Reverse Wiener Index
Theorem 2. 7]) The energy of a graph cannot be the square root of an odd integer
2. Experimental 1. Data set
2.2. Molecular Descriptors
All structures of molecules were drawn with Hyperchem (ver. 7.0) program and optimized by the Am1 semi-empirical method. The drawn molecules exported in a file format suitable for MOPAC (ver. 6.0) package. MOPAC output files were used by CODESSA program to calculate constitutional, topological, geometrical, electrostatic and quantum-chemical descriptors. In addition, Hyperchem output files were used by DRAGON (ver. 3.0) package to calculate topological descriptors. Then the most relevant descriptors selected by stepwise multiple linear regression (MLR) method. This descriptors are: HOMA (Harmonic Oscillator Model of Aromaticity) index, nOH (number of hydroxyl groups) and HOMO (Highest Occupied Molecular Orbital) energy. These descriptors were used as inputs to generate ANN model.
Table 1. The data set and corresponding experimental and ANN predicted values of the redox potential at pH=7.0 (E7 (V))
No. Name E7exp E7cal No. Name E7exp E7cal
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
4- O2NphOH 4-NCphOH 3,5-Cl2phOH 4-F3CphOH 4-phOCphOH 3-NCphOH 4-IphOH 4-HOOCphOH 3-H3COCphOH 4-HphOH 4-ClphOH 4-FphOH Tyrosine 4-H3CphOH 3,5-(H3CO)2phOH 3-H3CphOH 3,5-(H3C)2phOH 4-phphOH 2-H3CphOH 4-tert-butylphOH 2-H3COphOH
1.23 1.17 1.15 1.13 1.12 1.11 1.09 1.04 0.98 0.97 0.94 0.93 0.89 0.87 0.85 0.85 0.84 0.84 0.82 0.80 0.77
1.21 1.10 1.10 1.17 1.04 1.13 1.02 1.00 1.08 0.96 0.96 0.91 0.79 0.85 0.84 0.90 0.89 0.71 0.89 0.86 0.73
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
2,6-(H3C)2phOH 2-H3CO, 4-H3CphOH 3,4-(H3CO)2phOH 3,4,5-(H3CO)3phOH 2-HO, 4-HOOCphOH 2,6-(H3CO)2phOH 2,3-(HO)2phOH
3,4-Dihydrocynnamic acid 2-HOphOH
2-HO, 4-H3CphOH 4-H2NphOH 3-O2NphOH 4-BrphOH
3-HO, 5-H3COphOH 4-H3COphOH
2,3-(HO)2, 5-H3COOCphOH 4-H3COCphOH
3-HO, 4-H3COCphOH 3-HOphOH
Sesamol 4-HOphOH
0.77 0.68 0.67 0.66 0.60 0.58 0.58 0.54 0.53 0.52 0.41 1.13 0.96 0.84 0.73 0.56 1.06 0.89 0.81 0.62 0.46
0.84 0.69 0.67 0.57 0.65 0.54 0.53 0.53 0.62 0.58 0.39 1.21 0.98 0.76 0.74 0.61 1.07 0.86 0.81 0.65 0.62
50
In the above table, compounds 33-37 and 38-42 constitute the external and internal test sets, respectively.
.3. Result and discussion
The ANN model was constructed using statistica neural networks (SNNs) (ver. 7.0).
Among many types of ANN, Multilayer perceptron was applied. We used a two-phase algorithm namely, back-propagation and levenberg-marquardt for training the network. The ANN inputs are three molecular descriptors, which were selected by stepwise MLR method, while its output signal represents the redox potential of interested molecules. The number of nodes in hidden layer, learning rate and momentum has been optimized before training the network. Table 2 shows the architecture and specification of the optimized network. The root mean square error in prediction of E7 for training, internal and external test sets are: 0.048, 0.050 and 0.051 respectively. Other statistical parameters obtained from this model are shown in table 3. These results reveal the robustness and capability of ANN based on QSPR model in prediction of redox potential of various antioxidants. Fig. 1 shows the plot of ANN calculated versus experimental values of redox potential. A good correlation for this plot confirms the ability of the ANN model in prediction of redox potential. The residuals of ANN calculated values of redox potential plotted against their experimental values in Fig. 2. The propagation of the residuals on both sides of zero line indicates that there isn't any systematic error in the developed ANN model.
Table 2. Architecture and Specification of optimized ANN model. .
Table3. Statistical parameters obtained using the ANN model.
