51) 5(50.96)

21.95( 26.19 42.97 53.6 C82 17.76) (25.83) (38.28) 4(46.58)

18.17( 21.37 41.67 52.0 C86 15.22) (21.45) (38.74) 5(47.04)

10 55 60 65 70 75 80 85 90 n

Conclusion:

[Sn(Tpp)(Fc-COO)2] 1 and fullerenes have important pharmaceutical and physicochemical properties. The electrochemical data of [Sn(Tpp)(Fc-COO)2]@Cn were reported here. These include the four free-energies of electron transfer (Get(1) to Get(4 Using the number of carbon atoms (n), along with the equations of the model, one can derive sound structural relationships between the aforementioned physicochemical data. By using the equations of this model, was calculated a good approximation for Get(1) to Get(4) for [Sn(Tpp)(Fc-COO)2]@Cn of the fullerenes C60, C70, C76, C82,

C86. The novel supramolecular complexes discussed have neither been synthesized nor reported previously.

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References:

[1](a) H. W. Kroto, J. R. Heath, S. C. O'Brien, R. F. Curl and R. E. Smalley, Nature, 318 (1985) 162. (b) H. W. Kroto, Nature, 329 (1987) 529.

[2] H. Shen, Molecular Physics, 105(17-18) (2007) 2405–2409.

[3] a) K. Kimura, N. Ikeda, Y. Maruyama, T. Okazaki, H. Shinohara, S. Bandow, S. Iijima, Chem. Phys. Letters, 379 (2003) 340–344; b) B.W. Smith, M. Monthioux, D. E. Luzzi, Nature, 396 (1998) 3239. c) T. Miyake, S. Saito, Solid State Commun., 125 (2003) 201. d) M. Zhang, M. Yudasaka, S. Bandow, S. Iijima, Chem. Phys. Lett., 369 (2003) 680.

[4] L. Kavan, L. Dunsch, H. Kataura, Carbon, 42 (2004) 1011–1019.

[5] B. S. Sherigara, W. Kutner, F. D’Souza, Electroanalysis, 15 (2003) 753–72

[6] R. E. Haufler, J. Conceicao, L. P. F. Chibante, Y. Chai, N. E. Byrne, S. Flanagan, et al., J. Phys. Chem., 94 (1990) 8634–6.

[7] Q. Xie, E. Perez-Codero, L. Echegoyen, J. Am. Chem. Soc., 114 (1992) 3978–80.

[8] C. Jehoulet, Y. O. Obeng, Y. T. Kim, F. Zhou, A. J. Bard, J. Am. Chem. Soc., 114 (1992) 4237–47.

[9] J. H. Jang et al., Bull. Korean Chem. Soc., 28(11) (2007) 1967.

[10] Molina P., Pastor A., M. Jesu´ s Vilaplana M., and Desamparados Velasco M., Organometallics 16 (1997) 5836-5843.

[11] A.A.Taherpour, Chem. Phys. Lett., 469 (2009) 135-139.

[12] A.A.Taherpour, J. Phys. Chem. C 113 (2009) 5402.

[13] A.A.Taherpour, Chem. Phys. Lett., 483 (2009) 233-240.

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Polarizability Study of Fullerene Nano- structures C20 to C300 by Using Monopole- Dipole Interactions Theorem

Avat Arman Taherpour1* and Nosratollah Mahdizadeh1,2

1 Chemistry Department, Faculty of Science, Islamic Azad University, P.O. Box 38135-567, Arak, Iran 2 Chemistry Department, Payame-Noor Universit, Islamabad-Ghar, Iran.

Abstract:

