Topological Invariants of Möbius-Like Graphenic Nanostructures

  • Mihai V. Putz
  • Marzio De Corato
  • Giorgio Benedek
  • Jelena Sedlar
  • Ante Graovac
  • Ottorino Ori
Part of the Carbon Materials: Chemistry and Physics book series (CMCP, volume 7)


Topological invariants are computed for some carbon zigzag nanoribbons in the limit of infinite carbon atoms N by applying standard and Möbius-like periodicity. Topological modeling considerations allow then to assign to the half-twisted molecules a certain grade of chemical stability based on the actions of two basic topological properties, compactness and efficiency, which represent the influence that long-range connectivity has on lattice stability. Conclusions about Möbius-nanoribbon topological dimensionality are also presented.


Topological Insulator Wiener Index Topological Modeling Topological Invariant Graphenic Nanoribbons 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



MVP thanks the Romanian Ministry of Education and Research for support through the CNCS-UEFISCDI project Code TE-16/2010-2013.


  1. Ajami D, Oeckler O, Simon A, Herges R (2003) Nature 426:819CrossRefGoogle Scholar
  2. Ayme J-F, Beves JE, Leigh DA, McBurney RT, Rissanen K, Schultz D (2012) Nat Chem 4:15. doi: 10.1038/nchem.1193 CrossRefGoogle Scholar
  3. Benedek G, Bernasconi M, Cinquanta E, D’Alessio L, De Corato M (2011) In: Cataldo F, Graovac A, Ori O (eds) The mathematics and topology of fullerenes. Carbon materials: chemistry and physics 4. Springer, Dordrecht, p 217. doi: 10.1007/978-94-007-0221-9_12 CrossRefGoogle Scholar
  4. Boeyens JC, Levendis DC (2012) Int J Mol Sci 13:9081CrossRefGoogle Scholar
  5. Bonchev D, Mekenyan O (1980) Z Naturforsch 35a:739Google Scholar
  6. Caetano EWS, Freire VN, dos Santos SG, Galvao DS, Sato F (2008) J Chem Phys 128:164719CrossRefGoogle Scholar
  7. Caetano EWS, Freire VN, dos Santos SG, Albuquerque EL, Galvao DS, Sato F (2009) Langmuir 25(8):4751CrossRefGoogle Scholar
  8. Cataldo F, Ori O, Iglesias-Groth S (2010) Mol Simul 36(5):341CrossRefGoogle Scholar
  9. Chattaraj PK, Liu GH, Parr RG (1995) Chem Phys Lett 237:171CrossRefGoogle Scholar
  10. Clementi E, Raimondi DL (1963) J Chem Phys 38:2686CrossRefGoogle Scholar
  11. Clementi E, Raimondi DL, Reinhardt WP (1967) J Chem Phys 47:1300CrossRefGoogle Scholar
  12. Estrada E, Simón-Manso Y (2012) Chem Phys Lett 548:80CrossRefGoogle Scholar
  13. Gardner M (1978) Möbius Bands. In: Mathematical magic show: more puzzles, games, diversions, illusions and other mathematical sleight-of-mind from Scientific American. Vintage, New York, pp 123–136, Ch. 9Google Scholar
  14. Graovac A, Ori O, Faghani M, Ashrafi AR (2011) Iran J Math Chem 2(1):99Google Scholar
  15. Gravesen J, Willatzen M (2005) Phys Rev A 72:032108CrossRefGoogle Scholar
  16. Guo ZL, Gong ZR, Dong H, Sun CP (2009) Phys Rev B 80:195310CrossRefGoogle Scholar
  17. Han D, Pal S, Liu Y, Yan H (2010) Nat Nanotechnol 5:712CrossRefGoogle Scholar
  18. Hawking S (2001) The universe in a nutshell. Bantam Books, New YorkGoogle Scholar
  19. Hayashi M, Ebisawa H (2001) J Phys Soc Jpn 70:3495CrossRefGoogle Scholar
  20. Heilbronner E (1964) Tetrahedron Lett 29:1923CrossRefGoogle Scholar
  21. Koopmans T (1934) Physica 1:104CrossRefGoogle Scholar
  22. Möbius AF (1858) Werke 2:519Google Scholar
  23. Ori O, Putz MV (2013) Fuller Nanotub Carbon Nanostruct 21:000Google Scholar
  24. Ori O, Cataldo F, Graovac A (2009) Fuller Nanotub Carbon Nanostruct 17:308CrossRefGoogle Scholar
