Table of Contents
Journal of Nanoscience
Volume 2016 (2016), Article ID 1028031, 5 pages
http://dx.doi.org/10.1155/2016/1028031
Research Article

On Molecular Topological Properties of TiO2 Nanotubes

Department of Basic Sciences and Humanities (Mathematics), Calcutta Institute of Engineering and Management, Kolkata, India

Received 28 June 2016; Accepted 30 October 2016

Academic Editor: Zhengjun Zhang

Copyright © 2016 Nilanjan De. 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.

Linked References

  1. I. Gutman and N. Trinajstić, “Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons,” Chemical Physics Letters, vol. 17, no. 4, pp. 535–538, 1972. View at Publisher · View at Google Scholar · View at Scopus
  2. G. H. Fath-Tabar, “Old and new Zagreb indices of graphs,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 65, no. 1, pp. 79–84, 2011. View at Google Scholar · View at MathSciNet
  3. M. H. Khalifeh, H. Yousefi-Azari, and A. R. Ashrafi, “The first and second Zagreb indices of some graph operations,” Discrete Applied Mathematics, vol. 157, no. 4, pp. 804–811, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. B. Zhou, “Upper bounds for the Zagreb indices and the spectral radius of series-parallel graphs,” International Journal of Quantum Chemistry, vol. 107, no. 4, pp. 875–878, 2007. View at Publisher · View at Google Scholar · View at Scopus
  5. B. Zhou and I. Gutman, “Further properties of Zagreb indices,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 54, no. 1, pp. 233–239, 2005. View at Google Scholar · View at MathSciNet
  6. K. Xu, K. Tang, H. Liu, and J. Wang, “The Zagreb indices of bipartite graphs with more edges,” Journal of Applied Mathematics & Informatics, vol. 33, no. 3-4, pp. 365–377, 2015. View at Publisher · View at Google Scholar
  7. K. C. Das, K. Xu, and J. Nam, “Zagreb indices of graphs,” Frontiers of Mathematics in China, vol. 10, no. 3, pp. 567–582, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. B. Furtula and I. Gutman, “A forgotten topological index,” Journal of Mathematical Chemistry, vol. 53, no. 4, pp. 1184–1190, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. N. De, S. M. A. Nayeem, and A. Pal, “F-index of some graph operations,” Discrete Mathematics, Algorithms and Applications, vol. 8, no. 2, Article ID 1650025, 17 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  10. N. De, S. M. A. Nayeem, and A. Pal, “The F-coindex of some graph operations,” SpringerPlus, vol. 5, article 221, 2016. View at Publisher · View at Google Scholar
  11. H. Abdoa, D. Dimitrov, and I. Gutman, “On extremal trees with respect to the F-index,” https://arxiv.org/abs/1509.03574.
  12. M. O. Albertson, “The irregularity of a graph,” Ars Combinatoria, vol. 46, pp. 219–225, 1997. View at Google Scholar · View at MathSciNet · View at Scopus
  13. M. Tavakoli, F. Rahbarnia, and A. R. Ashrafi, “Some new results on irregularity of graphs,” Journal of Applied Mathematics & Informatics, vol. 32, no. 5-6, pp. 675–685, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  14. N. De, A. Pal, and S. M. A. Nayeem, “The irregularity of some composite graphs,” International Journal of Applied and Computational Mathematics, vol. 2, no. 3, pp. 411–420, 2016. View at Publisher · View at Google Scholar
  15. H. Abdo and D. Dimitrov, “The irregularity of graphs under graph operations,” Discussiones Mathematicae Graph Theory, vol. 34, no. 2, pp. 263–278, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. A. Miličević, S. Nikolić, and N. Trinajstić, “On reformulated Zagreb indices,” Molecular Diversity, vol. 8, no. 4, pp. 393–399, 2004. View at Publisher · View at Google Scholar