Since the discovery of fullerenes (Cn), one of the main classes of carbon compounds, the unusual structures and physiochemical properties of these molecules have been discovered, and many potential applications and physicochemical properties have been introduced. Up to now, various empty carbon fullerenes with different numbers “n,” such as C20 through C300 (like C60, C70, C76, C82,…, C300) have been obtained. The linear uniform field electric dipole polarizability tensors of 46 fullerenes in the range C20 through C240 were calculated by the atom monopole-dipole interaction (AMDI) theory, using the monopole and dipole polarizabilities of the carbon atom found previously to fit polarizability tensors of aromatic hydrocarbons. The structures are taken to be those predicted by molecular dynamics energy optimization. The isotropic mean polarizabilities calculated for C60 and C70 are comparable to experimental data from solid film studies and to quantum mechanical calculations. Topological indices are digital values that are assigned based on chemical composition. These values are purported to correlate chemical structures with various chemical and physical properties. They have been successfully used to construct effective and useful mathematical methods to establish clear relationships between structural data and the physical properties of these materials. In this study were extended the calculation of the parameters concern to atom monopole-dipole moment such as Ellipsoid (α1 to α3 and ā), AMDI (Atom monopole-dipole interaction theory; α1 to α3 and ā) and semi-axes a,b,c of a thin ellipsoidal shell of uniform thickness (in Å, ABC α1 to α3) by QSAR for C20 through C300 .

Keywords: Fullerenes; Polarizability; AMDI theory; Ellipsoid; Semi-axes thin ellipsoidal shell.

Introduction:

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The electrochemical properties of the fullerene C60 have been studied since the early 1990s, when these materials became available in macroscopic quantities (for a review see [1]).[1-3] In – 2– 1990, have shown that C60 is electrochemically reducible in the CH2Cl2 medium to C60 and C60 . + In 1992, have cathodically reduced both C60 in six reversible one-electron steps for -0.97 vs. Fc/Fc (Fc=ferrocene). This fact, along with the absence of anodic electrochemistry of fullerenes, matches 6– the electronic structure of fullerenes: the LUMO of C60 can accept up to six electrons to form C60 , but the position of the HOMO does not allow for hole-doping under the usual electrochemical conditions. In 1991, Bard et al. [4-8] first reported on irreversible electrochemical and structural reorganization of solid fullerenes in acetonitrile medium. Dunsch et al. [5] have upgraded the experimental conditions by investigating highly organized C60 films on HOPG in aqueous medium. The reduction of such films manifested itself by re-structuring into conductive nanoclusters of ~102 nm in diameter. [5-9]

The linear uniform field electric dipole polarizability tensors of 46 fullerenes in the range C20 through C240 are calculated by the OlsonSundberg atom monopole-dipole interaction (AMDI) theory, using the monopole and dipole polarizabilities of the carbon atom found previously to fit polarizability tensors of aromatic hydrocarbons. The structures are taken to be those predicted by Zhang and co-workers by molecular dynamics energy optimization. The isotropic mean polarizabilities calculated for C60 and C70 are comparable to experimental data from solid film studies and to quantum mechanical calculations. Polarizability tensors are also calculated for conducting ellipsoidal shells which have the same moment of inertia tensor as the corresponding fullerenes. These are substantially smaller than the AMDI polarizabilities for the smaller fullerenes, but the two calculations tend to converge for the larger molecules.[7-15]

Graph theory has been found to be a useful tool in assessing the QSAR (Quantitative Structure Activity Relationship) and QSPR (Quantitative Structure Property Relationship). Numerous studies in the above areas have also used what are called topological indices (TI). It is important to use effective mathematical methods to make good correlations between several data properties of chemicals. Numerous studies have been performed related to the above mentioned fields by using the so-called topological indices (TI). The numbers of carbon atoms at the structures of the fullerenes were utilized here.[7-15]

In this study were extended the calculation of the parameters concern to atom monopole- dipole moment such as Ellipsoid (α1 to α3 and ā), AMDI (Atom monopole-dipole interaction theory; α1 to α3 and ā) and semi-axes a,b,c of a thin ellipsoidal shell of uniform thickness (in Å, ABC α1 to α3) by QSAR for C20 through C300 .

Graphs and Mathematical Method:

All graphing operations were performed using the Microsoft Office Excel 2003 program.

The numbers of carbon atoms at the structures of the fullerenes Cn were utilized to make the

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010. relationship and calculate the Ellipsoid, AMDI and thin ellipsoidal shell of uniform thickness. For modeling, both linear (MLR) and nonlinear (ANN) models were used in this study.

