Advances in Materials Science and Engineering

Advances in Materials Science and Engineering / 2019 / Article

Research Article | Open Access

Volume 2019 |Article ID 9062535 | 9 pages | https://doi.org/10.1155/2019/9062535

Eccentric Connectivity Index of t-Polyacenic Nanotubes

Academic Editor: Yuning Li
Received06 Oct 2018
Revised12 Dec 2018
Accepted10 Jan 2019
Published11 Feb 2019

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 well-known indices in case of anticonvulsant, anti-inflammatory, and diuretic activities. The ECI of an infinite class of 1-polyacenic (phenylenic) nanotubes has been recently studied. In this article, we extend this study to generalized polyacenic nanotubes and find ECI of t-polyacenic 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 structure-property and structure-activity relationships (QSPR/QSAR), the topological indices are very helpful in detecting the biological activities of a chemical compound [14].

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: first-generation 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]. Second-generation 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]. Third-generation 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 hyper-Wiener index [5], the Kirchhoff index [7], and electrotopological state indices [2]. Recently, fourth- and fifth-generation topological indices are placed as new generations topological indices. Fourth-generation topological indices are of highly discriminating power, i.e., . The examples of fourth-generation topological indices include eccentric connectivity index [8], superaugmented eccentric connectivity index [9], and superaugmented eccentric connectivity topochemical indices [10]. Detour matrix-based adjacent path eccentric distance sum indices [11] belong to the fifth-generation 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 distance-based chemical structure descriptor, called the eccentric connectivity index (ECI), which is presented as

It is recorded in [1216] 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 anti-inflammatory 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 [2126] and for several molecular graphs [2729].

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 [3033]. In the first organic solid-state injection laser, the lasing was discovered by using the single crystals of tetracene [34, 35]. They have application in rechargeable Li-ion 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.

Recently, the Zagreb indices of 3-polyacenic (anthracenic), 4-polyacenic (tetracenic), and 5-polyacenic (pentacenic) nanotubes have been studied in [3941], respectively. The ECI of 1-polyacenic (phenylenic) nanotubes has been presented in [25]. In this paper, we generalize these results to t-polyacenic nanotubes for and present the ECI for these nanotubes.

2. Main Results

The generalized molecular graph of the t-polyacenic 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 t-polyacenic nanotube, where a polyacene unit consists of t hexagons. The molecular graph of the t-polyacenic nanotube has rows and q columns. For , the t-polyacenic 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:

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 .


Vertex partitionRowsCardinality for each row


Theorem 1. Let be the graph of the t-polyacenic 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 t-polyacenic 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 t-polyacenic 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 anti-inflammatory 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 t-polyacenic 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

