Advertisement

Distance Under Symmetry: (3,6)-Fullerenes

  • Ali Reza AshrafiEmail author
  • Fatemeh Koorepazan − Moftakhar
  • Mircea V. Diudea
Chapter
Part of the Carbon Materials: Chemistry and Physics book series (CMCP, volume 9)

Abstract

A (3,6)-fullerene is a planar 3-connected cubic graph whose faces are triangles and hexagons. In this chapter, the modified Wiener and hyper–Wiener indices of three infinite classes of (3,6)-fullerenes are considered into account. Some open questions are also presented.

Keywords

(3,6)-fullerene Wiener index Modified Wiener index Hyper–Wiener index Modified hyper–Wiener index 

Notes

Acknowledgment

The first and second authors are partially supported by the University of Kashan under Grant No. 464092/20.

References

  1. Ashrafi AR, Mehranian Z (2013) Topological study of (3,6)– and (4,6)–fullerenes. In: Ashrafi AR, Cataldo F, Iranmanesh A, Ori O (eds) Topological modelling of nanostructures and extended systems, Carbon materials: chemistry and physics. Springer, Dordrecht/New York, pp 487–510CrossRefGoogle Scholar
  2. Ashrafi AR, Sabaghian − Bidgoli H (2009) A numerical method for computing PI index of fullerene molecules containing carbon atoms. J Comput Theor Nanosci 6:1706–1708CrossRefGoogle Scholar
  3. Ashrafi AR, Ghorbani M, Jalali M (2008) The vertex PI and Szeged indices of an infinite family of fullerenes. J Theor Comput Chem 7:221–231CrossRefGoogle Scholar
  4. Ashrafi AR, Ghorbani M, Jalali M (2009) Study of IPR fullerenes by counting polynomials. J Theor Comput Chem 8:451–457CrossRefGoogle Scholar
  5. Ashrafi AR, Cataldo F, Iranmanesh A, Ori O (eds) (2013) Topological modelling of nanostructures and extended systems, vol 7, Carbon materials: chemistry and physics. Springer Science + Business Media, DordrechtGoogle Scholar
  6. Behmaram A, Yousefi − Azari H, Ashrafi AR (2013) On the number of matchings and independent sets in (3,6) − fullerenes. MATCH Commun Math Comput Chem 70:525–532Google Scholar
  7. Bosma W, Cannon J, Playoust C (1997) The magma algebra system. I. The user language. J Symb Comput 24:235–265CrossRefGoogle Scholar
  8. Cataldo F, Graovac A, Ori O (eds) (2011) The mathematics and topology of fullerenes, vol 4, Carbon materials: chemistry and physics. Springer Science + Business Media B.V, DordrechtGoogle Scholar
  9. Devos M, Goddyn L, Mohar B, Samal R (2009) Cayley sum graphs and eigenvalues of (3,6) − fullerenes. J Combin Theory Ser B99:358–369CrossRefGoogle Scholar
  10. Deza M, Dutour Sikiric M, Fowler PW (2009) The symmetries of cubic polyhedral graphs with face size no larger than 6. MATCH Commun Math Comput Chem 61:589–602Google Scholar
  11. Diudea MV (1996a) Walk numbers WM: Wiener numbers of higher rank. J Chem Inf Comput Sci 36:535–540CrossRefGoogle Scholar
  12. Diudea MV (1996b) Wiener and hyper–Wiener numbers in a single matrix. J Chem Inf Comput Sci 36:833–836CrossRefGoogle Scholar
  13. Diudea MV (1997) Cluj matrix invariants. J Chem Inf Comput Sci 37:300–305CrossRefGoogle Scholar
  14. Diudea MV, Katona G, Pârv B (1997) Delta number, Dde, of dendrimers. Croat Chem Acta 70:509–517Google Scholar
  15. Diudea MV, Ursu O, Nagy LCs (2002) TOPOCLUJ. Babes − Bolyai University, ClujGoogle Scholar
  16. Djafari S, Koorepazan − Moftakhar F, Ashrafi AR (2013) Eccentric sequences of two infinite classes of fullerenes. J Comput Theor Nanosci 10:2636–2638CrossRefGoogle Scholar
  17. Firouzian S, Faghani M, Koorepazan − Moftakhar F, Ashrafi AR (2014) The hyper − Wiener and modified hyper − Wiener indices of graphs with an application on fullerenes. Studia Universitatis Babes − Bolyai Chemia 59:163–170Google Scholar
