Research Article  Open Access
Eccentric Connectivity Index of tPolyacenic Nanotubes
Abstract
The eccentric connectivity index ECI is a chemical structure descriptor that is currently being used for the modeling of biological activities of a chemical compound. This index has been proved to provide a high degree of predictability as compared to some other wellknown indices in case of anticonvulsant, antiinflammatory, and diuretic activities. The ECI of an infinite class of 1polyacenic (phenylenic) nanotubes has been recently studied. In this article, we extend this study to generalized polyacenic nanotubes and find ECI of tpolyacenic nanotubes for .
1. Introduction
A basic concept of chemistry is that the properties/activities of a molecule depend upon its structural characteristics. Molecular graphs can be used to model the chemical structures of molecules and molecular compounds, by considering atoms as vertices and the chemical bonds between the atoms as edges. In the study of quantitative structureproperty and structureactivity relationships (QSPR/QSAR), the topological indices are very helpful in detecting the biological activities of a chemical compound [1–4].
A topological index is a numerical graph invariant that is used to correlate the chemical structure of a molecule with its physicochemical properties and biological activities. Generally, topological indices are classified into five generations: firstgeneration topological indices are integer numbers obtained by simple operations from local vertex invariants involving only one vertex at a time. Some of the famous topological indices of this class are Wiener index, Hosoya index, and Centric indices of Balaban [5]. Secondgeneration topological indices are real numbers based on integer graph properties. These indices were obtained via structural operations from integer local vertex invariants, involving more than one vertex at a time. Some examples of this class include molecular connectivity indices, Balaban J index, bond connectivity indices, and kappa shape indices [5]. Thirdgeneration topological indices are real numbers which are based on local properties of the molecular graph. These indices are of recent introduction and have very low degeneracy. These are based on information theory applied to the terms of distance sums or on newly introduced nonsymmetrical matrices. Some examples include information indices [6], the hyperWiener index [5], the Kirchhoff index [7], and electrotopological state indices [2]. Recently, fourth and fifthgeneration topological indices are placed as new generations topological indices. Fourthgeneration topological indices are of highly discriminating power, i.e., . The examples of fourthgeneration topological indices include eccentric connectivity index [8], superaugmented eccentric connectivity index [9], and superaugmented eccentric connectivity topochemical indices [10]. Detour matrixbased adjacent path eccentric distance sum indices [11] belong to the fifthgeneration topological indices.
Let G be a connected molecular graph with vertex set and edge . Let be the set of those edges of G that are incident to a vertex , and then the degree of k is denoted by and is defined as the cardinality of . The distance from a vertex to a vertex is denoted by and is defined as the minimum number of edges lying between them. The eccentricity of a given vertex is defined as the largest distance between k and any vertex l of G.
Sharma et al. in [8] have presented a distancebased chemical structure descriptor, called the eccentric connectivity index (ECI), which is presented as
It is recorded in [12–16] that ECI provides good correlations with regard to physicochemical properties and biological activities. This index is reported as a highly discriminating descriptor for QSPR/QSAR studies [8, 9, 17]. The degree of prediction of ECI is better than the Wiener index in case of diuretic activity [18] and antiinflammatory activity in [19]. Also, this index has been proved to provide a high degree of predictability with regard to anticonvulsant activity [20] in comparison to Zagreb indices. Recently, the eccentric connectivity index has been studied for certain nanotubes [21–26] and for several molecular graphs [27–29].
Polyacenes relate to a family of polycyclic aromatic hydrocarbon (PAH) compounds which are formed by the linearly fused benzene rings. Numerous molecules of this class have interesting optical, thermodynamic, electronic, ferromagnetic, and photoconductive properties [30–33]. In the first organic solidstate injection laser, the lasing was discovered by using the single crystals of tetracene [34, 35]. They have application in rechargeable Liion batteries [36] and also have presence in various celestial objects like planetary nebulae [37]. In this sense, the polyacenes have received much attention. The index of linear polyacenes has been studied in [38]. The molecular graphs of certain linear polyacene molecules are given in Figure 1.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Recently, the Zagreb indices of 3polyacenic (anthracenic), 4polyacenic (tetracenic), and 5polyacenic (pentacenic) nanotubes have been studied in [39–41], respectively. The ECI of 1polyacenic (phenylenic) nanotubes has been presented in [25]. In this paper, we generalize these results to tpolyacenic nanotubes for and present the ECI for these nanotubes.