  1. J. V. de Julián-Ortiz, C. de Gregorio Alapont, I. Rı́os-Santamarina, R. Garcı́a-Doménech, 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
  2. 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
  3. J.-B. Liu, X.-F. Pan, F.-T. Hu, and F.-F. Hu, “Asymptotic Laplacian-energy-like invariant of lattices,” Applied Mathematics and Computation, vol. 253, pp. 205–214, 2015. View at: Publisher Site | Google Scholar
  4. L. Pogliani, “Modeling enthalpy and hydration properties of inorganic compounds,” Croatica Chemica Acta, vol. 3, pp. 803–817, 1997. View at: Google Scholar
  5. 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
  6. E. S. Soofi and J. J. Retzer, “Information indices: unification and applications,” Journal of Econometrics, vol. 107, no. 1-2, pp. 17–40, 2002. View at: Publisher Site | Google Scholar
  7. 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
  8. 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
  9. H. Dureja and A. K. Madan, “Superaugmented eccentric connectivity indices: new-generation 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
  10. H. Dureja, S. Gupta, and A. K. Madan, “Predicting anti-HIV-1 activity of 6-arylbenzonitriles: 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
  11. M. Singh, H. Jangra, P. V. Bharatam, and A. K. Madan, “Detour matrix-based 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
  12. A. R. Ashrafi and M. Ghorbani, “Eccentricity connectivity index,” in Novel Molecular Structure Descriptors-Theory and Applications II, I. Gutman and B. Furtula, Eds., pp. 169–182, University of Kragujevac, Kragujevac, Serbia, 2010. View at: Google Scholar
  13. T. Došliś and M. Saheli, “Eccentricity connectivity index of fullerenes,” in Novel Molecular Structure Descriptors-Theory and Applications II, I. Gutman and B. Furtula, Eds., pp. 183–192, University of Kragujevac, Kragujevac, Serbia, 2010. View at: Google Scholar
  14. A. Ilić, “Eccentricity connectivity index of benzenoid graphs,” in Novel Molecular Structure Descriptors-Theory and Applications II, I. Gutman and B. Furtula, Eds., pp. 139–168, University of Kragujevac, Kragujevac, Serbia, 2010. View at: Google Scholar
  15. A. K. Madan and H. Dureja, “Eccentricity based descriptors for QSAR/QSPR,” in Novel Molecular Structure Descriptors-Theory and Applications II, I. Gutman and B. Furtula, Eds., pp. 91–138, University of Kragujevac, Kragujevac, Serbia, 2010. View at: Google Scholar
  16. A. K. Madan and H. Dureja, “Applications of eccentricity connectivity index,” in Novel Molecular Structure Descriptors-Theory and Applications II, I. Gutman and B. Furtula, Eds., pp. 247–268, University of Kragujevac, Kragujevac, Serbia, 2010. View at: Google Scholar
  17. V. Kumar, S. Sardana, and A. K. Madan, “Predicting anti-HIV activity of 2,3-diaryl-1,3-thiazolidin-4-ones: computational approach using reformed eccentric connectivity index,” Journal of Molecular Modeling, vol. 10, no. 5-6, pp. 399–407, 2004. View at: Publisher Site | Google Scholar
  18. 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
  19. S. Gupta, M. Singh, and A. K. Madan, “Application of graph theory: relationship of eccentric connectivity index and wiener’s index with anti-inflammatory activity,” Journal of Mathematical Analysis and Applications, vol. 266, no. 2, pp. 259–268, 2002. View at: Publisher Site | Google Scholar
  20. S. Sardana and A. K. Madan, “Predicting anticonvulsant activity of benzamides/benzylamines: computational approach using topological descriptors,” Journal of Computer-Aided Molecular Design, vol. 16, no. 8-9, pp. 545–550, 2002. View at: Publisher Site | Google Scholar
  21. A. R. Ashrafi, T. Došlić, and M. Saheli, “The eccentric connectivity index of TUC4C8(R) nanotubes,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 65, no. 1, pp. 221–230, 2011. View at: Google Scholar
  22. 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
  23. A. Iranmanesh and Y. Alizadeh, “Eccentric connectivity index of HAC5C7[p, q] and nHAC5C6C7[p, q] anotubes,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 69, pp. 175–182, 2013. View at: Google Scholar
  24. I. Nadeem and H. Shaker, “On eccentric connectivity index of TiO2 nanotubes,” Acta Chimica Slovenica, vol. 63, no. 2, pp. 363–368, 2016. View at: Publisher Site | Google Scholar
  25. N. P. Rao and K. L. Lakshmi, “Eccentric connectivity index of V-phenylenic nanotubes,” Digest Journal of Nanomaterials and Biostructures, vol. 6, no. 1, pp. 81–87, 2010. View at: Google Scholar
  26. 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
  27. 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
  28. 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
  29. 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
  30. 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
  31. 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
  32. 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
  33. 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
  34. 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
  35. 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
  36. 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
  37. L. Biennier, M. Alsayed-Ali, A. Foutel-Richard 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
  38. 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
  39. 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
  40. N. Soleimani, M. J. Nikmehr, and H. A. Tavallaee, “Theoretical study of nanostructures using topological indices,” Studia Universitatis Babes-Bolyai Chemia, vol. 59, no. 4, pp. 139–148, 2014. View at: Google Scholar
  41. 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

Copyright © 2019 Jia-Bao 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.


More related articles

885 Views | 279 Downloads | 0 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.