Skip to main content

Geometry and Topology of Nanotubes and Nanotori

  • Chapter
  • First Online:

Part of the book series: Carbon Materials: Chemistry and Physics ((CMCP,volume 8))

Abstract

A molecular graph is the graph of a molecule in which the vertices are atoms and edges are chemical bonds. We review the recent results on computing symmetry of series of armchair polyhex, zig-zag polyhex and C4C8(R/S) nanotubes and nanotori. The topological properties of these nanostructures are also investigated.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  • Arezoomand M, Taeri B (2009) The full symmetry and irreducible representations of nanotori. Acta Cryst A 65:249–252

    Article  CAS  Google Scholar 

  • Asadpour J, Mojarad R, Safikhani L (2011) Computing some topological indices of nanostructures. Dig J Nanomat Biostruct 6:937–941

    Google Scholar 

  • Ashrafi AR, Loghman A (2006a) PI index of armchair polyhex nanotubes. Ars Combin 80:193–199

    Google Scholar 

  • Ashrafi AR, Loghman A (2006b) PI index of zig-zag polyhex nanotubes. MATCH Commun Math Comput Chem 55:447–452

    Google Scholar 

  • Ashrafi AR, Loghman A (2006c) Padmakar-Ivan index of TUC4C8(S) nanotubes. J Comput Theor Nanosci 3:378–381

    Google Scholar 

  • Ashrafi AR, Loghman A (2008) Computing Padmakar-Ivan index of a TC4C8(R) Nanotorus. J Comput Theor Nanosci 5:1431–1434

    Article  CAS  Google Scholar 

  • Ashrafi AR, Rezaei F (2007) PI index of polyhex nanotori. MATCH Commun Math Comput Chem 57:243–250

    Google Scholar 

  • Ashrafi AR, Yousefi S (2007a) A new algorithm for computing distance matrix and Wiener index of zig-zag polyhex nanotubes. Nanoscale Res Lett 2:202–206

    Google Scholar 

  • Ashrafi AR, Yousefi S (2007b) Computing the wiener index of a TUC4C8(S) nanotorus. MATCH Commun Math Comput Chem 57:403–410

    Google Scholar 

  • Ashrafi AR, Rezaei F, Loghman A (2009) PI index of the C4C8(S) nanotorus. Revue Roum Chim 54:823–826

    Google Scholar 

  • Ashrafi AR, Došslić T, Saheli M (2011a) The eccentric connectivity index of TUC4C8(R) nanotubes. MATCH Commun Math Comput Chem 65:221–230

    Google Scholar 

  • Ashrafi AR, Saheli M, Ghorbani M (2011b) The eccentric connectivity index of nanotubes and nanotori. J Comput Appl Math 16:4561–4566

    Google Scholar 

  • Balaban AT (1982) Distance connectivity index. Chem Phys Lett 89:399–404

    Article  CAS  Google Scholar 

  • Balaban AT (1983) Topological indices based on topological distances in molecular graphs. Pure Appl Chem 55:199–206

    Article  CAS  Google Scholar 

  • Bosma W, Cannon J, Playoust C (1997) The Magma algebra system I: the user language. J Symb Comput 24:235–265

    Article  Google Scholar 

  • Das KC (2010) Atom-bond connectivity index of graphs. Discrete Appl Math 158:1181–1188

    Article  Google Scholar 

  • Das KC, Trinajstić N (2010) Comparison between first geometric-arithmetic index and atom-bond connectivity index. Chem Phys Lett 497:149–151

    Article  CAS  Google Scholar 

  • Diudea MV, Ursu O, Nagy LCs (2002) TOPOCLUJ. Babes-Bolyai University, Cluj

    Google Scholar 

  • Diudea MV, Stefu M, Pârv B, John PE (2004) Wiener index of armchair polyhex nanotubes. Croat Chem Acta 77:111–115

    CAS  Google Scholar 

  • Dobrynin A, Gutman I (1994) On a graph invariant related to the sum of all distances in a graph. Publ Inst Math (Beograd) (N.S.) 56:18–22

    Google Scholar 

  • Dobrynin A, Gutman I (1995) Solving a problem connected with distances in graphs. Graph Theor Notes NY 28:21–23