Discussion:

The polarizabilities of the ellipoids are simply correlated with their geometry, as can be seen from the fact that the principal polarizabilities are approximately proportional to the lengths of the corresponding axes. To some extent this holds for the AMDI model as well, but for the smaller members of the series there are cases where the principal polarizabilities are not in the same ratio as the axes. his is apparently because the atom dipole contribution, which is not simply related to the axis lengths, is relatively larger for the smaller members. A further measure of the correspondence between the molecules and the ellipsoids is found in the comparison of the principal polarizability axes found by the AMDI theory with the principal geometric axes of the ellipsoids. Where the three semi-axes are distinct, the axes directions are the same to within a few tenths of a degree. The numbers of carbon atoms at the structures of the fullerenes Cn were utilized and extended the calculation of the parameters concern to atom monopole-dipole moment such as Ellipsoid (α1 to α3 and ā), AMDI (Atom monopole-dipole interaction theory; α1 to α3 and ā) and semi-axes a,b,c of a thin ellipsoidal shell of uniform thickness (in Å, ABC α1 to α3) by QSAR for C20 through C300.

Ellipsoid

0 0 50 100 150 200 250 300 n

References:

[1] R. E. Haufler, J. Conceicao, L. P. F. Chibante, Y. Chai, N. E. Byrne, S. Flanagan, et al., J. Phys. Chem., 1990, 94, 8634–6.

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[2] Q. Xie, E. Perez-Codero, L. Echegoyen, J. Am. Chem. Soc., 1992,114, 3978–80.

[3] C. Jehoulet, Y. O. Obeng, Y. T. Kim, F. Zhou, A. J. Bard, J. Am. Chem. Soc., 1992,114, 4237

[4] P. Janda, T. Krieg, L. Dunsch, Adv. Mater., 1998, 17, 1434–8.

[5] A. Touzik, H. Hermann, P. Janda, L. Dunsch, K. Wetzig, Europhys. Lett., 2002, 60, 411–7.

[6] T. Suzuki, K. Kikuchi, F. Oguri, Y. Nakao, S. Suzuki, Y. Achiba, K. Yamamoto, H. Funasaka, and T. Takahashi, Tetrahedron, 1996, 52(14), 4973-4982.

[7] A. A. Taherpour, Fullerenes, Nanotubes and Carbon Nanostructures, 2007, 15, 279–289.

[8] A. A. Taherpour, Fullerenes, Nanotubes and Carbon Nanostructures, 2007, 15, 405–415.

[9] Y. P. Du, Y. Z. Liang, B. Y. Li and C. J. Xu, J. Chem. Inf. Cmput. Sci., 2002, 42, 1128-1138.

[10] M. Randić , J. Am. Chem. Soc., 1975, 97, 6609-6615.

[11] S. D. Bolboaca and L. Jantschi, Int. J. Mol. Sci., 2007, 8, 335-345.

[12] Z. Slanina, F. Uhlik, S. L. Lee, E. Osawa, MATCH Commun. Math. Comput. Chem., 2001, 44, 335-348.

[13] A. A. Taherpour, Fullerenes, Nanotubes, and Carbon Nanostructures, 2008, 16, 196–205.

[14] A. A. Taherpour, F. Shafiei, J. Mol. Struct. THEOCHEM, 2005, 726, 183-188.

[15] A. A. Taherpour, Fullerenes, Nanotubes and Carbon Nanostructures, 2009, 17(1), 26-37.

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A Classification of Upper and Lower Bounds of Energy in Graphs

S. Rahimi Sharebaf , A. Afzal Shahidi Department of Mathematics Shahrood University of Technology , Shahrood , Iran [email protected] [email protected] Abstract

The energy of a graph , denoted by , is defined as the sum of the absolute values of all eigenvalues of . In this paper we show that a classification of upper and lower bounds of energy and compared each other. A summary of proposed classification of graph energy bounds shown in table 1.

Keywords: Graph energy ,classification, upper and lower bounds of energy.

1. Introduction Let be a graph. If is the set of vertices of , then The adjacency matrix of , , is an matrix, where if and are adjacent and otherwise. The ,,…, spectrum of graph , Spec , is the set of the eigenvalues of A, denoted by 1 0 .The energy of graph , is defined by

|| The search of the upper and lower bounds for the energy of graph and minimal and maximal energy, is a wide field of spectral graph theory ( See [1] ,[2], [3]).

Many bounds of graph energy have been identified but was not provided to any category. In this paper we attempt to provide a classification of upper and lower bounds of graph energy. This classification is included following categories.