  25. Ori O, Cataldo F, Vukicevic D, Graovac A (2010) Iran J Math Chem 1(2):5Google Scholar
  26. Parr RG, Bartolotti LJ (1982) J Am Chem Soc 104:3801CrossRefGoogle Scholar
  27. Parr RG, Chattaraj PK (1991) J Am Chem Soc 113:1854CrossRefGoogle Scholar
  28. Parr RG, Pearson RG (1983) J Am Chem Soc 105:7512CrossRefGoogle Scholar
  29. Parr RG, Yang W (1989) Density functional theory of atoms and molecules. Oxford University Press, New YorkGoogle Scholar
  30. Pickover CA (2006) The Möbius strip: Dr. August Mobius’s marvelous band in mathematics, games, literature, art, technology, and cosmology. Thunder’s Mouth Press, New YorkGoogle Scholar
  31. Putz MV (2008) MATCH Commun Math Comput Chem 60:845Google Scholar
  32. Putz MV (2010) Int J Mol Sci 11:4227CrossRefGoogle Scholar
  33. Putz MV (2012a) Quantum theory: density, condensation, and bonding. Apple Academics/CRC Press, Toronto/NewarkGoogle Scholar
  34. Putz MV (2012b) Chemical orthogonal spaces. Mathematical chemical monographs, vol 14. Faculty of Sciences, University of Kragujevac, KragujevacGoogle Scholar
  35. Putz MV (2012c) Valence atom with bohmian quantum potential: the golden ratio approach. Chem Cent J 6:135. doi: 10.1186/1752-153X-6-135 CrossRefGoogle Scholar
  36. Putz MV, Ori O (2012) Chem Phys Lett 548:95CrossRefGoogle Scholar
  37. Sedlar J, Vukicevic D, Cataldo F, Ori O, Graovac A (2013) available on-line at (printed edn. 2014) ARS MATHEMATICA CONTEMPORANEA 7:1, ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
  38. Slater JC (1930) Phys Rev 36:57CrossRefGoogle Scholar
  39. Starostin EL, van der Heijden GHM (2007) Nat Mater 6:563CrossRefGoogle Scholar
  40. Tanda S, Tsuneta T, Okajima Y, Inagaki K, Yamaya K, Hatakenaka N (2002) Nature 417:397CrossRefGoogle Scholar
  41. Tanda S, Tsuneta T, Toshima T, Matsuura T, Tsubota M (2005) J Phys IV (France) 131:289CrossRefGoogle Scholar
  42. Todeschini R, Consonni V (2000) Handbook of molecular descriptors. Wiley-VCH, WeinheimCrossRefGoogle Scholar
  43. Vukicevic D, Cataldo F, Ori O, Graovac A (2011) Chem Phys Lett 501:442CrossRefGoogle Scholar
  44. Wakabayashi K, Harigaya K (2003) J Phys Soc Jpn 72:998CrossRefGoogle Scholar
  45. Weisstein EW (2012) Moebius Strip. From MathWorld-A Wolfram Web Resource.
  46. Xing SK, Li Y, Zhao XZ, Cai ZS, Shang ZF, Wang GC (2011) Acta Phys Chim Sin 27(5):1000Google Scholar
  47. Yakubo K, Avishai Y, Cohen D (2003) Phys Rev B 67:125319CrossRefGoogle Scholar
  48. Yang Y, Han D, Nangreave J, Liu Y, Yan H (2012) ACS Nano 6(9):8209CrossRefGoogle Scholar
  49. Yoon ZS, Osuka A, Kim D (2009) Nat Chem 1:113CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Mihai V. Putz
    • 1
  • Marzio De Corato
    • 2
    • 3
    • 4
  • Giorgio Benedek
    • 2
    • 5
  • Jelena Sedlar
    • 6
  • Ante Graovac
  • Ottorino Ori
    • 7
  1. 1.Laboratory of Computational and Structural Physical Chemistry, Biology-Chemistry DepartmentWest University of TimişoaraTimişoaraRomania
  2. 2.Dipartimento di Scienza dei MaterialiUniversitá di Milano-BicoccaMilanItaly
  3. 3.Centro S3, CNR-Istituto NanoscienzeModenaItaly
  4. 4.Dipartimento di FisicaUniversità di Modena e Reggio EmiliaModenaItaly
  5. 5.Donostia International Physics Center (DIPC)Donostia/San SebastiánSpain
  6. 6.Faculty of Civil Engineering, Architecture and GeodesyUniversity of SplitSplitCroatia
  7. 7.Actinium Chemical ResearchRomeItaly

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