    Google Scholar 

  • Dobrynin A, Gutman I, Domotor GA (1995) Wiener-type graph invariant for some bipartite graphs. Appl Math Lett 8(5):57–62

    Google Scholar 

  • Estrada E (2008) Atom-bond connectivity and the energetic of branched alkanes. Chem Phys Lett 463:422–425

    Article  CAS  Google Scholar 

  • Estrada E, Torres L, Rodriguez L, Gutman I (1998) An atom-bond connectivity index: modelling the enthalpy of formation of alkanes. Indian J Chem 37A:849–855

    Google Scholar 

  • Farahani MR (2012) Some connectivity indices and Zagreb index of polyhex nanotubes. Acta Chim Slov 59:779–783

    CAS  Google Scholar 

  • Fath-Tabar GH, Vaez-Zadeh B, Ashrafi AR, Graovac A (2011) Some inequalities for the atom-bond connectivity index of graph operations. Discret Appl Math 159:1323–1330

    Article  Google Scholar 

  • Furtula B, Graovac A, Vukičević D (2009) Atom-bond connectivity index of trees. Discret Appl Math 157:2828–2835

    Article  Google Scholar 

  • Gutman I (1994) A formula for the Wiener number of trees and its extension to graphs containing cycles. Graph Theor Notes NY 27:9–15

    Google Scholar 

  • Gutman I, Das KC (2004) The first Zagreb index 30 years after. Commun Math Comput Chem 50:83–92

    CAS  Google Scholar 

  • Gutman I, Trinajstić N (1972) Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons. Chem Phys Lett 17:535–538

    Article  CAS  Google Scholar 

  • Heydari A, Taeri B (2009) Szeged index of TUC4C8(S) nanotubes. Eur J Combin 30:1134–1141

    Article  Google Scholar 

  • HyperChem package Release 7.5 for Windows (2002) Hypercube Inc., Florida, USA

    Google Scholar 

  • Iranmanesh A, Ashrafi AR (2007) Balaban index of an armchair polyhex, TUC4C8(R) and TUC4C8(S) nanotorus. J Comput Theor Nanosci 4:514–517

    CAS  Google Scholar 

  • John PE, Diudea MV (2004) Wiener index of zig-zag polyhex nanotubes. Croat Chem Acta 77:127–132

    CAS  Google Scholar 

  • Khadikar PV, Deshpande NV, Kale PP, Dobrynin A, Gutman I, Domotor G (1995) The Szeged index and an analogy with the wiener index. J Chem Inf Compute Sci 35:545–550

    Google Scholar 

  • Khadikar PV, Karmarkar S, Agrawal VK (2001) A novel PI index and its applications to QSPR/QSAR studies. J Chem Inf Comput Sci 41:934–949

    Article  CAS  Google Scholar 

  • Khalifeh MH, Yousefi-Azari H, Ashrafi AR (2009) The first and second Zagreb indices of some graph operations. Discret Appl Math 157:804–811

    Article  Google Scholar 

  • Khodashenas H, Nadjafi-Arani MJ, Ashrafi AR, Gutman I (2011) A new proof of the Szeged-Wiener theorem. Kragujev J Math 35:165–172

    Google Scholar 

  • Klarner DA (1997) Polyominoes. In: Goodman JE, O’Rourke J (eds) Handbook of discrete and computational geometry, CRC Press, Boca Raton, pp 225–242 (Chaper 12)

    Google Scholar 

  • Klavžar S (2007) On the PI index: PI-partitions and Cartesian product graphs. MATCH Commun Math Comput Chem 57:573–586

    Google Scholar 

  • Klavžar S, Rajapakse A, Gutman I (1996) The Szeged and the wiener index of graphs. Appl Math Lett 9:45–49

    Article  Google Scholar 

  • Manoochehrian B, Yousefi-Azari H, Ashrafi AR (2008) Szeged index of a zig-zag polyhex nanotube. Ars Combin 86:371–379

    Google Scholar 

  • Randić M (1974) On the recognition of identical graphs representing molecular topology. J Chem Phys 60:3920–3928

    Article  Google Scholar 

  • Randić M (1975) On characterization of molecular branching. J Am Chem Soc 97:6609–6615

    Article  Google Scholar 

  • Randić M (1976) On discerning symmetry properties of graphs. Chem Phys Lett 42:283–287