1. General bounds on the number of vertices and edges. 2. Bounds on the bipartite graphs. 3. Comparative bounds.

1.General energy bounds

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The following bounds are general energy bounds. In these bounds n and m are the number of vertices and number of edges of graph Ssee [2].

i. . ii. / 21|| √2 iii. 2 2 √ 2. Energy bounds 2√1 in the bipartite graphs

The following bounds are respectively upper and lower bounds for the energy of the bipartite graph. In these bounds, p and q are each the number of vertices of parts of graph Also r is rank of graph [3], [4], [5], [6]. . i. ii. √ 21

iii. 1 iv. 2 22 2 i. √ √ 2

ii. 2/ iii. 1 5

Bellow we show1 a new bound in bipartite graph.

Theorem 1. if is a bipartite graph with vertices and edgs, then

(1)

Proof. Suppose that 22 are the2 eigenvalues 2 of . Then with Lemma1, we have (2)

For bipartite graphs, we have 1.

Moreover, since 1

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Then using (3) and (4) with the Cauchy-Schwartz inequality, applied to the vectors and (1, …, 1) with entries, we obtain the inequality

| |,…,| | 2 (5) Now, since the function |2 | 22 2decreases on the intervals , we see that must hold. Since Lemma 2, we have, belongs to interval 2 22 2 , and hence must hold as well. From this fact, and Inequalities (4) and 1 1 (5), it immediately√ follows that Inequalities (1) hold. 1 √ 1 3. Comparative bounds

3.1 Energy bounds in tree

A connected acyclic graph is called a tree. A tree with n vertices contains n-1 edges. A tree possesses necessarily vertices of degree one. Such vertices are called terminal or pendent vertices. The tree with maximal number n-1 of terminal vertices is the path ( ) and the tree with minimal number 2 of terminal vertices is the star ( ). Let is arbitrary tree with n vertices. Below we compare wieu energy of tree groups. , so we have [7],

Denote by the tree obtained by joining k terminal vertices to the terminal vertex of . In the set of all trees without and , is the tree with minimal energy[8]. , 3.2 Energy bounds in unit cycle,

Let be the graph obtained from the star graph with n vertices by adding an edge and be obtained by connecting a vertex of the circuit with a terminal vertex of the path . In the set of all unicyclic graph, is minimal energy and is maximal energy [9,10]. Table 1 is an illustration of classification of graph energy bounds that in here just discussed.

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Table 1- A summary of proposed classification of upper and lower bounds of energy

Classes Comparative bounds bounds of energy Graph Graph lower bounds upper bounds with with

maximal minimal energy energy

Strong 1) General ly regular 2) 2) bounds graphs 2 1| | 3) 2 √ 2√1 2 1) 1) √2) 2 1 Energy 2) Bounds on --- 2/ 1 the bipartite 2 1 5 graphs , 3) 2 1 2 4)2 2 2 √ √ 2 Energy ------bounds on the trees Energy bounds on ------the unit cycle graphs

References

[1] N.Biggs, Algebriac Graph Theory, Cambridge Univercity Press, Cambridge, 1974.

[2] I. Gutman, The energy of graph: old and new results, in: A. Betten, A. Kohnert, R. Laue, A. Wasserman(Eds.) Algebric Combinatorics and applications, Spinger, Berlin, 2001, pp. 196-211.

[3]. S. Akbari, E. Ghorbani, S. Zare, Some relations between rank, chromatic number and energy of graphs, Discr. Math, 309 (2009) 601-605.

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[4] J. Koolen, V. Moulton, Maximal energy bipartite graph, Graph and Combinatories (2003)19 :131-135.

[5]. Xueliang Li, Jianbin Zhang, Lusheng Wang, on bipartite graphs with minimal energy, Discrete App Mathematics (2008).

[6]. Juan Rada, Antonio Tino, Upper and lower bounds for the energy of bipartite graphs, J. Math. Appl, 289 (2004) 446-455. [7] I.Gutman, Acyclic Systems With Extermal Huckel -Electron Energy, Theoret. Chem.Acta (Berl.) 45(1977)79-87. [8] Aimei Yu, Xuezheng Lv, Minimum energy on trees with k pendent vertices, Linear Algebra and its App, 418(2006) 625-633.

