2012
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94
Chebyshev finite difference method for a two−point boundary value problems with applications to chemical reactor theory
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2
In this paper, a Chebyshev finite difference method has been proposed in order to solve nonlinear twopoint boundary value problems for second order nonlinear differential equations. A problem arising from chemical reactor theory is then considered. The approach consists of reducing the problem to a set of algebraic equations. This method can be regarded as a nonuniform finite difference scheme. The method is computationally attractive and applications are demonstrated through an illustrative example. Also a comparison is made with existing results.
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1
7


A.
Saadatmandi
University of Kashan
University of Kashan
I R Iran
saadatmandi@kashanu.ac.ir


M.
Azizi
Shariaty Technical College
Shariaty Technical College
I R Iran
Chemical reactor
Chebyshev finite difference method
Numerical methods
Boundary value problems
Gauss–Lobatto nodes
Study of fullerenes by their algebraic properties
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2
The eigenvalues of a graph is the root of its characteristic polynomial. A fullerene F is a 3 connected graphs with entirely 12 pentagonal faces and n/2 10 hexagonal faces, where n is the number of vertices of F. In this paper we investigate the eigenvalues of a class of fullerene graphs.
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9
24


M.
Ghorbani
Shahid Rajaee Teacher Training
University
Shahid Rajaee Teacher Training
University
I R Iran
ghorbani30@gmail.com