2. Main Results
The generalized molecular graph of the tpolyacenic nanotube is shown in Figure 2. In this graphical representation, q counts the number of polyacene units in a row and p counts the number of alternative polyacene units in a column of the tpolyacenic nanotube, where a polyacene unit consists of t hexagons. The molecular graph of the tpolyacenic nanotube has rows and q columns. For , the tpolyacenic nanotube is known as phenylenic, naphthalenic, anthacenic, tetracenic, pentacenic, and hexacenic nanotubes, respectively. The molecular graphs of these nanotubes are presented in Figure 3. Let G be a molecular graph of the nanotube and then we can observe that for each . So, we have the vertex partitions of G as follows:
(a)
(b)
(c)
(d)
(e)
(f)
The vertex partitions of G along with their cardinalities corresponding to each row are presented in Table 1. In the following theorems, we formulate the eccentric connectivity index for nanotubes for .

Theorem 1. Let be the graph of the tpolyacenic nanotube, and then for q even, we have
Proof. Consider . Let represents the vertices in the row. With respect to or , we have the following cases.
Case 1 (when and ). In this case, the eccentricity of each vertex in each row is . Hence, from Table 1 and (1), we have
Case 2 (when and ). In this case,where . Hence, from Table 1 and (1), we have
Case 3 (when and ). In this case,Also,where . Hence, from Table 1 and (1), we have
Case 4 (when , is even and ). In this case,Hence, from Table 1 and (1), we have
Case 5 (when , is odd and ). In this case, we use the eccentricities of vertices as given in case 4. From Table 1 and (1), we have
Theorem 2. Let be the graph of the tpolyacenic nanotube, and then for q odd, we have
Proof. Consider . Let represent the vertices in the row of G. With respect to or , we have the following cases.
Case 1 (when , and ). In this case, the eccentricity of each vertex in each row is . Hence, from Table 1 and (1), we have
Case 2 (when , and ). In this case, the eccentricity of each vertex in each row is . Hence, from Table 1 and (1), we have
Case 3 (when and ). In this case,where . Hence, from Table 1 and (1), we have
Case 4 (when and is even). In this case,Also,where . Hence, from Table 1 and (1), we have
Case 5 (when and is odd). In this case,Also,where . Hence, from Table 1 and (1), we have
Case 6 (when and is odd). In this case,Also,where . Hence, from Table 1 and (1), we have
Case 7 (when and is odd). In this case,Also,Hence, from Table 1 and (1), we have
Case 8 (when , is even and ). In this case,Hence, from Table 1 and (1), we have
Case 9 (when , is odd and ). In this case, we use the eccentricities of vertices as given in case 8. From Table 1 and (1), we have
Remark 1. The results presented by Rao and Lakshmi in [25] become special cases of the results given in Theorems 1 and 2 for .
3. Conclusion
In this paper, we present generalized formulae of ECI for tpolyacenic nanotubes. The comparability about biological activities of chemical compounds is of immense interest in QSAR/QSPR studies. The eccentric connectivity index ECI provides the best prediction accuracy rate compared to other indices in various biological activities of diverse nature such as antiinflammatory activity, anticonvulsant activity, and diuretic activity. In this sense, this index can be very helpful in QSAR/QSPR studies, and by using the given results, we can present mathematical models of several biological activities of all chemical compounds, which correspond to tpolyacenic nanotubes such as phenylenic nanotubes, naphthalenic nanotubes, anthracenic nanotubes, tetracenic nanotubes, pentacenic nanotubes, and hexacenic nanotubes.