Set R SE
Training Internal External
0.968 0.961 0.957
0.047 0.057 0.066
Parameter Value
Number of nodes in the input layer Number of nodes in the hidden layer Number of nodes in the output layer Learning rate
Momentum
3 4 1 0.4 0.5
Keywords: Quantitative s neural network
References:
1. M.S. Cooke, M.D. Evans 2. R. Bosque, J. Sales, J. Ch 3. M. Reis, B.Lobato, J. Lam
Figure 1. Plot of predic experimental values of redo ANN model.
51
structure-property relationship, Phenolic ant
ns, M. Dizdaroglu, J. Lunec, FASEB J. 17 (200 Chem. Inf. Comput. Sci. 43 (2003) 637.
ameira, A.S. Santos, C.N. Alves. J. Med. Chem dicted versus
dox potential for .
Figure 2. Plot of res experimental of redox po
antioxidant, Artificial
003) 1195.
em. 42 (2007) 440.
residual versus potential for ANN
52
Eccentric Connectivity Polynomial of an Infinite Family of Fullerenes
M. Ghasemi, M. Ghorbani and A. Azad
Abstract. The eccentric connectivity index of molecular graph G,ζc( )G ,was proposed by Sharma , Goswami and Madan. It is defined as ζc( )G =
∑
u V G∈ ( )deg ( )G u ecc u( ),wheredeg ( )G x denotes the degree of vertex x in G and ecc u( )=Max d x u
{
( , ) |x∈V G( )}
.The eccentricity connectivity polynomial of a molecular graph G defined as
( )
( , ) a V G( )deg ( )G ecc a
ECP G x a x
=
∑
∈ where ecc(a) is defined as the length of a maximal path connecting a to another vertex of G. in this paper this polynomial is computed fpr triangular benzenoid graphs.Keywords: Eccentric connectivity index
References
[1]V. Sharma, R.Goswami, A.K.Madan,j.chem. Inf. Comput.sci.,37,273(1997).
[2]B. Zhou,Z.Du, MATCH commun. Math.Comput. Chem, 63(2010)(in press).
53
The Enumeration Of IPR Hetero-Fullerenes Constructed By Leapfrog Principle
M. Ghazi and S. Shakeraneh
Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785 – 136, I R. Iran
Abstract. Hetero-fullerenes are fullerene molecules in which one or more carbon atoms are replaced by hetero-atoms such as boron or nitrogen, whose formation is a kind of “on-ball”
doping of the fullerene cage. In this paper by using the Pólya's theorem we compute the number of possible positional isomers and chiral isomers of the hetero-fullerenes.
(Received June 04, 2010; accepted July 15, 2010)
Keywords: Fullerene, Hetero-fullerene, Leapfrog Principle, Cycle Index.
Reference
[1] H. W. Kroto, J. R. Heath, S. C.O’Brien, R. F.Curl, R.E. Smalley, Nature, 318, 162 (1985).
[2] P. W. Fowler. Chem. Phys. Letters, 131, 444 (1986).
[3] P. W. Fowler, D. E., Manolopoulos, An Atlas of Fullerenes, Oxford Univ. Press, Oxford, 1995.
[4] M. Ghorbani and A. R. Ashrafi, J. Comput. Theor. Nanosci., 3, 803 (2006).
54
Computing Omega and PI Polynomials of Graphs
M. Ghazi and M. Ghorbani
Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785 – 136, I. R. Iran;mghorbani@srttu.edu
Abstract. A new counting polynomial, called Omega Ω( , )G x , was recently proposed by Diudea. It is defined on the ground of “opposite edge strips” ops. The Sadhana polynomial
( , )
Sd G x can also be calculated by ops counting. In this paper we compute these polynomials for some classes of 8 – cycle graphs.
Key words: Omega polynomial, Sadhana Polynomial, 8 -Cycles Graph.
References
1. E. Osawa, Kagaku (Kyoto), 25 (1970) 854.
2. H. Kroto, J. R. Heath, S. C. O’Brian, R. F. Curl, and R. E. Smalley, Nature (London), 318 (1985) 162.
3. W. Kraetschmer, L. D. Lamb, K. Fostiropoulos, and D. R. Huffman, Nature (London), 347 (1990) 354.
4. R. Tenne, Chem. Eur. J., 8 (2002) 5296.
55
Omega polynomial of Polyomino Chains
Mahsa Ghazi
Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785 – 136, I. R. Iran;
Abstract. A new counting polynomial, called Omega Ω( , )G x , was recently proposed by Diudea. It is defined on the ground of “opposite edge strips” ops. The Sadhana polynomial
( , )
Sd G x can also be calculated by ops counting. In this paper we compute these polynomials for 4 – cycles graph.
