Distance-based topological indices of nanosheets, nanotubes and nanotori of \(\hbox {SiO}_2\)

  • Micheal Arockiaraj
  • Sandi Klavžar
  • Shagufa Mushtaq
  • Krishnan Balasubramanian
Original Paper


We have computed distance-based topological indices of nanosheets, nanotubes and nanotori of \(\hbox {SiO}_2\) which find potential applications in drug, food, and cosmetic industry. As topological indices correlate with physico-chemical properties and estimating efficiency of drug deliveries of these species, we compute the topological indices based on their degrees and distances of the associated molecular graphs. We have obtained exact analytical expressions of various topological indices such as the Wiener, vertex-Szeged, edge-Szeged, edge-vertex-Szeged, Padmakar-Ivan, Schultz and Gutman indices of \(\hbox {SiO}_2\) nanosheet, nanotube and torus using the cut method which involves decomposing a molecular graph by means of the transitive closure property of Djoković–Winkler relation to smaller strength-weighted quotient graphs.


Silicon dioxide nanostructures Drug delivery QSAR/QSPR Cut method Topological indices 


  1. 1.
    F. Alali, I.H. Karampelas, Y.H. Kim, E.P. Furlani, Photonic and thermofluidic analysis of colloidal plasmonic nanorings and nanotori for pulsed-laser photothermal applications. J. Phys. Chem. C 117(39), 20178–20185 (2013)CrossRefGoogle Scholar
  2. 2.
    M. Arockiaraj, J. Clement, K. Balasubramanian, Topological indices and their applications to circumcised donut benzenoid systems, kekulenes and drugs. Polycycl. Aromat. Comp. (2017).
  3. 3.
    A.T. Balaban, Applications of graph theory in chemistry. J. Chem. Inf. Comput. Sci. 25(3), 334–343 (1985)CrossRefGoogle Scholar
  4. 4.
    A.T. Balaban, I. Motoc, D. Bonchev, O. Mekenyan, Topological indices for structure-activity correlations. Top. Curr. Chem. 114, 21–55 (1983)CrossRefGoogle Scholar
  5. 5.
    K. Balasubramanian, A method for nuclear spin statistics in molecular spectroscopy. J. Chem. Phys. 74(12), 6824–6829 (1981)CrossRefGoogle Scholar
  6. 6.
    K. Balasubramanian, Spectra of chemical trees. Int. J. Quantum Chem. 21(3), 581–590 (1982)CrossRefGoogle Scholar
  7. 7.
    K. Balasubramanian, Operator and algebraic methods for NMR spectroscopy. I. Generation of NMR spin species. J. Chem. Phys. 78(11), 6358–6368 (1983)CrossRefGoogle Scholar
  8. 8.
    K. Balasubramanian, Applications of combinatorics and graph theory to spectrosocpy and quantum chemistry. Chem. Rev. 85(6), 599–618 (1985)CrossRefGoogle Scholar
  9. 9.
    K. Balasubramanian, Characteristic polynomials of organic polymers and periodic structure. J. Comput. Chem. 6(6), 656–661 (1985)CrossRefGoogle Scholar
  10. 10.
    K. Balasubramanian, Tree pruning and lattice statistics on Bethe lattices. J. Math. Chem. 2(1), 69–82 (1988)CrossRefGoogle Scholar
  11. 11.
    K. Balasubramanian, Nuclear-spin statistics of \(\text{ C }_{60}\), \(\text{ C }_{60}\text{ H }_{60}\) and \(\text{ C }_{60}\text{ D }_{60}\). Chem. Phys. lett. 183(3–4), 292–296 (1991)Google Scholar
  12. 12.
    K. Balasubramanian, Exhaustive generation and analytical expressions of matching polynomials of fullerenes \(\text{ C }_{20}\)-\(\text{ C }_{50}\). J. Chem. Inf. Comput. Sci. 34(2), 421–427 (1994)Google Scholar
  13. 13.
    K. Balasubramanian, Matching polynomials of fullerene clusters. Chem. Phys. Lett. 201(1–4), 306–314 (1994)Google Scholar
  14. 14.