Data Availability
All data generated or analyzed during this study are included in this article.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
The authors would like to express their sincere gratitude to the anonymous referees and the editor for many valuable, friendly, and helpful suggestions, which led to a great deal of improvement of the original manuscript. This work was done under the project supported by the Higher Education Commission, Pakistan, via Grant no. 5331/Federal/NRPU/R&D/HEC/2016. This research was funded by the China Postdoctoral Science Foundation under Grant no. 2017M621579, the Postdoctoral Science Foundation of Jiangsu Province under Grant no. 1701081B, and Project of Anhui Jianzhu University under Grant nos. 2016QD116 and 2017dc03.
References
 J. V. de JuliánOrtiz, C. de Gregorio Alapont, I. Rı́osSantamarina, R. Garcı́aDoménech, and J. Gálvez, “Prediction of properties of chiral compounds by molecular topology,” Journal of Molecular Graphics and Modelling, vol. 16, no. 1, pp. 14–18, 1998. View at: Publisher Site  Google Scholar
 L. B. Kier and L. H. Hall, “An electrotopological state index for atoms in molecules,” Pharmaceutical Research, vol. 7, no. 8, pp. 801–807, 1990. View at: Publisher Site  Google Scholar
 J.B. Liu, X.F. Pan, F.T. Hu, and F.F. Hu, “Asymptotic Laplacianenergylike invariant of lattices,” Applied Mathematics and Computation, vol. 253, pp. 205–214, 2015. View at: Publisher Site  Google Scholar
 L. Pogliani, “Modeling enthalpy and hydration properties of inorganic compounds,” Croatica Chemica Acta, vol. 3, pp. 803–817, 1997. View at: Google Scholar
 Q.N. Hu, Y.Z. Liang, and K.T. Fang, “The matrix expression, topological index and atomic attribute of molecular topological structure,” Journal of Data Science, vol. 1, pp. 361–389, 2003. View at: Google Scholar
 E. S. Soofi and J. J. Retzer, “Information indices: unification and applications,” Journal of Econometrics, vol. 107, no. 12, pp. 17–40, 2002. View at: Publisher Site  Google Scholar
 J.B. Liu and X.F. Pan, “Minimizing Kirchhoff index among graphs with a given vertex bipartiteness,” Applied Mathematics and Computation, vol. 291, pp. 84–88, 2016. View at: Publisher Site  Google Scholar
 V. Sharma, R. Goswami, and A. K. Madan, “Eccentric connectivity index: a novel highly discriminating topological descriptor for structure–property and structure–activity studies,” Journal of Chemical Information and Computer Sciences, vol. 37, no. 2, pp. 273–282, 1997. View at: Publisher Site  Google Scholar
 H. Dureja and A. K. Madan, “Superaugmented eccentric connectivity indices: newgeneration highly discriminating topological descriptors for QSAR/QSPR modeling,” Medicinal Chemistry Research, vol. 16, no. 7–9, pp. 331–341, 2007. View at: Publisher Site  Google Scholar
 H. Dureja, S. Gupta, and A. K. Madan, “Predicting antiHIV1 activity of 6arylbenzonitriles: computational approach using superaugmented eccentric connectivity topochemical indices,” Journal of Molecular Graphics and Modelling, vol. 26, no. 6, pp. 1020–1029, 2008. View at: Publisher Site  Google Scholar
 M. Singh, H. Jangra, P. V. Bharatam, and A. K. Madan, “Detour matrixbased adjacent path eccentric distance sum indices for QSAR/QSPR. Part I: development and evaluation,” International Journal of Computational Biology and Drug Design, vol. 7, no. 4, pp. 295–318, 2014. View at: Publisher Site  Google Scholar