S.
Heidari Rad
Shahid Rajaee Teacher Training
University
Shahid Rajaee Teacher Training
University
I R Iran
Molecular Graph
Adjacency matrix
Eigenvalue
Fullerene
[1. I. Gutman, The energy of a graph, Ber. Math.Statist. Sekt. Forsch. Graz 103 (1978) 1–##2. I. Gutman, The Energy of a Graph: Old and New Results, Algebraic Combinatorics and##Applications, SpringerVerlag, Berlin, 2001.##3. I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry,##SpringerVerlag, Berlin, 1986.##4. H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl and R. E. Smalley, C60:##Buckminsterfullerene. Nature 318 (1985) 162–163.##5. H. W. Kroto, J. E. Fichier and D. E. Cox, The Fullerene, Pergamon Press, New York,##6. D. Cvetković, M. Doob and H. Sachs, Spectra of Graphs: Theory and Applications (Pure##and Applied Mathematics), Academic Press, 1997.##7. S. L. Lee, Y. L. Luo, B. E. Sagan and Y.N. Yeh, Eigenvectors and eigenvalues of some##special graphs, IV multilevel circulants. Int. J. Quant. Chem. 41 (1992) 105 – 116.##]
On discriminativity of Zagreb indices
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2
Zagreb indices belong to better known and better researched topological indices. We investigate here their ability to discriminate among benzenoid graphs and arrive at some quite unexpected conclusions. Along the way we establish tight (and sometimes sharp) lower and upper bounds on various classes of benzenoids.
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T.
Doslic
University of Zagreb
University of Zagreb
Croatia
doslic@master.grad.hr
Zagreb index
Benzenoid graph
Catacondensed benzenoid
[1. A. R. Ashrafi, T. Došlić, A. Hamzeh, The Zagreb coindices of graph operations,##Discrete Appl. Math. 158 (2010) 1571–1578.##2. J. Braun, A. Kerber, M. Meringer, C. Rucker, Similarity of molecular descriptors:##the equivalence of Zagreb indices and walk counts, MATCH Commun. Math.##Comput. Chem. 54 (2005), 163–176.##3. D. de Caen, An upper bound on the sum of squares of degrees in a graph, Discrete##Math. 185 (1998) 245–248.##4. S. J. Cyvin, I. Gutman, Kekulé Structures in Benzenoid Hydrocarbons, Lec. Notes##in Chemistry, Springer, Heidelberg, 1988.##5. K. Ch. Das, Maximizing the sum of the squares of the degrees of a graph, Discrete##Math. 285 (2004) 57–66.##6. K. Ch. Das, I. Gutman, Some properties of the second Zagreb index, MATCH##Commun. Math. Comput. Chem. 52 (2004) 103–112.##7. T. Došlić, VertexWeighted Wiener Polynomials for Composite Graphs, Ars Math.##Contemp. 1 (2008) 66–80.##8. I. Gutman, K. Ch. Das, The first Zagreb index 30 years after, MATCH Commun.##Math. Comput. Chem. 50 (2004) 83–92.##9. M. H. Khalifeh, H. YousefiAzari, A. R. Ashrafi, The first and second Zagreb##indices of graph operations, Discrete Appl. Math. 157 (2009) 804–811.##10. M. H. Khalifeh, H. YousefiAzari, A. R. Ashrafi, S.Wagner, Some new results on##distancebased graph invariants, Europ. J. Combin. 30 (2009) 1149–1163.##11. D. J. Klein, T. Došlić, D. Bonchev, Vertexweightings for distance moments and##thorny graphs, Discrete Appl. Math. 155 (2007) 2294–2302.##12. V. Nikiforov, The sum of the squares of degrees: an overdue assignment,##arXiv:math/0608660.##13. S. Nikolić, G. Kovačević, A. Miličević, N. Trinajstić, The Zagreb Indices 30 Years##After, Croat. Chem. Acta 76 (2003) 113–124.##14. D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River,##15. S. Yamaguchi, Estimating the Zagreb indices and the spectral radius of triangleand##quadranglefree connected graphs, Chem. Phys. Lett. 458 (2008) 396–398.##16. Y. S. Yoon, J. K. Kim, A relationship between bounds on the sum of squares of##degrees of graph, J. Appl. Math. & Comput. 21 (2006) 233–238.##17. B. Zhou, I. Gutman, Relations between Wiener, hyperWiener and Zagreb indices,##Chem. Phys. Lett. 394 (2004) 93–95.##18. B. Zhou, Upper bounds for the Zagreb indices and the spectral radius of seriesparallel##graphs, Int. J. Quant. Chem. 107 (2007) 875–878.##19. B. Zhou, N. Trinajstić, On reciprocal molecular topological index, J. Math. Chem.##44 (2008) 235–243.##]
Centric connectivity index by shell matrices
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2
Relative centricity RC values of vertices/atoms are calculated within the Distance Detour and ClujDistance criteria on their corresponding Shell transforms. The vertex RC distribution in a molecular graph gives atom equivalence classes, useful in interpretation of NMR spectra. Timed by vertex valences, RC provides a new index, called Centric Connectivity CC, which can be useful in the topological characterization of graphs and in QSAR/QSPR studies.
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M.
Diudea
BabesBolyai University
BabesBolyai University
Romania
diudea@gmail.com
Graph theory
Cluj matrix
Relative centricity
Centric connectivity index
