Journal of Mathematical Chemistry

, Volume 52, Issue 3, pp 805–819 | Cite as

Large-scale analysis of structural branching measures

Original Paper

Abstract

Structural branching of graphs has been investigated extensively. Yet, no method/model has yet been developed which captures all aspects of branching meaningfully. Another shortcoming of nearly all related work in this area is the fact that only small sets of example graphs have been used to perform those studies. Instead, we investigate structural branching of graphs statistically by using large sets of exhaustively generated graphs. Our findings explain some of the limits of existing branching measures as well as the search for novel branching measures by using correlation analysis.

Keywords

Molecular branching Graphs Graph analysis Quantitative graph measures 

References

  1. 1.
    A. Balaban, Highly discriminating distance-based topological index. Chem. Phys. Lett. 89, 399–404 (1982)CrossRefGoogle Scholar
  2. 2.
    S. Bertz, Branching in graphs and molecules. Discret. Appl. Math. 19, 65–83 (1988)CrossRefGoogle Scholar
  3. 3.
    T. Beyer, S. Hedetniemi, Constant time generation of rooted trees. SIAM J. Comput. 9, 706–712 (1980)CrossRefGoogle Scholar
  4. 4.
    D. Bonchev, Information Theoretic Indices for Characterization of Chemical Structures (Research Studies Press, Chichester, 1983)Google Scholar
  5. 5.
    D. Bonchev, D.H. Rouvray, Complexity in Chemistry, Biology, and Ecology. Mathematical and Computational Chemistry (Springer, New York, 2005)CrossRefGoogle Scholar
  6. 6.
    D. Bonchev, Topological order in molecules 1. Molecular branching revisited. J. Mol. Struct. (Theochem) 336, 137–156 (1995)CrossRefGoogle Scholar
  7. 7.
    D. Bonchev, E. Markel, A. Dekmezian, Topological analysis of long-chain branching patterns in polyolefins. J. Chem. Inf. Comput. Sci. 51(5), 1274–1285 (2001)CrossRefGoogle Scholar
  8. 8.
    D. Bonchev, N. Trinajstic, Information theory, distance matrix, and molecular branching. J. Chem. Phys. 67(10), 4517–4533 (1977)CrossRefGoogle Scholar
  9. 9.
    D. Bonchev, N. Trinajstic, On topological characterization of molecular branching. Int. J. Quant. Chem. 12, 293–303 (1978)Google Scholar
  10. 10.
    G. Chartrand, Introductory Graph Theory (Dover, New York, NY, 1985)Google Scholar
  11. 11.
    J. Claussen, Offdiagonal complexity: a computationally quick complexity measure for graphs and networks. Phys. Stat. Mech. Appl. 375(1), 365–373 (2007)CrossRefGoogle Scholar
  12. 12.
    M. Dehmer, Information processing in complex networks: graph entropy and information functionals. J. Appl. Math. Comput. 201, 82–94 (2008)CrossRefGoogle Scholar
  13. 13.
    M. Dehmer (ed.), Structural Analysis of Complex Networks (Birkhäuser, Cambridge, 2010)Google Scholar
  14. 14.
    M. Dehmer, Information theory of networks. Symmetry 3, 767–779 (2012)CrossRefGoogle Scholar
  15. 15.
    M. Dehmer, F. Emmert-Streib, T. Gesell, A comparative analysis of multidimensional features of objects resembling sets of graphs. Appl. Math. Comput. 196, 221–235 (2008)CrossRefGoogle Scholar
  16. 16.
    M. Dehmer, F. Emmert-Streib, Y. Tsoy, K. Varmuza, Quantifying structural complexity of graphs: information measures in mathematical chemistry, in Quantum Frontiers of Atoms and Molecules, ed. by M. Putz (Nova Publishing, New York, NY, 2011), pp. 479–498Google Scholar
  17. 17.
    M. Dehmer, L. Sivakumar, K. Varmuza, Uniquely discriminating molecular structures using eigenvalue-based descriptors. Match Commun. Math. Comput. Chem. 67(1), 147–172 (2012)Google Scholar
  18. 18.
    F. Emmert-Streib, M. Dehmer, Networks for systems biology: conceptual connection of data and function. IET Syst. Biol. 5, 185–207 (2011)CrossRefGoogle Scholar
  19. 19.
    M. Fischermann, I. Gutman, A. Hoffmann, D. Rautenbach, D. Vidovic, L. Volkmann, Extremal chemical trees. Z. Naturforsch. 57a, 49–52 (2002)Google Scholar
  20. 20.
    B. Furtula, A. Graovac, D. Vukicevic, Augmented Zagreb index. J. Math. Chem. 48(2), 370–380 (2010)CrossRefGoogle Scholar
  21. 21.
    I. Gutman, The energy of a graph. Ber. Math. Statist. Sekt. Forsch. Graz 103, 1–22 (1978)Google Scholar
  22. 22.
    I. Gutman, B. Furtula, M. Ivanovic, Notes on trees with minimal atom-bond connectivity index. Match Commun. Math. Comput. Chem. 67, 467–482 (2012)Google Scholar