 A. R. Ashrafi and M. Ghorbani, “Eccentricity connectivity index,” in Novel Molecular Structure DescriptorsTheory and Applications II, I. Gutman and B. Furtula, Eds., pp. 169–182, University of Kragujevac, Kragujevac, Serbia, 2010. View at: Google Scholar
 T. Došliś and M. Saheli, “Eccentricity connectivity index of fullerenes,” in Novel Molecular Structure DescriptorsTheory and Applications II, I. Gutman and B. Furtula, Eds., pp. 183–192, University of Kragujevac, Kragujevac, Serbia, 2010. View at: Google Scholar
 A. Ilić, “Eccentricity connectivity index of benzenoid graphs,” in Novel Molecular Structure DescriptorsTheory and Applications II, I. Gutman and B. Furtula, Eds., pp. 139–168, University of Kragujevac, Kragujevac, Serbia, 2010. View at: Google Scholar
 A. K. Madan and H. Dureja, “Eccentricity based descriptors for QSAR/QSPR,” in Novel Molecular Structure DescriptorsTheory and Applications II, I. Gutman and B. Furtula, Eds., pp. 91–138, University of Kragujevac, Kragujevac, Serbia, 2010. View at: Google Scholar
 A. K. Madan and H. Dureja, “Applications of eccentricity connectivity index,” in Novel Molecular Structure DescriptorsTheory and Applications II, I. Gutman and B. Furtula, Eds., pp. 247–268, University of Kragujevac, Kragujevac, Serbia, 2010. View at: Google Scholar
 V. Kumar, S. Sardana, and A. K. Madan, “Predicting antiHIV activity of 2,3diaryl1,3thiazolidin4ones: computational approach using reformed eccentric connectivity index,” Journal of Molecular Modeling, vol. 10, no. 56, pp. 399–407, 2004. View at: Publisher Site  Google Scholar
 S. Sardana and A. K. Madan, “Application of graph theory: relationship of molecular connectivity index, Wiener’s index and eccentric connectivity index with diuretic activity,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 43, pp. 85–98, 2011. View at: Google Scholar
 S. Gupta, M. Singh, and A. K. Madan, “Application of graph theory: relationship of eccentric connectivity index and wiener’s index with antiinflammatory activity,” Journal of Mathematical Analysis and Applications, vol. 266, no. 2, pp. 259–268, 2002. View at: Publisher Site  Google Scholar
 S. Sardana and A. K. Madan, “Predicting anticonvulsant activity of benzamides/benzylamines: computational approach using topological descriptors,” Journal of ComputerAided Molecular Design, vol. 16, no. 89, pp. 545–550, 2002. View at: Publisher Site  Google Scholar
 A. R. Ashrafi, T. Došlić, and M. Saheli, “The eccentric connectivity index of TUC_{4}C_{8}(R) nanotubes,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 65, no. 1, pp. 221–230, 2011. View at: Google Scholar
 A. R. Ashrafi, M. Saheli, and M. Ghorbani, “The eccentric connectivity index of nanotubes and nanotori,” Journal of Computational and Applied Mathematics, vol. 235, no. 16, pp. 4561–4566, 2011. View at: Publisher Site  Google Scholar
 A. Iranmanesh and Y. Alizadeh, “Eccentric connectivity index of HAC_{5}C_{7}[p, q] and nHAC_{5}C_{6}C_{7}[p, q] anotubes,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 69, pp. 175–182, 2013. View at: Google Scholar
 I. Nadeem and H. Shaker, “On eccentric connectivity index of TiO_{2} nanotubes,” Acta Chimica Slovenica, vol. 63, no. 2, pp. 363–368, 2016. View at: Publisher Site  Google Scholar
 N. P. Rao and K. L. Lakshmi, “Eccentric connectivity index of Vphenylenic nanotubes,” Digest Journal of Nanomaterials and Biostructures, vol. 6, no. 1, pp. 81–87, 2010. View at: Google Scholar
 M. Saheli and A. R. Ashrafi, “The eccentric connectivity index of armchair polyhex nanotubes,” International Journal of Chemistry and Chemical Engineering, vol. 29, no. 1, pp. 71–75, 2010. View at: Google Scholar