  18. Fowler PW, Manolopoulos DE (1995) An atlas of fullerenes. Oxford University Press, OxfordGoogle Scholar
  19. Fowler PW, John PE, Sachs H (2000) (3,6) − cages, hexagonal toroidal cages, and their spectra. DIMACS Ser Discrete Math Theoret Comput Sci 51:139–174Google Scholar
  20. Ghorbani M, Songhori M, Ashrafi AR, Graovac A (2015) Symmetry group of (3,6)-fullerenes. Fullerenes, Nanotubes, Carbon Nanostruct 23(9):788–791CrossRefGoogle Scholar
  21. Graovac A, Pisanski T (1991) On the Wiener index of a graph. J Math Chem 8:53–62CrossRefGoogle Scholar
  22. Gutman I, Šoltés L (1991) The range of the Wiener index and its mean isomer degeneracy. Z Naturforsch 46a:865–868Google Scholar
  23. Gutman I, Linert W, Lukovits I, Dobrynin AA (1997) Trees with extremal hyper–Wiener index: mathematical basis and chemical applications. J Chem Inf Comput Sci 37:349–354CrossRefGoogle Scholar
  24. HyperChem package Release 7.5 for Windows (2002) Hypercube Inc., Florida, USAGoogle Scholar
  25. John PE, Sachs H (2009) Spectra of toroidal graphs. Discret Math 309:2663–2681CrossRefGoogle Scholar
  26. Khalifeh MH, Yousefi–Azari H, Ashrafi AR (2008) The hyper − Wiener index of graph operations. Comput Math Appl 56:1402–1407CrossRefGoogle Scholar
  27. Klein DJ, Lukovits I, Gutman I (1995) On the definition of the hyper–Wiener index for cycle–containing structures. J Chem Inf Comput Sci 35:50–52CrossRefGoogle Scholar
  28. Koorepazan − Moftakhar F, Ashrafi AR (2013) Symmetry and PI index of C60+12n fullerenes. J Comput Theor Nanosci 10:2484–2486CrossRefGoogle Scholar
  29. Koorepazan − Moftakhar F, Ashrafi AR (2014) Fullerenes: topology and symmetry. In: Gutman I (ed) Topics in chemical graph theory. University of Kragujevac and Faculty of Science, Kragujevac, pp 163–176Google Scholar
  30. Koorepazan − Moftakhar F, Ashrafi AR (2015) Distance under symmetry. MATCH Commun Math Comput Chem 74:259–272Google Scholar
  31. Koorepazan − Moftakhar F, Ashrafi AR, Mehranian Z (2014a) Symmetry and PI polynomials of C50+10n fullerenes. MATCH Commun Math Comput Chem 71:425–436Google Scholar
  32. Koorepazan − Moftakhar F, Ashrafi AR, Mehranian Z, Ghorbani M (2014b) Automorphism group and fixing number of (3,6)– and (4, 6)–fullerene graphs. Elec Notes Disc Math 45:113–120CrossRefGoogle Scholar
  33. Myrvold W, Bultena B, Daugherty S, Debroni B, Girn S, Minchenko M, Woodcock J, Fowler PW (2007) FuiGui: a graphical user interface for investigating conjectures about fullerenes. MATCH Commun Math Comput Chem 58:403–422Google Scholar
  34. Schwerdtfeger P, Wirz L, Avery J (2013) Program fullerene: a software package for constructing and analyzing structures of regular fullerenes. J Comput Chem 34:1508–1526CrossRefGoogle Scholar
  35. The GAP Team (1995) GAP, groups, algorithms and programming. Lehrstuhl De für Mathematik. RWTH, AachenGoogle Scholar
  36. Wiener HJ (1947) Structural determination of paraffin boiling points. J Am Chem Soc 69:17–20CrossRefGoogle Scholar
  37. Yang R, Zhang H (2012) Hexagonal resonance of (3,6) − fullerenes. J Math Chem 50:261–273CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Ali Reza Ashrafi
    • 1
    • 2
    Email author
  • Fatemeh Koorepazan − Moftakhar
    • 1
    • 2
  • Mircea V. Diudea
    • 3
  1. 1.Department of Nanocomputing, Institute of Nanoscience and NanotechnologyUniversity of KashanKashanIran
  2. 2.Department of Pure Mathematics, Faculty of Mathematical SciencesUniversity of KashanKashanIran
  3. 3.Department of Chemistry, Faculty of Chemistry and Chemical EngineeringBabes-Bolyai UniversityCluj-NapocaRomania

Personalised recommendations