    K. Balasubramanian, K. Khokhani, S.C. Basak, Complex graph matrix representations and characterizations of proteomic maps and chemically induced changes to proteomes. J. Proteome Res. 5(5), 1133–1142 (2006)CrossRefGoogle Scholar
  15. 15.
    K. Balasubramanian, M. Randić, The characteristic polynomials of structures with pending bonds. Theor. Chim Acta 61(4), 307–323 (1982)CrossRefGoogle Scholar
  16. 16.
    M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale, Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study. Proc. Natl. Acad. Sci. 105(4), 1232–1237 (2008)CrossRefGoogle Scholar
  17. 17.
    S.C. Basak, G.D. Grunwald, B.D. Gute, K. Balasubramanian, D. Opitz, Use of statistical and neural net approaches in predicting toxicity of chemicals. J. Chem. Inf. Comput. Sci. 40(4), 885–890 (2000)CrossRefGoogle Scholar
  18. 18.
    S.C. Basak, D. Mills, M.M. Mumtaz, K. Balasubramanian, Use of topological indices in predicting aryl hydrocarbon receptor binding potency of dibenzofurans: a hierarchical QSAR approach. Indian J. Chem. 42A(6), 1385–1391 (2003)Google Scholar
  19. 19.
    A. Bharali, R. Bora, Computation of some degree based topological indices of silicates (\(\text{ SiO }_2\)) layer. Ann. Pure Appl. Math. 16(2), 287–293 (2018)Google Scholar
  20. 20.
    C. Cao, Y. Hua, Topological indices based on vertex, distance, and ring: on the boiling points of paraffins and cycloakanes. J. Chem. Inf. Comput. Sci. 41(4), 867–877 (2001)CrossRefGoogle Scholar
  21. 21.
    X. Chen, R. Klingeler, M. Kath, A.A.E. Gendy, K. Cendrowski, R.J. Kalenczuk, E. Borowiak-Palen, Magnetic silica nanotubes: synthesis, drug release, and feasibility for magnetic hyperthermia. ACS Appl. Mater. Interfaces 4(4), 2303–2309 (2012)CrossRefGoogle Scholar
  22. 22.
    M. Črepnjak, N. Tratnik, The Szeged index and the Wiener index of partial cubes with applications to chemical graphs. Appl. Math. Comput. 309, 324–333 (2017)Google Scholar
  23. 23.
    D. Djoković, Distance preserving subgraphs of hypercubes. J. Comb. Theory Ser. B 14(3), 263–267 (1973)CrossRefGoogle Scholar
  24. 24.
    E. Estrada, L. Torres, L. Rodríguez, I. Gutman, An atom-bond connectivity index: modelling the enthalpy of formation of alkanes. Indian J. Chem. 37A, 849–855 (1998)Google Scholar
  25. 25.
    S. Fajtlowicz, On conjectures of Graffiti-II. Congr. Numer. 60, 187–197 (1987)Google Scholar
  26. 26.
    F. Farrukh, S. Hafi, R. Farooq, M.R. Farahani, Calculating some topological indices of \(\text{ SiO }_2\) layer structure. J. Inform. Math. Sci. 8(3), 181–187 (2016)Google Scholar
  27. 27.
    F. Farrukh, R. Farooq, M.R. Farahani, On the atom-bond connectivity and geometric-arithmetic indices of \(\text{ SiO }_2\) layer structure. Moroc. J. Chem. 5(2), 384–390 (2017)Google Scholar
  28. 28.
    W. Gao, W. Wang, D. Dimitrov, Y. Wang, Nano properties analysis via fourth multiplicative ABC indicator calculating. Arab. J. Chem. (2017).
  29. 29.
    I. Gutman, A formula for the Wiener number of trees and its extension to graphs containing cycles. Graph Theory Notes NY. 27, 9–15 (1994)Google Scholar
  30. 30.
    I. Gutman, A.R. Ashrafi, The edge version of the Szeged index. Croat. Chem. Acta 81(2), 263–266 (2008)Google Scholar
  31. 31.