  17. B. Zhou and N. Trinajstić, “Some properties of the reformulated Zagreb indices,” Journal of Mathematical Chemistry, vol. 48, no. 3, pp. 714–719, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. A. Ilić and B. Zhou, “On reformulated Zagreb indices,” Discrete Applied Mathematics, vol. 160, no. 3, pp. 204–209, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. G. Su, L. Xiong, L. Xu, and B. Ma, “On the maximum and minimum first reformulated Zagreb index of graphs with connectivity at most k,” Filomat, vol. 25, no. 4, pp. 75–83, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. N. De, “Some bounds of reformulated Zagreb indices,” Applied Mathematical Sciences, vol. 6, no. 101, pp. 5005–5012, 2012. View at Google Scholar · View at MathSciNet · View at Scopus
  21. N. De, “Reformulated Zagreb indices of dendrimers,” Mathematica Aeterna, vol. 3, no. 2, pp. 133–138, 2013. View at Google Scholar · View at MathSciNet
  22. S. Ji, X. Li, and B. Huo, “On reformulated Zagreb indices with respect to acyclic, unicyclic and bicyclic graphs,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 72, no. 3, pp. 723–732, 2014. View at Google Scholar · View at MathSciNet
  23. S. Ji, X. Li, and Y. Qu, “On reformulated zagreb indices with respect to tricyclic graphs,” https://arxiv.org/abs/1406.7169.
  24. N. De, S. M. A. Nayeem, and A. Pal, “Reformulated first Zagreb index of some graph operations,” Mathematics, vol. 3, no. 4, pp. 945–960, 2015. View at Publisher · View at Google Scholar
  25. G. H. Shirdel, H. Rezapour, and A. M. Sayadi, “The hyper Zagreb index of graph operations,” Iranian Journal of Mathematical Chemistry, vol. 4, no. 2, pp. 213–220, 2013. View at Google Scholar
  26. M. Veylaki, M. J. Nikmehr, and H. A. Tavallaee, “The third and hyper-Zagreb coindices of some graph operations,” Journal of Applied Mathematics and Computing, vol. 50, no. 1-2, pp. 315–325, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. B. Basavanagoud and S. Patil, “A note on Hyper-Zagreb index of graph operations,” Iranian Journal of Mathematical Chemistry, vol. 7, no. 1, p. 8992, 2016. View at Google Scholar
  28. M. R. Farahani, “Computing the hyper-Zagreb index of hexagonal nanotubes,” Journal of Chemistry and Materials Research, vol. 2, no. 1, pp. 16–18, 2015. View at Google Scholar
  29. R. A. Evarestov, Y. F. Zhukovskii, A. V. Bandura, and S. Piskunov, “Symmetry and models of single-walled TiO2 nanotubes with rectangular morphology,” Central European Journal of Physics, vol. 9, no. 2, pp. 492–501, 2011. View at Publisher · View at Google Scholar · View at Scopus
  30. V. V. Ivanovskaya, A. N. Enyashin, and A. L. Ivanovskiǐ, “Nanotubes and fullerene-like molecules based on TiO2 and ZrS2: electronic structure and chemical bond,” Russian Journal of Inorganic Chemistry, vol. 49, no. 2, pp. 244–251, 2004. View at Google Scholar · View at Scopus
  31. A. N. Enyashin and G. Seifert, “Structure, stability and electronic properties of TiO2 nanostructures,” Physica Status Solidi, vol. 242, no. 7, pp. 1361–1370, 2005. View at Publisher · View at Google Scholar
  32. M. A. Malik and M. Imran, “On multiple Zagreb index of TiO2 nanotubes,” Acta Chimica Slovenica, vol. 62, no. 4, pp. 973–976, 2015. View at Publisher · View at Google Scholar · View at Scopus
  33. I. Nadeem and H. Shaker, “On eccentric connectivity index of TiO2 nanotubes,” Acta Chimica Slovenica, vol. 63, pp. 363–368, 2016. View at Publisher · View at Google Scholar