Drawing Diamond Structures with Eigenvectors

  • István László
  • Ante Graovac
  • Tomaž Pisanski
Part of the Carbon Materials: Chemistry and Physics book series (CMCP, volume 6)


Very often the basic information about a nanostructure is a topological one. Based on this topological information, we have to determine the Descartes coordinates of the atoms. For fullerenes, nanotubes and nanotori, the topological coordinate method supplies the necessary information. With the help of the bi-lobal eigenvectors of the Laplacian matrix, the position of the atoms can be generated easily. This method fails, however, for nanotube junctions and coils and other nanostructures. We have found recently a matrix W which could generate the Descartes coordinates for fullerenes, nanotubes and nanotori and also for nanotube junctions and coils as well. Solving, namely, the eigenvalue problem of this matrix W, its eigenvectors with zero eigenvalue give the Descartes coordinates. There are nanostructures however, whose W matrices have more eigenvectors with zero eigenvalues than it is needed for determining the positions of the atoms in 3D space. In this chapter, we have studied this problem in the case of diamond structures. We have found that this extra degeneracy is due to the fact that the first and second neighbour interactions do not determine the geometry of the structure. It was found that including the third neighbour interaction as well, diamond structures were described properly.


Adjacency Matrix Null Space Atomic Arrangement Zero Eigenvalue Neighbour Interaction 
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.



I. László thanks for the support of grants TAMOP-4.2.1/B-09/1/KONV-2010-0003, TAMOP-4.2.1/B-09/1/KMR-2010-0002 and for the support obtained in the frame work of bilateral agreement between the Croatian Academy of Science and Art and the Hungarian Academy of Sciences. The research of T. Pisanski has been financed by ARRS project P1-0294 and within the EUROCORES Programme EUROGIGA (project GReGAS N1-0011) of the European Science Foundation.