    I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total \(\varphi \)-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17(4), 535–538 (1972)Google Scholar
  32. 32.
    B. Hemmateenejad, A. Mohajeri, Application of quantum topological molecular similarity descriptors in QSPR study of the O-methylation of substituted phenols. J. Comput. Chem. 29(2), 266–274 (2008)CrossRefGoogle Scholar
  33. 33.
    P. Horcajada, A. Rámila, J. Pérez Pariente, M. Vallet-Regi, Influence of pore size of MCM-41 matrices on drug delivery rate. Microporous Mesoporous Mater. 68(1–3), 105–109 (2004)CrossRefGoogle Scholar
  34. 34.
    S. Kasani, P. Zheng, N. Wu, Tailoring optical properties of a large-area plasmonic gold nanoring array pattern. J. Phys. Chem. C 122(25), 13443–13449 (2018)CrossRefGoogle Scholar
  35. 35.
    P.V. Khadikar, S. Karmarkar, V.K. Agrawal, A novel PI index and its applications to QSPR/QSAR studies. J. Chem. Inf. Comput. Sci. 41(4), 934–949 (2001)CrossRefGoogle Scholar
  36. 36.
    S. Klavžar, On the canonical metric representation, average distance, and partial Hamming graphs. Eur. J. Comb. 27, 68–73 (2006)CrossRefGoogle Scholar
  37. 37.
    S. Klavžar, I. Gutman, Wiener number of vertex-weighted graphs and a chemical application. Discret. Appl. Math. 80(1), 73–81 (1997)CrossRefGoogle Scholar
  38. 38.
    S. Klavžar, I. Gutman, B. Mohar, Labeling of benzenoid systems which reflects the vertex-distance relation. J. Chem. Inf. Comput. Sci. 35(3), 590–593 (1995)CrossRefGoogle Scholar
  39. 39.
    S. Klavžar, A. Lipovec, Partial cubes as subdivision graphs and as generalized Petersen graphs. Discret. Math. 263(1–3), 157–165 (2003)CrossRefGoogle Scholar
  40. 40.
    S. Klavžar, P. Manuel, M.J. Nadjafi-Arani, R.S. Rajan, C. Grigorious, S. Stephen, Average distance in interconnection networks via reduction theorems for vertex-weighted graphs. Comput. J. 59(12), 1900–1910 (2016)CrossRefGoogle Scholar
  41. 41.
    S. Klavžar, M.J. Nadjafi-Arani, Wiener index in weighted graphs via unification of \(\Theta ^\ast \)-classes. Eur. J. Comb. 36, 71–76 (2014)Google Scholar
  42. 42.
    S. Klavžar, M.J. Nadjafi-Arani, Cut method: update on recent developments and equivalence of independent approaches. Curr. Org. Chem. 19(4), 348–358 (2015)CrossRefGoogle Scholar
  43. 43.
    M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, I. Gutman, The edge Szeged index of product graphs. Croat. Chem. Acta. 81(2), 277–281 (2008)Google Scholar
  44. 44.
    M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, S.G. Wagner, Some new results on distance-based graph invariants. Eur. J. Comb. 30(5), 1149–1163 (2009)CrossRefGoogle Scholar
  45. 45.
    M. Llusar, C. Sanchez, Inorganic and hybrid nanofibrous materials templated with organogelators. Chem. Mater. 20(3), 782–820 (2008)CrossRefGoogle Scholar
  46. 46.
    A. Mahmiani, O. Khormali, A. Iranmanesh, M. Yousefidaz, The new version of Szeged index. Optoelectron. Adv. Mater.-Rapid Commun. 4(12), 2182–2184 (2010)Google Scholar
  47. 47.
    M. Manoharan, M.M. Balakrishnarajan, P. Venuvanalingam, K. Balasubramanian, Topological resonance energy predictions of the stability of fullerene clusters. Chem. Phys. Lett. 222(1–2), 95–100 (1994)CrossRefGoogle Scholar
  48. 48.
    A. Mohajeri, M.H. Dinpajooh, Structure-toxicity relationship for aliphatic compounds using quantum topological descriptors. J. Mol. Struct. -Theochem 855(1–3), 1–5 (2008)CrossRefGoogle Scholar
  49. 49.