[9]. Yaoping Hou, I Gutman, Ching-Wah Woo, Unicyclic graphs with maximal energy, Linear Algebra and its App, 365(2002) 27-36.

[10]. Yaoping Hou, Unicyclic graphs with minimal energy, Mathematical chemistry, 29 (2001) No. 3.

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Szeged Index of some Dendrimers

A. Seyed Mirzaei1, A. R. Ashrafi2 and G. H. Fath-Tabar2 1Department of science, Islamic Azad University, Qom Branch, Qom ,Iran 2Department of Mathematics, Faculty of Science, University of Kashan, Kashan 87317-51167, Iran

Abstract

Suppose G be a simple connected graph. Let e = uv be an edge of G and nu(e) is the number of vertices closer to u than v and nv(e) is the number of vertices closer to v than u then the Szeged index of G is defined as Sz(G) = e=uv nu(e)nv(e) . In this paper, the Szeged index of some dendrimers is computed for the first time.

1. Introduction

Dendrimers are large and complex molecules with very well-defined chemical structures. They consist of three major architectural components: core, branches and end groups. Nanostar dendrimers are part of a new group of macromolecules. The topological study of these macromolecules is the subject of some recent papers.1,2

Let G be a connected simple molecular graph with vertex and edge sets V(G) and E(G), respectively. As usual, the distance between the vertices u and v of G is denoted by dG(u,v) (or d(u,v) for short) and it is defined as the number of edges in a minimal path connecting vertices u and v.3

The Szeged index of a graph G is defined as Sz(G) = e=uv[nu(e)nv(e)], where nu(e) is the 4,5 number of vertices lying closer to u than to v and nv(e) is defined analogously. The mathematical properties of this topological index can be found in some recent papers.6-13

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Figure 1. The Suzuki's Bi-branched Dendrimer.

In this paper our notation is standard and taken mainly from the standard book of graph theory. The goal of this article is to compute the Szeged index of two classes of dendrimeric nanostars.

2. Main results

In this section the Szeged index of a class of nanostar dendrimers, S[n], is computed. If A and B are graphs such that V(A)  V(B) and E(A)  E(B) then A is called a subgraph of B, A ≤ B. To compute this topological index, we partition the edge set of S[n] into the classes with the same n(e)

= nu(e)nv(e), where e = uv is an edge of S[n]. We first notice that the graph S[n] can be constructed from subgraphs isomorphic to H and the core of S[n], see Figures 2 and 3.

Figure 2. The Peace H of Figure 3. The Core K of

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S[n]. S[n].

We now explain our method for constructing S[n]. In the first step, we join a subgraph isomorphic to H from vertex A to a subgraph isomorphic to K in the vertex B. Since, K has exactly three branches, we can construct S[1] by K and three isomorphic subgraphs. In the second step, six subgraphs isomorphic to H join with S[1]. Finally, in the step r, 3 × 2r subgraphs isomorphic to H are joined to S[r-1]. Begin by a simple calculation. Clearly, Sz(K) = 3564 and ])1[S(Sz  57600.

By these calculations, Sz(S[2]) = 370152. We are now ready to calculate the step n of this nanostar. In Table 1, the values of nu(e)nv(e) for types of edges are computed. We have:

Table 1. The Type of Edges in S[n] and Calculation of nu(e)nv(e).

Edges nu(e)nv(e) e 1n ((30  27 n(n  ))1  2(3)1 n2  )1 e e, e, 2n 3n 6n (((30  27 n(n  ))1  )2  3)(3 2  2n ) e 5n e, 8n (((30  27n n(  ))1  )5  3)(6  2 1n ) e 4n e, 7n (((30  27 n(n  ))1  )6  3)(7  2 1n ) e  1)1n( n e  2)1n( e,  6)1n( e, n(  (((30  27 n(n  ))1  )9 10 3)(  2 ) (((30  27 n(n  ))1  )20  21 3)(  )3 e  5)1n( e,  8)1n( (((30  27 n(n  ))1  )23  24 3)(  )2 e  4)1n( e,  7)1n( (((30  27 n(n  ))1  )24  25 3)(  )2 e n(  1)2 (((30  27 n(n  ))1  )27  28 3)(  )2