Key words: Omega polynomial, Sadhana Polynomial, 8 -Cycles Graph.
Reference
1.E. Osawa, Kagaku (Kyoto), 1970, 25, 854; C.A. 1971, 74, 75698v.
2.H. Kroto, J. R. Heath, S. C. O’Brian, R.F. Curl, and R. E. Smalley, Nature (London), 1985, 318, 162.
3.W. Kraetschmer, L. D. Lamb, K. Fostiropoulos, and D. R. Huffman, Nature (London), 1990, 347, 354.
4. R. Tenne, Chem. Eur. J., 2002, 8, 5296.
56
On The Wiener and Szeged Indices of Some Nano Structures
M. Ghazi and M. Ghorbani
Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785-136, I. R. Iran
Abstract. An important invariant of connected graphs is the Wiener index, which is equal to the sum of distances between all pairs of vertices of the respective graph. In this paper by using a new method introduced by klavzar we compute the Wiener and Szeged indices of some partial cube graphs.
Key words: Wiener index, Szeged index, Polyomino Chains, Triangular Benzenoid
Reference
[1] A. A. Dobrynin, R. Entringer, I. Gutman, Acta Appl. Math. 66 (2001) 211–249.
[2] A. A. Dobrynin, L. S. Mel’nikov, MATCH Commun. Math. Comput. Chem. 50 (2004) 146–164.
[3] I. Gutman, Indian J. Chem. 36A (1997) 128–132.
[4] S. Klavžar, I. Gutman, Discrete Appl. Math. 80 (1997) 73–81.
[5] H. Y. Zhu, D. J. Klein, I. Lukovits, J. Chem. Inf. Comput. Sci. 36 (1996) 420–428.
[6 H. Wiener, J. Amer. Chem. Soc. 69 (1947) 17–20
57
Computing ABC Index of Nanostar Dendrimers
Fatemeh Gheshlaghi and Modjtaba Ghorbani
Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785-136, I. R. Iran
Abstract. The ABC index is a topological index was defined as
d ( ) d ( ) 2
( ) d ( ) d ( )
G G
uv E
G G
u v
A BC G
u v
∈
+ −
=∑ , where d ( )G u denotes degree of vertex u. In this paper we compute this new topological index for C4C8 nanotubes.
Key words: Topological Index, ABC Index, Nanotubes.
References
[1] E. Estrada, Chem. Phys. Lett. 463, 422 (2008).
[2] E. Estrada, L. Torres, L. Rodríguez, I. Gutman, Indian J. Chem. 37A, 849 (1998).
[3] R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, Weinheim, 2000.
[4] M. Randić, J. Am. Chem. Soc., 97, 6609 (1975).
[5] B. Furtula, A. Graovac, D. Vukičević, Disc. Appl. Math., 157, 2828 (2009).
58
[6] N. Trinajstić , Chemical Graph Theory, CRC Press, Boca Raton, FL, 1992.
[7] A. Graovac and M. Ghorbani, Acta Chim. Slov., 57 (3), (2010) Accepted.
[8] I. Gutman, M. Ghorbani and M. A. Hosseinzadeh, Kragujevac J. Sci., 32, 47 (2010).
[9] M. A. Hosseinzadeh and M. Ghorbani, Optoelectron. Adv. Mater.- Rapid Comm, 4(3), 378 (2010).
Hosoy Azam Giahtaze
Departme
Abstract. In the fields o topological index also kno that is calculated based on indices are numerical param graph invariant. Topologi quantitative structure-activ other properties of molecule The Hosoya polynomial of H(G,x) = dG(u,v) distance between a pair o chemical applications [1,2 index and hyper-Wiener ind The Wiener index of a con all pairs of vertices in G [3]
The hyper-Wiener index of
59
oya Polynomials Of TUC4C8(R) Nanotours zeh, Neda Sari, Leila Jadidolslam and Ali Iranm
ent of Mathematics, Tarbiat Modares Universi P. O. Box 14115-137, Tehran, Iran
a.giahtazeh@modares.ac.ir
of chemical graph theory and in mathema nown as a connectivity index is a type of a m on the molecular graph of a chemical comp
ameters of a graph which characterize its topol gical indices are used for example in th tivity relationships (QSARs) in which the bio
ules are correlated with their chemical structure of a connected graph G, denoted by H (G, x) an
(1) where V(G) is the set of vertices of G of vertices u and v in G. The Hosoya pol
,2], especially two well-known topological i index, can be directly obtained from this polyn onnected graph G ,W(G) is equal to the sum o
]. That is
W (G) = (2)
of gragh G ,WW(G) is defined as [4]:
anmanesh
rsity
matical chemistry, a molecular descriptor mpound. .Topological ology and are usually the development of biological activity or ure.