 A. Ilić and I. Gutman, “Eccentric connectivity index of chemical trees,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 65, pp. 731–744, 2011. View at: Google Scholar
 M. J. Morgan, S. Mukwembi, and H. C. Swart, “On the eccentric connectivity index of a graph,” Discrete Mathematics, vol. 311, no. 13, pp. 1234–1299, 2011. View at: Publisher Site  Google Scholar
 B. Zhou and Z. Du, “On eccentric connectivity index,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 63, pp. 181–198, 2010. View at: Google Scholar
 J. E. Anthony, “Functionalized acenes and heteroacenes for organic electronics,” Chemical Reviews, vol. 106, no. 12, pp. 5028–5048, 2006. View at: Publisher Site  Google Scholar
 M. Bendikov, F. Wudl, and D. F. Perepichka, “Tetrathiafulvalenes, oligoacenenes, and their buckminsterfullerene derivatives: the brick and mortar of organic electronics,” Chemical Reviews, vol. 104, no. 11, pp. 4891–4946, 2004. View at: Publisher Site  Google Scholar
 J. L. Bredas, J. P. Calbert, D. A. da Silva Filho, and J. Cornil, “Organic semiconductors: a theoretical characterization of the basic parameters governing charge transport,” Proceedings of the National Academy of Sciences, vol. 99, no. 9, pp. 5804–5809, 2002. View at: Publisher Site  Google Scholar
 R. Firouzi and M. Zahedi, “Polyacenes electronic properties and their dependence on molecular size,” Journal of Molecular Structure: THEOCHEM, vol. 862, no. 1–3, pp. 7–15, 2008. View at: Publisher Site  Google Scholar
 J. H. Schon, C. Kloc, A. Dodabalapur, and B. Batlogg, “An organic solid state injection laser,” Science, vol. 289, no. 5479, pp. 599–601, 2000. View at: Publisher Site  Google Scholar
 J. H. Schön, C. Kloc, and B. Batlogg, “Retraction note to: superconductivity in molecular crystals induced by charge injection,” Nature, vol. 406, no. 6797, pp. 702–704, 2000. View at: Publisher Site  Google Scholar
 T. Yamabe, S. Yata, and S. Wang, “The structures and properties of conjugated hydrocarbons such as polyacenic materials and polycyclic aromatic hydrocarbons (PAHs) doped with lithium,” Synthetic Metals, vol. 137, no. 1–3, pp. 949–951, 2003. View at: Publisher Site  Google Scholar
 L. Biennier, M. AlsayedAli, A. FoutelRichard et al., “Laboratory measurements of the recombination of PAH ions with electrons: implications for the PAH charge state in interstellar clouds,” Faraday Discussions, vol. 133, pp. 289–301, 2006. View at: Publisher Site  Google Scholar
 P. V. Khadikar, S. Karmarkar, and R. G. Varma, “On the estimation of PI index of polyacenes,” Acta Chimica Slovenica, vol. 49, pp. 755–771, 2002. View at: Google Scholar
 N. Soleimani, M. J. Nikmehr, and H. A. Tavallaee, “Computation of the different topological indices of nanostructures,” Journal of the National Science Foundation of Sri Lanka, vol. 43, no. 2, pp. 127–133, 2015. View at: Publisher Site  Google Scholar
 N. Soleimani, M. J. Nikmehr, and H. A. Tavallaee, “Theoretical study of nanostructures using topological indices,” Studia Universitatis BabesBolyai Chemia, vol. 59, no. 4, pp. 139–148, 2014. View at: Google Scholar
 M. Veylaki and M. J. Nikmehr, “Some degree based topological indices of nanostructures,” Bulgarian Chemical Communications, vol. 47, no. 3, pp. 872–875, 2015. View at: Google Scholar
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Copyright © 2019 JiaBao Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.