    Article  Google Scholar 

  • Saheli M, Ashrafi AR (2010a) The eccentric connectivity index of zig-zag polyhex nanotubes and nanotori. J Comput Theor Nanosci 7:1900–1903

    Google Scholar 

  • Saheli M, Ashrafi AR (2010b) The eccentric connectivity index of armchair polyhex nanotubes. Maced J Chem Chem Eng 29:71–75

    Google Scholar 

  • Sardana S, Madan AK (2001) Application of graph theory: relationship of molecular connectivity index, Wiener’s index and eccentric connectivity index with diuretic activity. MATCH Commun Math Comput Chem 43:85–98

    CAS  Google Scholar 

  • Sharma V, Goswami R, Madan AK (1997) Eccentric connectivity index: a novel highly discriminating topological descriptor for structure-property and structure-activity studies. J Chem Inf Comput Sci 37:273–282

    Article  CAS  Google Scholar 

  • Staic MD, Petrescu-Nita A (2013) Symmetry group of two special types of carbon nanotori. Acta Cryst A 69:1–5

    Article  Google Scholar 

  • The GAP Team (1995) GAP, groups, algorithms and programming. Lehrstuhl De für Mathematik, RWTH, Aachen

    Google Scholar 

  • Trinajstic N (1992) Chemical graph theory. CRC Press, Boca Raton

    Google Scholar 

  • Wiener H (1947) Structural determination of paraffin boiling points. J Am Chem Soc 69:17–20

    Article  CAS  Google Scholar 

  • Yavari M, Ashrafi AR (2009) On the symmetry of a zig-zag and an armchair polyhex carbon nanotorus. Symmetry 1:145–152

    Article  CAS  Google Scholar 

  • Yousefi S, Ashrafi AR (2006) An exact expression for the wiener index a polyhex nanotoruse. MATCH Commun Math Comput Chem 56:169–178

    CAS  Google Scholar 

  • Yousefi S, Ashrafi AR (2007) An exact expression for the wiener index of a TUC4C8(R) nanotorus. J Math Chem 42:1031–1039

    Article  CAS  Google Scholar 

  • Yousefi S, Ashrafi, AR (2008a) Distance matrix and wiener index of armchair polyhex nanotubes. Stud Univ Babes-Bolyai Chem 53:111–116

    Google Scholar 

  • Yousefi S, Ashrafi AR (2008b) An algorithm for constructing wiener matrix of TUC4C8(R) nanotubes. Curr Nanosci 4:161–165

    Google Scholar 

  • Yousefi S, Ashrafi AR (2011) 3-dimensional distance matrix of a TC4C8(R) nanotoruse. MATCH Commun Math Comput Chem 65:249–254

    CAS  Google Scholar 

  • Yousefi S, Yousefi-Azari H, Khalifeh MH, Ashrafi AR (2008c) Computing distance matrix and related topological indices of an achiral polyhex nanotube. Int J Chem Mod 1:149–156

    Google Scholar 

  • Yousefi S, Yousefi-Azari H, Ashrafi AR, Khalifeh MH (2008d) Computing wiener and Szeged indices of a polyhex Nanotorus. J Sci Univ Tehran 33:7–11

    Google Scholar 

  • Yousefi-Azari H, Ashrafi AR, Khalifeh MH (2008e) Computing vertex-PI index of single and multiwalled nanotubes. Dig J Nanomat Bios 3:315–318

    Google Scholar 

  • Zhou B (2004) Zagreb indices. MATCH Commun Math Comput Chem 52:113–118

    CAS  Google Scholar 

  • Zhou B, Gutman I (2005) Further properties of Zagreb indices. MATCH Commun Math Comput Chem 54:233–239

    CAS  Google Scholar 

Download references

Acknowledgements

The first and second authors are partially supported by the university of Kashan under grant number 364988/99.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Fatemeh Koorepazan-Moftakhar .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Koorepazan-Moftakhar, F., Ashrafi, A., Ori, O., Putz, M. (2015). Geometry and Topology of Nanotubes and Nanotori. In: Putz, M., Ori, O. (eds) Exotic Properties of Carbon Nanomatter. Carbon Materials: Chemistry and Physics, vol 8. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9567-8_6

Download citation

Publish with us

Policies and ethics