Abstract
The complex network models have been widely used to understand mathematical characteristics of Titanium Dioxide Nanotube. Titanium Dioxide Nanotube is a well-known semiconductor and has several industrial and scientific applications. A nanostructure belong to a significant and an extensively investigated compounds in chemical science. It has been derived through engineering mechanism at the molecular scale. The most significant of these new materials are single-walled Titanium Dioxide Nanotube (SWTNT). They have remarkable electronic properties and many other unique characteristics. To compute and study topological indices of nanostructures is a respected problem in nanotechnology. Quantitative structure-property and structure-activity relationships of the single-walled Titanium Dioxide Nanotube (SWTNT) compounds necessitate expressions for the molecular topological features of these compounds. Topological indices are vital devices for investigating chemical compounds to comprehend the fundamental topology of chemical structures. Ev-degree and ve-degree based topological indices are two novel degrees based indices as of late defined in graph theory. Ev-degree and ve-degree based topological indices have been defined as corresponding to their relating partners. In this paper, we have computed topological indices based on ev-degree and ve-degree for the two dimensional lattice of three-layered single-walled Titanium Dioxide Nanotube (SWTNT).
Graphic Abstract
Titanium nanotube is a well-known semiconductor and has several industrial and scientific applications. A nanostructure belong to a significant and an extensively investigated compounds in chemical science. It has been derived through engineering mechanism at the molecular scale. Quantitative structure-property and structure-activity relationships of the single-walled Titanium nanotubes (SWTNT) compounds necessitate expressions for the molecular topological features of these compounds. Ev-degree and ve-degree based topological indices have been defined as corresponding to their relating partners. In this paper, we have computed topological indices based on ev-degree and ve-degree for the two dimensional lattice of three-layered single-walled Titanium nanotubes (SWTNT).
Similar content being viewed by others
References
Z. Shao, P. Wu, X. Zhang, D. Dimitrov and J. B. Liu (2018). IEEE Access 6, 27604–27616.
Z. Shao, P. Wu, Y. Gao, I. Gutman and X. Zhang (2017). Appl. Math. Comput. 315, 298–312.
C. A. Grimes and K. G. Mor TIO2 Nanotube Arrays: Synthesis, Properties, and Applications (Springer Science & Business Media, Berlin, 2009).
L. Menon, et al. (2012). J. Nanosci. Nanotechnol. 12, 7658–7676.
D. B. Strukov, G. S. Snider, D. R. Stewart and R. S. Williams (2008). Nature 453, 80–93.
L. Zhong (2012). Appl. Math. Lett. 25, (3), 561–566.
M. Chellali, T. W. Haynes, S. T. Hedetniemi and T. M. Lewis (2017). Discrete Math. 340, (2), 31–38.
B. Horoldagva, K. C. Das and T. A. Selenge (2019). Discrete Optim. 31, 1–7.
S. Ediz (2017). Celal Bayar Univ. J. Sci. 13, (3), 615–618.
B. Sahin and S. Ediz (2018). Iran. J. Math. Chem. 9, (4), 263–277.
S. Ediz (2018). Int. J. Comput. Sci. Math. 9, (1), 1–12.
H. Wiener (1947). J. Am. Chem. Soc. 69, (1), 17–20.
I. Gutman and N. Trinajsti (1972). Chem. Phys. Lett. 17, (4), 535–538.
S. Ediz, M. R. Farahani and M. Imran (2017). Int. J. Adv. Biotechnol. Res. 8, (4), 277–282.
S. Ediz (2017). Int. J. Syst. Sci. Appl. Math. 2, (5), 87–98.
M. S. Anjum and M. U. Safdar (2019). Eng. Appl. Sci. Lett. 21, 19–37.
Z. Shao, M. K. Siddiqui and M. H. Muhammad (2018). Symmetry 10, (7), 244–260.
Z. Shao, et al. (2019). Eng. Appl. Sci. Lett. 2, (1), 1–11.
A. U. R. Virk, M. N. Jhangeer and M. A. Rehman (2018). Eng. Appl. Sci. Lett. 1, (2), 37–50.
W. Gao, M. Asif and W. Nazeer (2018). Open J. Math. Anal. 2, (2), 10–26.
M. Munir, et al. (2016). Symmetry 97, 1–15.
Y. C. Nah, I. Paramasivam and P. Schmuki (2010). ChemPhysChem 11, 2698–2713.
A. Raheem, et al. (2019). J. Inform. Optim. Sci. 14, 1–11.
D. Mardare and P. Hones (1999). Mater. Sci. Eng. B 68, 42–47.
D. V. Bavykin, J. M. Friedrich and F. C. Walsh (2006). Adv. Mater. 18, 2807–2824.
Y. Li, et al. (2004). Chem. Phys. Lett. 389, 124–128.
J. Zhao, et al. (2005). Nanotechnology 16, 24–50.
J.-B. Liu, C. Wang, S. Wang and Bing Wei (2019). Bull. Malays. Math. Sci. Soc. 42, 67–78.
M. Baa, et al. (2015). Can. J. Chem. 93, (10), 1157–1160.
W. Gao, M. R. Farahani and M. R. Rajesh Kanna (2016). Open J. Discrete Math. 6, (2), 82–88.
J.-B. Liu, J. Zhao, J. Min and J. D. Cao (2019). Fractals 27, (8), 19–35.
W. Gao and M. R. Farahani (2017). J. Interdiscipl. Math. 20, (5), 1341–8.
A. W. Bharati Rajan, C. Grigorious and S. Stephen (2012). J. Comput. Math. Sci. 3, (5), 498–556.
M. Randic (1975). J. Am. Chem. Soc. 97, (23), 6609–6615.
I. Gutman, B. Ruscic, N. Trinajsti and C. F. Wilcox Jr. (1975). J. Chem. Phys. 62, (9), 3399–3405.
J.-B. Liu, J. Zhao, H. He and Z. Shao (2019). J. Stat. Phys. 177, 1131–1147.
W. Gao, M. K. Siddiqui, M. Naeem and N. A. Rehman (2017). Molecules 22, (9), 1496–1510.
Acknowledgements
The authors would like to thanks the two anonymous reviewers for their very constructive comments that helped us to enhance the quality of this manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhang, J., Siddiqui, M.K., Rauf, A. et al. On Ve-Degree and Ev-Degree Based Topological Properties of Single Walled Titanium Dioxide Nanotube. J Clust Sci 32, 821–832 (2021). https://doi.org/10.1007/s10876-020-01842-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10876-020-01842-3