e  2)in( e,  6)in( e, n(  (((30  27 n(n  ))1  20(  i(  18)1 )) 21(  i(  18)1 ))(3 )3 e e, (((30  27 n(n  ))1  23(  i(  18)1 )) 24(  i(  18)1 ))(3 )2  5)in(  8)in( (((30  27 n(n  ))1  24(  i(  18)1 )) 25(  i(  18)1 ))(3 )2 e  4)in( e,  7)in( (((30  27 n(n  ))1  27(  i(  18)1 )) 28(  i(  18)1 ))(3 )2 e   1))1i(n(

e12 e, 16 e, 13 (((30  27 n(n  ))1  20(  n(  18)2 )) 21(  n(  18)2 ))(3 )3 e e, (((30  27 n(n  ))1  23(  n(  18)2 )) 24(  n(  18)2 ))(3 )2 15 18 e14 e, 17 (((30  27 n(n  ))1  24(  n(  18)2 ))  25(  n(  18)2 ))(3 )2 e11 (((30  27 n(n  ))1  27(  n(  18)2 ))  28(  n(  18)2 ))(3 )2

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e02 e, 06 e, 03 ((30  27 n(n  ))1  20(  n(  18)1 )(21 n(  18)1 3)(  )3 e e, ((30  27 n(n  ))1  23(  n(  18)1 )(24  n(  18)1 3)(  )2 05 08 e04 e, 07 ((30  27 n(n  ))1  24(  n(  18)1 )(25  n(  18)1 3)(  )2 e01 ((30  27 n(n  ))1  27(  n(  18)1 )(28  n(  18)1 3)(  )1

Theorem 1. 2 3 4 2 n n n SZ [( nS ])  720  5544n  9828n  13203n  6561n  3969n 2  3969n2  36542 .

Proof. By calculating S1, S2, S3, S4 and S5 its easy to see that

SZ nS n )][(  S1  S2  S3  S4  S5 thus by straightforward calculation we have:

SZ [( nS ])   720  5544n  9828n 2  13203n 3  6561n 4  3969n 2 2 n  3969n2 n  36542 n .

References

[1] A. R. Ashrafi and M. Mirzargar, Indian J. Chem., 47A (2008) 538-541.

[2] A. Karbasioun and A. R. Ashrafi, Macedonian J. Chem. Chem. Eng., 28 (2009) 49-54.

[3] D. B. West, Introduction to Graph Theory, Prentice Hall, NJ, 1996.

[4] A. R. Ashrafi and F. Gholami-Nezhaad, Current Nanoscience, 5 (2009) 51-53 . [5] G. H. Fath-Tabar, B. Furtula and I. Gutman, J. Math. Chem., Doi:10.1007/s10910-009-9584-

[6] G. H. Fath-Tabar, M. J. Nadjafi-Arani, M. Mogharrab and A. R. Ashrafi, MATCH Commun. Math. Comput. 63 (2010) 145150.

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Balaban Index of Dendrimers

H. Shabani and A. R. Ashrafi Department of Mathematics, Faculty of Science, University of Kashan, Kashan 87317-51167, Iran

A graph G is defined as a pair G = (V,E), where V is a finite, non-empty set of vertices and E is a set of edges. The term chemical graph was introduced by Cullen in 1758. He used those graphs for affinity diagrams showing a relationship between chemical substances. Those results have never been published officially. In a chemical graph, vertices represent atoms and edges represent bonds. Numbers reflecting certain structural features of a molecule that are obtained from its chemical graph are usually called topological indices. The Wiener index W, one of widely used descriptors of molecular topology, was introduced in 1947 by Wiener as the half-sum of all topological distances in the hydrogen-depleted graph representing the skeleton of the molecule [1]. Here, we denote by d(u,v), the topological distance between vertices u and v of the graph G, which is the length of a minimum path between these vertices.

Let G be a graph and x  V(G). We denote by d(x), the summation of topological distances between x and all vertices of G. The Balaban index [2-6] of a graph G is defined as m/( +

-0.5 1)e=uv[d(u)d(v)] , where m is the number of edges of G and (G) = |E(G)| - |V(G)| + 1 is the cyclomatic number of G. Balaban and co-authors5 defined and then investigated this index even for infinite graphs.

