Advertisement

Journal of Mathematical Chemistry

, Volume 1, Issue 2, pp 153–174 | Cite as

Applications of caterpillar trees in chemistry and physics

  • Sherif El-Basil
Review

Abstract

The relations of caterpillar trees (which are also known as Gutman trees and benzenoid trees) to other mathematical objects such as polyhex graphs, Clar graphs, king polyominos, rook boards and Young diagrams are discussed. Potential uses of such trees in data reduction, computational graph theory, and in the ordering of graphs are considered. Combinatorial and physical properties of benzenoid hydrocarbons can be studied via related caterpillars. It thus becomes possible to study the properties of large graphs such as benzenoid (i.e. polyhex) graphs in terms of much smaller tree graphs. Generation of the cyclic structures of wreath and generalized wreath product groups through the use of caterpillar trees is illustrated.

Keywords

Hydrocarbon Graph Theory Data Reduction Product Group Small Tree 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    F. Harary and A.J Schwenk, Discrete Math. 6 (1973)359.Google Scholar
  2. [2]
    Ref. [1],p. 361.Google Scholar
  3. [3]
    F. Harary and A.J. Schwenk, Mathematika 18 (1971)138.Google Scholar
  4. [4]
    F. Harary and A.J. Schwenk, Utilitas Math. 1 (1972)203.Google Scholar
  5. [5]
    I. Gutman, Theor. Chim. Acta 45 (1977)309.Google Scholar
  6. [6]
    S. El-Basil, J. Chem. Soc., Faraday Trans. 2, 82 (1986)299.Google Scholar
  7. [7]
    S. El-Basil, Theor. Chim. Acta 65 (1984)199.Google Scholar
  8. [8]
    S. El-Basil, Theor. Chim. Acta 65 (1984)191.Google Scholar
  9. [9]
    Ordering of structures has been explicitly considered in both the mathematical and chemical literature. Thus, J.F. Nagle, J. Math. Phys. 7(1966)J88. considered a general linear ordering relation for graphs with the same number of vertices, while E. Ruch, Theor. Chim. Acta 38(1975)167, considered ordering of Young diagrams. M. Randić and C.L. Wilkins, Chem. Phys. Lett. 63(1979)332; J. Phys. Chem. 83(1979)1525, considered ordering of alkanes and how it reflects on their physical properties.Google Scholar
  10. [10]
    Here, the termdata reduction means studying the properties of a large molecule in terms of those of a smaller one (see refs. [4041]).Google Scholar
  11. [11]
    The term polyhex graphs means the molecular graph of a benzennid hydroearbon. This term was bu used in: N. Ohkami. A. Motoyama, T. Yamaguchi, H. Hosoya and I. Gutman,Tetrahedron 37(1981)1113.Google Scholar
  12. [12]
    A recent account mar be found in: N. Trinajstić,Chemical Graph Theory, Vol. 2 (CRC Boca Raton, Florida, 1983)Google Scholar
  13. [13]
    Cf. H. Sachs,Combinatozioa4, 1 (1984)89.Google Scholar
  14. [14]
    C. Domb and M.S. Green,Phase Transitions and Critical Phenomena, Vols. 1–3, ed. M.S. Green (Academic Press, London, 1972).Google Scholar
  15. [15]
    I. Gutman, Z. Naturforsch. a37 (1982)69.Google Scholar
  16. [16]
    I. Gutman and S. EI-Basil, Z. Naturforsch. a39 (1984)276.Google Scholar
  17. [17]
    Line graphs are defined in: F. Hamry,Graph Theory (Addison-Wesley, Reading, 1969)Ch. 8.Google Scholar
  18. [18]
    King polyomino graphs were first defined in: A. Motoyama and H. Hosoya, J. Math. Phys. 18 (1977)1485.Google Scholar
  19. [19]
    Cf. C.D. Godsil and I. Gutman, Croat. Chem. Acta 54 (1981)53.Google Scholar
  20. [20]
    A bipartite graph (bigraph bicolarable graph)G is a graph whose vertex setV can be partitioned into two subsetsV 1 andV 2 such that every line ofG joinsV 1 withV 2 (see ref. [14], p. 17).Google Scholar