[1. N. Trinajstić, Chemical Graph Theory, CRC Press: Boca Raton, FL, 1983.##2. F. Harary, Graph Theory, AddisonWesley, Reading, M.A., 1969.##3. M. V. Diudea, I. Gutman, and L. Janschi, Molecular Topology, Nova Science,##Huntington, N. Y., 2001.##4. M. V. Diudea, Nanomolecules and Nanostructures, Polynomials and Indices, MCM,##No. 10, Univ. Kragujevac and Fac. Sci. Kragujevac, Serbia, 2010.##5. M. V. Diudea, M. S. Florescu, and P. V. Khadikar, Molecular Topology and Its##Applications, EFICON, Bucharest, 2006.##6. H. Wiener, Structural Determination of Paraffin Boiling points, J.Am.Chem.Soc.##1947, 69, 1720.##7. I. Lukovits, The Detour Index, Croat. Chem. Acta, 1996, 69, 873882.##8. I. Lukovits and M. Razinger, On Calculation of the Detour Index,##J. Chem. Inf. Comput. Sci., 1997, 37, 283286.##9. M. V. Diudea, Cluj Matrix Invariants. J. Chem. Inf. Comput. Sci. 1997, 37, 300##10. M. V. Diudea, Cluj Matrix CJu: source of various graph descriptors, Commun.##Math. Comput. Chem. (MATCH), 1997, 35, 169183.##11. M. V. Diudea and I. Gutman, WienerType Topological Indices. Croat. Chem. Acta,##1998, 71, 2151.##12. M. V. Diudea, Valencies of Property. Croat. Chem. Acta, 1999, 72, 835851.##13. M. V. Diudea, B. Parv, and I. Gutman, DetourCluj Matrix and Derived Invariants##J. Chem. Inf. Comput. Sci. 1997, 37, 11011108.##14. C. Y. Hu and L. Xu, Algorithm for Computer Perception of Topological Symmetry.##Anal. Chim. Acta, 1994, 295, 127134.##15. G. S. Ezra, Symmetry Properties of Molecules, Lecture Notes in Chemistry 28,##Springer, 1982.##16. M. Razinger, K. Balasubramanian, and M. E. Munk, Graph Automorphism##Perception Algorithms in ComputerEnhanced Structure Elucidation. J. Chem. Inf.##Comput. Sci., 1993, 33, 197201.##17. Bonchev, D.; Balaban, A.T.; Randić, M. The Graph Center Concept for Polycyclic##Graphs, Int. J. Quantum Chem. 1981, 19, 6182.##18. Bonchev, D.; Mekenyan, O.; Balaban, A.T. Iterative Procedure for the Generalized##Graph Center in Polycyclic Graphs, J. Chem. Inf. Comput. Sci. 1989, 29, 9197.##19. M. V. Diudea, Layer Matrices in Molecular Graphs, J. Chem. Inf. Comput. Sci.##1994, 34, 10641071.##20. M. V. Diudea, M. Topan, and A. Graovac, Layer Matrices of Walk Degrees, J.##Chem. Inf. Comput. Sci. 1994, 34, 1071 1078.##21. M. V. Diudea and O. Ursu, Layer matrices and distance property descriptors.##Indian J. Chem., 42A, 2003, 12831294.##22. V. Sharma, R. Goswami, A. K. Madan, Eccentric connectivity index: A novel##highly discriminating topological descriptor for structure property and structure##activity studies, J. Chem. Inf. Comput. Sci. 37 (1997) 273282.##23. P. E. John and M. V. Diudea, The second distance matrix of the graph and its##characteristic polynomial, Carpath. J. Math., 2004, 20 (2), 235239.##24. T. Balaban, O. Mekenyan, and D. Bonchev, Unique Description of Chemical##Structures Based on Hierarchically Ordered Extended Connectivities (HOC##Procedures). I. Algorithms for Finding graph Orbits and Cannonical Numbering of##Atoms, J. Comput. Chem. 1985, 6, 538551.##25. A. T. Balaban, O. Mekenyan, and D. Bonchev, Unique Description of Chemical##Structures Based on Hierarchically Ordered Extended Connectivities (HOC##Procedures). II. Mathematical Proofs for the HOC Algorithm, J. Comput. Chem.##1985, 6, 552561.##26. O. Mekenyan, A. T. Balaban, and D. Bonchev, Unique Description of Chemical##Structures Based on Hierarchically Ordered Extended Connectivities (HOC##Procedures). VI. Condensed Benzenoid Hydrocarbons and Their 1HNMR##Chemical Shifts. J. Magn. Reson. 1985, 63, 113.##27. A.T. Balaban, A. T.; Moţoc, I.; Bonchev, D.; Mekenyan, O. Topological Indices for##Structure  Activity Correlations, Top. Curr. Chem. 1993, 114, 2155.##28. H Morgan, The generation of a unique machine description for chemical structures.##A technique developed at Chemical Abstracts Service, J. Chem. Doc. 1965, 5, 107##29. A. Ilić, I. Gutman, Eccentric Connectivity Index of Chemical Trees, MATCH##Commun. Math. Comput. Chem. 65 (2011), 731744.##30. A. Ilić, Eccentric connectivity index, in: I. Gutman, B. Furtula, Novel Molecular##Structure Descriptors Theory and Applications II, MCM 9, University of##Kragujevac, 2010.##31. G. Yu, L. Feng, A. Ilić, On the eccentric distance sum of trees and unicyclic graphs,##J. Math. Anal. Appl. 375 (2011), 934944.##32. M. V. Diudea, Centric connectivity index, Studia Univ. BabesBolyai, Chemia,##2010, 55 (4), 319324.##]
Distancebased topological indices of tensor product of graphs
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2
Let G and H be connected graphs. The tensor product G + H is a graph with vertex set V(G+H) = V (G) X V(H) and edge set E(G + H) ={(a , b)(x , y) ax ∈ E(G) & by ∈ E(H)}. The graph H is called the strongly triangular if for every vertex u and v there exists a vertex w adjacent to both of them. In this article the tensor product of G + H under some distancebased topological indices are investigated, when H is a strongly triangular graph. As a special case most of results given by Hoji, Luob and Vumara in [Wiener and vertex PI indices of Kronecker products of graphs, Discrete Appl. Math., 158 (2010), 18481855] will be deduced.
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M.
NadjafiArani
University of Kashan
University of Kashan
I R Iran
mjnajafiarani@gmail.com