    A. Mohajeri, P. Manshour, M. Mousaee, A novel topological descriptor based on the expanded Wiener index: applications to QSPR/QSAR studies. Iran. J. Math. Chem. 8(2), 107–135 (2017)Google Scholar
  50. 50.
    B. Munoz, A. Rámila, J. Pérez Pariente, I. Diaz, M. Vallet-Regi, MCM-41 organic modification as drug delivery rate regulator. Chem. Mater. 15(2), 500–503 (2003)CrossRefGoogle Scholar
  51. 51.
    D. Narducci, G. Cerofolini, E. ROMANO, Nanotori of semiconductor material for use in diagnostics and in the anti-tumor therapy and process for the production thereof.
  52. 52.
    T. Parsons-Moss, L.K. Schwaiger, A. Hubaud, Y.J. Hu, H. Tuysuz, P. Yang, K. Balasubramanian, H. Nitsche, in Plutonium complexation by phosphonate-functionalized mesoporous silica, 241th ACS National Meeting (Anaheim, CA, United States, 2011)Google Scholar
  53. 53.
    R. Ramaraj, K. Balasubramanian, Computer generation of matching polynimials of chemical graphs and lattices. J. Comput. Chem. 6(2), 122–141 (1985)CrossRefGoogle Scholar
  54. 54.
    M. Randić, Characterization of molecular branching. J. Am. Chem Soc. 97(3), 6609–6615 (1975)CrossRefGoogle Scholar
  55. 55.
    M. Randić, Novel molecular descriptor for structure-property studies. Chem. Phys. Lett. 211, 478–483 (1993)CrossRefGoogle Scholar
  56. 56.
    D.H. Rouvray, R.B. King, Topology in Chemistry: Discrete Mathematics of Molecules (Elsevier, Amsterdam, 2002)CrossRefGoogle Scholar
  57. 57.
    H.P. Schultz, Topological organic chemistry. 1. Graph theory and topological indices of alkanes. J. Chem. Inf. Comput. Sci. 29(3), 227–228 (1989)CrossRefGoogle Scholar
  58. 58.
    N. Tratnik, The edge-Szeged index and the PI index of benzenoid systems in linear time. MATCH Commun. Math. Comput. Chem. 77(2), 393–406 (2017)Google Scholar
  59. 59.
    M. Vallet-Regi, A. Rámila, R.P. del Real, J. Pérez Pariente, A new property of MCM-41: drug delivery system. Chem. Mater. 13(2), 308–311 (2001)CrossRefGoogle Scholar
  60. 60.
    D. Vukičević, B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges. J. Math. Chem. 46(4), 1369–1376 (2009)CrossRefGoogle Scholar
  61. 61.
    H. Wiener, Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69(1), 17–20 (1947)CrossRefGoogle Scholar
  62. 62.
    P. Winkler, Isometric embeddings in products of complete graphs. Discret. Appl. Math. 7, 221–225 (1984)CrossRefGoogle Scholar
  63. 63.
    H. Yuan, A.L. Parrill, QSAR development to describe HIV-1 integrase inhibition. J. Mol. Struct. -Theochem. 529(1–3), 273–282 (2000)CrossRefGoogle Scholar
  64. 64.
    B. Zhou, N. Trinajstić, On a novel connectivity index. J. Math. Chem. 46(4), 1252–1270 (2009)CrossRefGoogle Scholar
  65. 65.
    Y. Zhu, J. Shi, W. Shen, X. Dong, J. Feng, M. Ruan, Y. Li, Stimuli-responsive controlled drug release from a hollow mesoporous silica sphere/polyelectrolyte multilayer core-shell structure. Angew. Chem. Int. Ed. Engl. 44(32), 5083–5087 (2005)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsLoyola CollegeChennaiIndia
  2. 2.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia
  3. 3.Faculty of Natural Sciences and MathematicsUniversity of MariborMariborSlovenia
  4. 4.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia
  5. 5.School of Molecular SciencesArizona State UniversityTempeUSA

Personalised recommendations