  23. 23.
    I. Gutman, M. Randic, Algebraic characterization of skeletal branching. Chem. Phys. Lett. 47(1), 15–19 (1977)CrossRefGoogle Scholar
  24. 24.
    I. Gutman, B. Ruščić, T. Trinajstic, C. Wilcox Jr, Graph theory and molecular orbitals. XII. Acyclic polyenes. J. Chem. Phys. 62(9), 3399–3405 (1975)CrossRefGoogle Scholar
  25. 25.
    I. Gutman, N. Trinajstic, Graph theory and molecular orbitals. total \(\varphi \)-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17(4), 535–538 (1972)CrossRefGoogle Scholar
  26. 26.
    A. Hagberg, D. Schult, P. Swart, Exploring network structure, dynamics and function using NetworkX. In: G. Varoquaux, T. Vaught, J. Millman (eds.) Proceedings of the 7th python in science conference (SciPy2008) (Pasadena, CA, 2008), pp. 11–15Google Scholar
  27. 27.
    G. Hall, Eigenvalues of molecular graphs. Bull. Inst. Math. Appl. 17, 70–72 (1981)Google Scholar
  28. 28.
    F. Harary, Graph Theory (Addison-Wesley Publishing Company, Reading, MA, 1969)Google Scholar
  29. 29.
    H. Hosoya, Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons. Bull. Chem. Soc. Jpn. 44, 2332–2339 (1971)CrossRefGoogle Scholar
  30. 30.
    E. Kirby, Sensitivity of topological indices to methyl group branching in octanes and azulenes, or what does a topological index index? J. Chem. Inf. Comput. Sci. 34, 1030–1035 (1994)CrossRefGoogle Scholar
  31. 31.
    V. Kraus, M. Dehmer, F. Emmert-Streib, Probabilistic inequalities for evaluating structural network measures (2013). Submitted for publicationGoogle Scholar
  32. 32.
    V. Kraus, M. Dehmer, M. Schutte, On sphere-regular graphs and the extremality of information-theoretic network measures (2013). Accepted for publicationGoogle Scholar
  33. 33.
    J. Lozán, H. Kausch, Angewandte Statistik für Naturwissenschaftler, 4th edn. (Wissenschaftliche Auswertungen, Hamburg, 2007)Google Scholar
  34. 34.
    A. Mehler, Social ontologies as generalized nearly acyclic directed graphs: a quantitative graph model of social tagging, in Towards an Information Theory of Complex Networks: Statistical Methods and Applications, ed. by M. Dehmer, F. Emmert-Streib, A. Mehler (Birkhäuser, Boston/Basel, 2011), pp. 259–319CrossRefGoogle Scholar
  35. 35.
    L. Müller, K. Kugler, A. Dander, A. Graber, M. Dehmer, QuACN: an R package for analyzing complex biological networks quantitatively. Bioinformatics 27(1), 140–141 (2011). http://cran.r-project.org/web/packages/QuACN/ Google Scholar
  36. 36.
    A. Mowshowitz, Entropy and the complexity of graphs: I. An index of the relative complexity of a graph. Bull. Math. Biophys. 30(1), 175–204 (1968)CrossRefGoogle Scholar
  37. 37.
    S. Nikolic, G. Kovacevic, A. Milicevic, N. Trinajstic, The Zagreb indices 30 years after. Croat. Chem. Acta 76(2), 113–124 (2003)Google Scholar
  38. 38.
    T. Oliphant, Python for scientific computing. Comput. Sci. Eng. 90, 9 (2007)Google Scholar
  39. 39.
    A. Perdih, M. Perdih, On topological indices indicating branching Part I. The principal component analysis of alkane properties and indices. Acta Chim. Slov. 47, 231–259 (2000)Google Scholar
  40. 40.
    M. Randic, On characterization of molecular branching. J. Am. Chem. Soc. 97(23), 6609–6615 (1975)CrossRefGoogle Scholar
  41. 41.
    H. Schultz, E. Schultz, T. Schultz, Topological organic chemistry. 4. Graph theory, matrix permanents, and topological indices of alkanes. J. Chem. Inf. Comput. Sci. 32(1), 69–72 (1992)CrossRefGoogle Scholar
  42. 42.
    R. Taylor, Interpretation of the correlation coefficient: a basic review. J. Diagn. Med. Sonogr. 1, 35–39 (1990)CrossRefGoogle Scholar
  43. 43.
    R. Todeschini, V. Consonni, Molecular Descriptors for Chemoinformatics, Second, Revised and Enlarged edn. Methods and Principles in Medicinal Chemistry (Wiley, Weinheim, 2009)Google Scholar
  44. 44.
    H. Wiener, Structural determination of paraffin boiling points. J. Am. Chem. Soc. 1(69), 17–20 (1947)CrossRefGoogle Scholar
  45. 45.
    R. Wright, B. Richmond, A. Odlyzko, B. McKay, Constant time generation of free trees. SIAM J. Comput. 15, 540–548 (1986)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute for Bioinformatics and Translational ResearchUMITHall in TirolAustria

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