  21. [21]
    D. Hosoya, Bull. Chem. Soc. Japan 44 (1971)2332.Google Scholar
  22. [22]
    H. Hosoya and T. Yamaguchi, Tetrahedron Lett. (1975) 4659.Google Scholar
  23. [23]
    N. Ohkami and H. Hosoya, Theor. Chim. Acta 64 (1983)153.Google Scholar
  24. [24]
    Generating functions are defined in books on combinatorics: V. Krishnamurthy,Combinatorics: Theory and Applications (E. Horwood, New York, Halsted Press, 1986). The sextet polynomial [21] is a very special form of generating functions which generate all possible ways in which one can perfectly match [10] the edges of a polyhex graph [8].Google Scholar
  25. [25]
    A recent generalized approach to structure count is found in: B. Ruščić, N. Trinajstić and P. Křivka, Theor. Chim. Acta 69 (1986)107.Google Scholar
  26. [26]
    One of the earliest works pertaining to chemical combinatorics is that of: M. Gordon and W.T. Davison, J. Chem. Phys. 20 (1952)428.Google Scholar
  27. [27]
    Proper and improper sextets are defined in ref. [22] .Google Scholar
  28. [28]
    H. Hosoya and N. Ohkami, J. Comput. Chem. 4 (1983)585.Google Scholar
  29. [29]
    E. Heilbronner, Helv. Chim. Acta 36 (1953)171.Google Scholar
  30. [30]
    Details of some aspects of Clar sextet theory may be found in: I. Gutman, Bull. Soc. Chim. Beograd 47 (1982)453.Google Scholar
  31. [31]
    J. Aihara, Bull. Chem. Soc. Japan 49 (1976)1429.Google Scholar
  32. [32]
    A0 “Dewar-type” resonance theory depends on the concept of a reference structure: M.J.S. Dewar and C. de Llano, J. Amer. Chem. Soc. 91 (1969)789.Google Scholar
  33. [33]
    I. Gutman, Croat. Chem. Acta 56 (1983)365.Google Scholar
  34. [34]
    M. Randić, J. Amer. Chem. Soc. 99 (1977)444; Tetrahedron 33(1977)1905.Google Scholar
  35. [35]
    W.C. Herndon and M.L. Ellzey, Jr., J. Amer. Chem. Sac. 96 (1974)6631.Google Scholar
  36. [36]
    K. Balasubramanian and M. Randić, Theor. Chim. Acta 61 (1982)307; see also K. Balasubramanian, Int. J. Quantum Chem. 22(1982)581.Google Scholar
  37. [37]
    I. Gutman and M. Randić, Chem. Phys. Lett. 47 (1977)15.Google Scholar
  38. [38]
    G.H. Hardy, J.E. Littlewood and G. Pólya,Inequalities (Cambridge University Press, London, 1934) p. 44.Google Scholar
  39. [39]
    E. Ruch, Theor. Chim. Acta 38 (1975)167; E. Ruch and A. Schönhofer, Theor. Chim. Acta 19(1970)225.Google Scholar
  40. [40]
    A. Smolenskii, Russ. J. Phys. Chem. (English translation) 38 (1964)700.Google Scholar
  41. [41]
    M. Gordon and J.W. Kennedy, J. Chem. Soc., Faraday Trans. 2, 68 (1972)484.Google Scholar
  42. [42]
    A.T. Balaban and F. Harary, Tetrahedron 24 (1968)2505.Google Scholar
  43. [43]
    S. El-Basil, Chem. Phys. Lett. (1987), in press.Google Scholar
  44. [44]
    M. Randić, J. Amer. Chem. Soc. 97 (1975)6609.Google Scholar
  45. [45]
    F. Harary,Graph Theory (Addison-Wesley, Reading, 1969) p. 164.Google Scholar
  46. [46]
    K. Balasubramanian, Chem. Rev. 85 (1985)599; and numerous references cited therein.Google Scholar
  47. [47]
    G. Pólya, Acta Math. 65 (1937)145.Google Scholar
  48. [48]
    K. Balasubramanian, Studies in Physical and Theoretical Chemistry 23 (1982)149.Google Scholar
  49. [49]
    K. Balasubramanian, Theor. Chim. Acta 51 (1979)37.Google Scholar

Copyright information

© J.C. Baltzer AG, Scientific Publishing Company 1987

Authors and Affiliations

  • Sherif El-Basil
    • 1
  1. 1.Department of ChemistryUniversity of GeorgiaAthensUSA

Personalised recommendations