Throughout this paper, our notation is standard and taken mainly from the standard books of graph theory. In this paper we continue our earlier works on computing topological indices of dendrimers. The Balaban index of an infinite class of dendrimers is computed for the first time.

References

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

1. H. Wiener, Structural determination of the paraffin boiling points, J. Am. Chem. Soc., 69 (1947) 17-20.

2. A. Balaban, Highly discriminating distance-based topological index, Chem. Phys. Letters, 89 (1982) 399-404.

3. A. Balaban, Topological indices based on topological distances in molecular graphs, Pure Appl. Chem. 55 (1983) 199-206.

4. A.T. Balaban, P. Filip, Computer program for topological index J (average distance sum connectivity), MATCH Commun. Math. Comput. Chem., 16 (1984) 163-190.

5. A.T. Balaban, N. Ionescu-Pallas, T.S. Balaban, Asymptotic values of topological indices J and J' (average distance sum connectivities) for infinite cyclic and acyclic graphs, MATCH Commun. Math. Comput. Chem., 17 (1985) 121-146.

6. A.T. Balaban, Chemical graphs. 48. Topological index J for heteroatom-containing molecules taking into account periodicities of element properties, MATCH Commun. Math. Comput. Chem., 21 (1986) 115-122.

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The Third Conference and Workshop on Mathematical Chemistry (TCWMC 2010) Tarbiat Modares University, Tehran, Iran, February 22-24, 2010.

Computing Some Topological Indices of Tensor Product of Graphs

Z.Yarahmadi1 and S. Moradi2 1Department of Mathematics, Islamic Azad University, Khorramabad Branch Khorramabad, I. R. Iran [email protected] 2Department of Mathematics, Faculty of Science, Arak University, Arak, Iran [email protected]

Abstract

A topological index of a molecular graph G is a numeric quantity related to G which is invariant under symmetry properties of G. In this paper we obtain topological indices of this product.

1. Introduction

A graph G consists of a set of vertices V(G) and a set of edges E(G). For every vertex a V G)( , the edge connecting a and b is denoted by ab and degG a denotes the degree of a in G. The distance between two vertices in a connected graph G is the number of edges in a shortest path between them. This concept has been known for a very long time and recently has received considerable attention as a subject of its own. Suppose a and b are vertices of a graph G, their distance is shown by dG(a,b). A topological index of a molecular graph G is a numeric quantity related to G which is invariant under symmetry properties of G. The first and second Zagreb indices were originally defined as M G)(  deg 2 a and M G)(  deg a deg b 1   GVa )( G 2   GEab )( G G respectively. The first Zagreb index can be also expressed as a sum over edges of G M G)(  [deg a  deg b] ,see [1,2]. The first and second Zagreb coindices are defined as 1   GEab )( G G M G)(  [deg a  deg b] and M G)(  deg a deg b , see [3]. The Randić 1   GEab )( G G 2   GEab )( G G index of a graph G is R G)(  1 . The geometric-arithmetic index GA was conceived,  abe deg a deg b G G

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deg a deg b GA G)(  G G . The Schultz and modified Schultz indices are defined as   GEab )( 1 2 (degG a degG b) W G)(  (deg a  deg ) db ba ),( W G)(  deg a deg db ba ),( follows:    GVba )(},{ G G G , *   GVba )(},{ G G G , H G)(  1 respectively, see [4, 5]. The Hrary index H(G) is defined as   GVba )(},{ G bad ),( , see [6] for details. For any two simple graphs G and H, the tensor product G  H of G and H has vertex set V (G  H)  V G)( V H)( and edge set E(G  H)  ba dc |),)(,({ ac  E G)( and bd  E(H )}. It is easy to proof of this fact that | E(G  H  |2|) E G ||)( E(H |) , see [7]. First the Randic, GA, first and second Zagreb indices and first and second Zagreb coindices are computed. Finally, for computing topological indices which related to distance in graphs, we use the useful and simple equality that obtained in [8] for distance of vertices in tensor product of graphs. We use this equality for computing hyper Wiener, Schultz, modified Schultz and Hrary indices for tensor product of complete graph Kn and a graph G.