H.
Khodashenas
University of Kashan
University of Kashan
I R Iran
khodashenas@kashanu.ac.ir
tensor product
Wiener type invariant
Strongly triangular graph
On the edge reverse Wiener indices of TUC4C8(S) nanotubes
2
2
The edge versions of reverse Wiener indices were introduced by Mahmiani et al. very recently. In this paper, we find their relation with ordinary (vertex) Wiener index in some graphs. Also, we compute them for trees and TUC4C8(s) naotubes.
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55
65


A.
Mahmiani
Payame Noor University
Payame Noor University
I R Iran


O.
Khormali
Tarbiat Modares University
Tarbiat Modares University
I R Iran


A.
Iranmanesh
Tarbiat Modares University
Tarbiat Modares University
I R Iran
iranmanesh@modares.ac.i
Molecular Graph
Molecular matrix
Reveres Wiener indices
Edge reverse Wiener indices
Distance of graph
line graph
Nanotubes
Computing the Szeged index of 4,4 ׳bipyridinium dendrimer
2
2
Let e be an edge of a G connecting the vertices u and v. Define two sets N1 (e  G) and N2(e G) as N1(e  G)= {xV(G) d(x,u) d(x,v)} and N2(e  G)= {xV(G) d(x,v) d(x,u) }.The number of elements of N1(e  G) and N2(e  G) are denoted by n1(e  G) and n2(e  G) , respectively. The Szeged index of the graph G is defined as Sz(G) ( ) ( ) 1 2 n e G n e G e E . In this paper we compute the Szeged index of a 4,4 ׳Bipyridinium dendrimer.
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67
72


A.
ARJOMANFAR
ShareRay Branch,Iran
ShareRay Branch,Iran
I R Iran


N.
GHOLAMI
Islamic Azad University, Izeh Branch, Khouzestan, Iran
Islamic Azad University, Izeh Branch, Khouzestan,
I R Iran
Molecular Graph
Dendrimer
Szeged index
4
4 ׳Bipyridinium
Some topological indices of graphs and some inequalities
2
2
Let G be a graph. In this paper, we study the eccentric connectivity index, the new version of the second Zagreb index and the forth geometric–arithmetic index.. The basic properties of these novel graph descriptors and some inequalities for them are established.
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73
80


M.
MOGHARRAB
Persian Gulf University, Bushehr, Iran
Persian Gulf University, Bushehr, Iran
I R Iran


B.
KHEZRI–MOGHADDAM
Payame Noor University, Shiraz,
Iran
Payame Noor University, Shiraz,
Iran
I R Iran
topological index
Eccentric connectivity
Geometric–arithmetic
Zagreb index
Cauchy–Schwarz inequality
Automatic graph construction of periodic open tubulene ((5,6,7)3) and computation of its Wiener, PI, and Szeged indices
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2
The mathematical properties of nano molecules are an interesting branch of nanoscience for researches nowadays. The periodic open single wall tubulene is one of the nano molecules which is built up from two caps and a distancing nanotube/neck. We discuss how to automatically construct the graph of this molecule and plot the graph by spring layout algorithm in graphviz and netwrokx packages. The similarity between the shape of this molecule and the plotted graph is a consequence of our work. Furthermore, the Wiener, Szeged and PI indices of this molecule are computed.
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81
94


A.
YOOSOFAN
University of Kashan,
Iran
University of Kashan,
Iran
I R Iran


M.
NAMAZI−FARD
University of Kashan,
Iran
University of Kashan,
Iran
I R Iran
Open tubulene
topological index
Szeged index
Wiener index
PI index