  1. Biyikoglu T, Hordijk W, Leydold J, Pisanski T, Stadler PF (2004) Graph Laplacians, nodal domains, and hyperplane arrangements. Linear Algebra Appl 390:155–174CrossRefGoogle Scholar
  2. Biyikoglu T, Leydold J, Stadler PF (2007) Laplacian eigenvectors of graphs. Perron-Frobenius and Faber-Krahn type theorems. LNM 1915. Springer, Berlin/HeidelbergGoogle Scholar
  3. Brenner DW (1990) Empirical potentials for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. Phys Rev B 42:9458–9471CrossRefGoogle Scholar
  4. Colin de Verdière Y (1998) Spectres de graphes. Cours spécialisés 4. Société Mathématique de France, ParisGoogle Scholar
  5. Di Battista G, Eades P, Tamassia R, Tollis IG (1999) Graph drawing: algorithms for the visualization of graphs. Prentice Hall, Upper Saddle RiverGoogle Scholar
  6. Dresselhaus MS, Dresselhaus G, Eklund PC (1996) Science of fullerenes and carbon nanotubes: their properties and applications. Academic, New York/LondonGoogle Scholar
  7. Fowler PW, Manolopulos DE (1995) An atlas of fullerenes. Clarendon, OxfordGoogle Scholar
  8. Fowler PW, Pisanski T, Shawe-Taylor JS (1995) Molecular graph eigenvectors for molecular coordinates. In: Tamassia R, Tollis EG (eds) Graph drawing. DIMACS international workshop, GD’94, Princeton, New Jersey, USA, 10–12 October 1994. Lecture Notes in Computer Science 894. Springer, BerlinGoogle Scholar
  9. Godsil CD, Royle GF (2001) Algebraic graph theory. Springer, HeidelbergCrossRefGoogle Scholar
  10. Graovac A, Plavšić D, Kaufman M, Pisanski T, Kirby EC (2000) Application of the adjacency matrix eigenvectors method to geometry determination of toroidal carbon molecules. J Chem Phys 113:1925–1931CrossRefGoogle Scholar
  11. Graovac A, Lászlo I, Plavšić D, Pisanski T (2008a) Shape analysis of carbon nanotube junctions. MATCH Commun Math Comput Chem 60:917–926Google Scholar
  12. Graovac A, László I, Pisanski T, Plavšić D (2008b) Shape analysis of polyhex carbon nanotubes and nanotori. Int J Chem Model 1:355–362Google Scholar
  13. Hall KM (1970) An r-dimensional quadratic placement algorithm. Manag Sci 17:219–229CrossRefGoogle Scholar
  14. Kaufmann M, Wagner D (eds) (2001) Drawing graphs. Methods and models. LNCS 2025. Springer, New YorkGoogle Scholar
  15. Koren Y (2005) Drawing graphs by eigenvectors: theory and practice. Comput Math Appl 49:1867–1888CrossRefGoogle Scholar
  16. László I (2004a) Topological coordinates for nanotubes. Carbon 42:983–986CrossRefGoogle Scholar
  17. László I (2004b) The electronic structure of nanotubes and the topological arrangements of carbon atoms. In: Buzaneva E, Scharff P (eds) Frontiers of multifunctional integrated nanosystems. NATO Science Series, II. Mathematics, Physics and Chemistry, 152. Kluwer Academic Publishers, Dordrecht, p 11Google Scholar
  18. László I (2005) Topological coordinates for Schlegel diagrams of fullerenes and other planar graphs. In: Diudea MV (ed) Nanostructures: novel architecture. Nova, New York, pp 193–202Google Scholar
  19. László I (2008) Hexagonal and non-hexagonal carbon surfaces. In: Blank V, Kulnitskiy B (eds) Carbon nanotubes and related structures. Research Singpost, Kerala, pp 121–146Google Scholar
  20. László I, Rassat A (2003) The geometric structure of deformed nanotubes and the topological coordinates. J Chem Inf Comput Sci 43:519–524CrossRefGoogle Scholar
  21. Lovász L, Schrijver A (1999) On the null space of the Colin de Verdière matrix. Annales de l’Institute Fourier (Grenoble) 49:1017–1026CrossRefGoogle Scholar
  22. Lovász L, Vesztergombi K (1999) Representation of graphs. In: Halász G, Lovász L, Simonovits M, Sós VT (eds) Paul Erdős and his mathematics. Bolyai Society. Springer, New YorkGoogle Scholar
  23. László I, Rassat A, Fowler PW, Graovac A (2001) Topological coordinates for toroidal structures. Chem Phys Lett 342:369–374CrossRefGoogle Scholar
  24. László I, Graovac A, Pisanski T, Plavšić D (2011) Graph drawing with eigenvectors. In: Putz MV (ed) Carbon bonding and structures. Advances in physics and chemistry. Springer, Dordrecht, pp 95–115CrossRefGoogle Scholar
  25. László I, Graovac A, Pisanski T (2012) Nanostructures and eigenvectors of matrices. In: Ashrafi AR, Cataldo F, Graovac A, Iranmanesh A, Ori O, Vukicevic D (eds) Carbon materials chemistry and physics: topological modelling of nanostructures and extended systems. Springer, Dordrecht/Heidelberg/London/New YorkGoogle Scholar
  26. Manolopoulos DE, Fowler PW (1992) Molecular graphs, point groups, and fullerenes. J Chem Phys 96:7603–7614CrossRefGoogle Scholar
  27. Pisanski T, Shawe-Taylor JS (1993) Characterising graph drawing with eigenvectors. In: Technical report CSD-TR-93-20, Royal Holloway, University of London, Department of Computer Science, Egham, Surrey TW200EX, EnglandGoogle Scholar
  28. Pisanski T, Shawe-Taylor JS (2000) Characterising graph drawing with eigenvectors. J Chem Inf Comput Sci 40:567–571CrossRefGoogle Scholar
  29. Pisanski T, Žitnik A (2009) Representing graphs and maps. In: Beineke LW, Wilson RJ (eds) Encyclopedia of mathematics and its applications, 128. Cambridge University Press, Cambridge, pp 151–180Google Scholar
  30. Pisanski T, Plestenjak B, Graovac A (1995) NiceGraph and its applications in chemistry. Croat Chim Acta 68:283–292Google Scholar
  31. Rassat A, László I, Fowler PW (2003) Topological rotational strengths as chirality descriptors for fullerenes. Chem Eur J 9:644–650CrossRefGoogle Scholar
  32. Stone AJ (1981) New approach to bonding in transition-metal clusters and related compounds. Inorg Chem 20:563–571CrossRefGoogle Scholar
  33. Trinajstić N (1992) Chemical graph theory. CRC Press, Boca Raton/Ann Arbor/London/TokyoGoogle Scholar
  34. Tutte WT (1963) How to draw a graph. Proc Lond Math Soc 13:743–768CrossRefGoogle Scholar
  35. van der Holst H (1996) Topological and spectral graph characterizations. Ph.D. thesis, University of Amsterdam, AmsterdamGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • István László
    • 1
  • Ante Graovac
  • Tomaž Pisanski
    • 2
  1. 1.Department of Theoretical Physics, Institute of PhysicsBudapest University of Technology and EconomicsBudapestHungary
  2. 2.Department of Mathematics, Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

Personalised recommendations