2. Main Results

In this section at first we present new simple proofs for some important theorems related to tensor product of graphs and then as an application of pervious results, we compute the Wiener index for tensor product of K n and G where K n is complete graph of order n and G is a connected graph. Finally Zagreb indices and Zagreb coindices are computed for tensor product of graphs.

Theorem 2.1. Let G and H be graphs. The first and second Zagreb indices and coindices of tensor product of G and H are given by:

M1 (G  H) M1 G)( M1 (H),

M 2 (G  H)2M 2 G)( M 2 (H),

M 1 (G  H )  E G (|)(|2 M 1 H )(  M 1 (H ))  E H (|)(|2 M 1 G)(  M 1 (G))  M 1 (G  H )  M 1 (G  H ),

M 2 (G  H )  2M 1 G ()( M 2 (H )  M 2 (H ))  2 M 1 (H )(M 2 G)( M 2 (G))  M 2 (G  H )  M 2 (G  H ).

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Theorem 2.2. Let G and H be graphs. The Randic index of tensor product of G and H is computed as follows:

R(G  H )  2 R G)( R(H ).

Theorem 2.3. Let G and H be graphs and G be regular. The GA index of tensor product of G and H is computed as follows:

GA(G  H )  2 E G)( GA(H ).

In what follows at first for a graph G we define the set TG  E G)( as follows:

TG  {ab  E G |)( ab is contained in a triangle}.

Theorem2.4. Let G be a graph and K n be a complete graph of order n. The hyper Wiener index of tensor product of K n and G are given by:

n 1 n WW (K n  G)   WW G)(  3  V G)( 5 En G)( 3 Tn G .  2  2

Theorem 2.5. Let G be a graph and K n be a complete graph of order n. The Schultz and modified Schultz indices of tensor product of K n and G are given by:

n W (K  G)   2[ Wn G)(  (8 nn  )1 E G)(  2M G)(  2 (deg b  deg d ]) ,  n    1   Tbde G G 2 G

W (K  G)  (n  2 [)1 n 2W G)(  (4 nn  )1 M G)(  2( n  )1 M G)(  n deg b deg d ].) . * n * 1 2   Tbde G G G

Theorem 2.6. Let G be a graph and K n be a complete graph of order n. The Harary index of tensor product of K n and G is computed as follows:

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n 2 1   2 1 H (K n  G) n H G)(  2   V G)(  3 En G)(  6 Tn G . 2

References

[1] I. Gutman and K. C. Das, The first Zagreb index 30 years after, MATCH Commun.

Math. Comput. Chem., 50 (2004), 8392.

[2] M. Khalifeh, H. Yousefi-Azari and A. R. Ashrafi, The first and second Zagreb

indices of some graph operations, Discrete Appl. Math., 157 (4)(2009), 804811.

[3] T. Došlić, A. Hamzeh and A. R. Ashrafi, The Zagreb Coindices of Graph Operations, submitted.

[4] I. Gutman, W. Yan , B.-Y. Yang and Y.-N. Yeh, Generalized Wiener indices of ZigZagging Pentachains, J. Math. Chem., 42(2)(2007), 103-117.

[5] H.P. Schultz, Topolodical organic chemistry. 1. Graph theory and topological indices of alkanes, J. Chem. Inf. Comput. Sci., 29(1989) 227-228.

[6] D. Pliavšić, S. Nikolić, N. Tranistić, On the Harary undex for the characterization of chemical graph, J. Math. Chem. 12(1993) 235-250.

[7] W. Imrich and S. Klavžar, Product Graphs, Structure and Recognition, John Wiley & Sons, New York, USA, 2000.

[8] S. Moradi, A note on tensor product of graphs, MATCH Commun. Math. Comput.

Chem., (to appear).

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Automorphism Group and Topological Indices of the Chemical Graph of Fullerenes

Mehdi Zabihi Eslamic Azad university Tafresh branch Mohammad Reza Ahmadi

Abstract

In an earlier paper, the authors of this paper designed a MATLAB program for computing symmetry of molecules. They applied this program to calculate the symmetry of the fullerene C80.

In this paper, using a well-known result on graphs, we write another MATLAB program for computing the automorphism group of some fullerene graphs, which has better running time. The PI, Wiener and Schultz indices of these chemical graphs are also computed.

Keywords: Fullerene, automorphism group of a graph, topological index

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