and is defined as:
G and d (u , v) is the olynomial has many l indices, i:e; Wiener
ynomial.
of distances between
WW(G) = ( From Eqs (1) and (2) the fi Wiener index, i.e;
Also, from Eqs (1,3) the fol W We described TUC4C8(R) n [p, q] which p is the numbe The main result of this pape Theorem: If 1 ≤ q ≤
H(p,q,x)=pq(
-
where a= ,
Keywords: Hosoya polyno
60
) first derivative of the Hosoya polynomial at x
W(G) = H'(G,x) (4) following relation holds:
WW(G) = (5)
) nanotorus by two parameters p and q and simp ber of rhombs on the level 1 and the length of to
per, is the following theorem:
, then:
– )
.
nomial, Wiener index, hyper-Wiener index, TU ) (3)
t x = 1 is equal to the
mply denoted by T f torus is q.
UC4C8(R) nanotorus.
61 References
[1] I. Gutman, S. Klavzar, M. Petkovsek, P. Zigert, MATCH Commun. Math. Comput.
Chem. 43, (2001), 49.
[2] E. Estrada, O. Ivancius, I. Gutman, A. Gutierrez, L. Rodriguez, New J. Chem., 22, (1998), 819.
[3] H. Hosoya, Bull. Chem. Soc. Jpn., 44, (1971), 2332.
[4] D. J. Kelin, I. Lukovist, I. Gutman, J. Chem. Inf. Comput. Sci., 35, (1995), 50.
62
Computing Eccentric Connectivity Polynomial Of Fullerenes
M. Ghorbani and M. A Iranmanesh
Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785 – 136, I. R. Iran;
Department of Mathematics, Faculty of Science, Yazd University, Yazd, I. R. Iran
Abstract The eccentricity connectivity polynomial of a molecular graph G is defined as ξ(x)= Σa∈V(G)xε(a), where ε(a) is defined as the length of a maximal path connecting a to another vertex of G. Fullerenes are 3 cubic graphs with exactly 12 pentagonal faces. In this paper this polynomial is computed for an infinite family of fullerenes.
Keywords: Eccentricity Connectivity Polynomial, Eccentricity Connectivity Index, Fullerene Graphs.
References
1. Sharma, V.; Goswami, R.; Madan, A. K. J. Chem. Inf. Comput. Sci. 1997, 37, 273 - 282.
2. Zhou, B.; Du, Z. MATCH Commun. Math. Comput. Chem. 2010, 63, in press.
3. Dobrynin, A.; Kochetova, A. J. Chem., Inf., Comput. Sci. 1994, 34, 1082 - 1086.
4. Gutman, I. J. Chem. Inf. Comput. Sci. 1994, 34, 1087 - 1089.
63
5. Gutman, I.; Polansky O. E. Mathematical Concepts in Organic Chemistry, Springer- Verlag, New York, 1986.
6. Johnson, M. A.; Maggiora, G. M. Concepts and Applications of Molecular Similarity, Wiley Interscience, New York, 1990.
7. Kroto, H. W.; Heath, J. R.; O’Brien, S. C.; Curl, R. F.; Smalley, R.E. Nature, 1985, 318 162 - 163.
8. Kroto, H. W.; Fichier, J. E.; Cox, D. E. The Fulerene, Pergamon Press, New York, 1993.
64
On Omega and Sadhana Polynomials Of Leapfrog Fullerenes
F
363n×
Modjtaba Ghorbani and Mohsen Jaddi
Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785 – 136, I. R. Iran;
Department of Mathematics, Payame Noor University (PNU), Aran&Bidgol, 87415141, I. R. Iran;
Abstract. A leapfrog transform Gl of G is a graph on 3n vertices obtained by truncating the dual of G. Hence, Gl= Tr(G*), where G* denotes the dual of G. It is easy to check that Gl itself is a fullerene graph. In this paper, the Omega and Sadhana polynomials of a new infinite class of Leapfrog fullerenes are computed for the first time.
Key words: Omega Polynomial, Sadhana Polynomial, Fullerene Graph, Leapfrog Fullerene.
References
[1] H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl, R.E. Smalley, Nature, 318, 162 (1985).
[2] H. W. Kroto, J. E. Fichier, D. E Cox, The Fulerene, Pergamon Press, New York, 1993.
65