Combinatorial Self-Similarity

Croatica Chemica Acta

Volumn 69, Issue 3, 1996, Pages 1117-1148

  El Basil, Sherif ^a  

^a CAIRO UNIVERSITY   (Egypt)

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[No Author keywords available]

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EID: 0346435619     PISSN: 00111643     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (55)

1. See, e.g. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, San Francisco, 1983.
2. »Classical Fractals« mean the early ones such as the Cantor set, the Sierpinski triangle and the Koch curve, (see Ref. 3).
3. These fractals and their methods of generation are described in simple and detailed language in: H. O. Peitgen, H. Jürgens and D. Saupe, Chaos and Fractals, Springer-Verlag, New York, 1992, Chapter 2.
4. H. O. Peitgen, H. Jürgens and D. Saupe, Chaos and Fractals, Springer-Verlag, New York, 1992, p. 137, p. 203.
5. S. G. Hoggar, Mathematics for Computer Graphics, Cambridge University Press, Cambridge, 1992, p. 222.
6. R. L. Devaney, A First Course in Chaotic Dynamical Systems, Theory and Experiments, Addison-Wesley Publishing Company, Inc., New York, 1992. pp. 185-190.
7. In p. 185 of the above reference, it is stated that when the author (R. L. Decaney) took a vote among his students on the dimension of the Sierpinski triangle, many said that the latter is two-dimensional, but a sizeable minority voted for one-dimensional; the average vote is 1.6, which is very nearly the accurate fractal dimension of that object!
8. D. H. Rouvray and R. B. Pandey, J. Chem. Phys. 85 (1986) 2286.
9. D. J. Klein, J. J. Cravey, and G. E. Hite, Polycyclic Aromat. Compd. 2 (1991) 165.
10. D. J. Klein, T. P. Živković, and A. T. Balavan, MATCH 29 (1993) 107.
11. D. Plavšić, N. Trinajstić, and D. J. Klein, Croat. Chem. Acta 65 (1992) 279.
12. S. El-Basil, J. Chem. Soc. Faraday Trans. 90 (1994) 2201.
13. S. El-Basil, J. Chem. Soc. Faraday Trans. 89 (1993) 909; J. Mol. Struct. (Theochem) 228 (1993) 67; J. Math. Chem. 14 (1993) 305; J. Mol. Struct. (Theochem) 313 (1994) 237.
14. S. El-Basil, J. Chem. Soc. Faraday Trans. 89 (1993) 909; J. Mol. Struct. (Theochem) 228 (1993) 67; J. Math. Chem. 14 (1993) 305; J. Mol. Struct. (Theochem) 313 (1994) 237.
15. S. El-Basil, J. Chem. Soc. Faraday Trans. 89 (1993) 909; J. Mol. Struct. (Theochem) 228 (1993) 67; J. Math. Chem. 14 (1993) 305; J. Mol. Struct. (Theochem) 313 (1994) 237.
16. S. El-Basil, J. Chem. Soc. Faraday Trans. 89 (1993) 909; J. Mol. Struct. (Theochem) 228 (1993) 67; J. Math. Chem. 14 (1993) 305; J. Mol. Struct. (Theochem) 313 (1994) 237.
17. M. Feigenbaum, J. Stat. Phys. 19 (1978) 25.
18. D. J. Klein and M. Randić, J. Comput. Chem. 8 (1987) 516; S. El-Basil, J. Chem. Soc. Faraday Trans. 90 (1994) 2201.
19. D. J. Klein and M. Randić, J. Comput. Chem. 8 (1987) 516; S. El-Basil, J. Chem. Soc. Faraday Trans. 90 (1994) 2201.
20. E. Clar, The Aromatic Sextet, John Wiley & Sons, London, 1972, p. 41.
21. K. H. Rosen, Elementary Number Theory and its Applications, Addison-Wesley, New York, 1993, Ch. 10 (See also ref. 12).
22. S. J. Cyvin and I. Gutrman, Lecture Notes in Chemistry, Springer, Berlin, 1988, vol. 46.
23. M. Randić, Chem. Phys. Lett. 38 (1976) 68; J. Am. Chem. Soc. 99 (1977) 444; Int. J. Quantum Chem. 17 (1980) 549;
24. M. Randić, Chem. Phys. Lett. 38 (1976) 68; J. Am. Chem. Soc. 99 (1977) 444; Int. J. Quantum Chem. 17 (1980) 549;
25. M. Randić, Chem. Phys. Lett. 38 (1976) 68; J. Am. Chem. Soc. 99 (1977) 444; Int. J. Quantum Chem. 17 (1980) 549;
26. M. Randić, S. Nikolić, and N. Trinajstić, in Graph Theory and Topology in Chemistry, R. B. King and D. H. Rouvray (Eds.), Elsevier, Amsterdam 1987, pp. 429-447.
27. D. J. Klein and W. A. Seitz, J. Mol. Struct. (Theochem) 169 (1988) 167.
28. H. Hosoya, Discrete Appl. Math. 19 (1988) 239 and references therein.
29. C. L. Liu, Elements of Discrete Mathematics, McGraw Hill, 1977, p. 73.
30. This »novel« description of the aromatic sextet is abstracted from the intriguing book: M. Shroeder, Fractals, Chaos Power laws, W. H. Freeman, New York, 1991, H p. 302; The terms proper and improper sextets seem to have been first introduced H in: N. Ohkami and H. Hosoya, Theor. Chim. Acta 64 (1983) 153, to describe the two resonance structures of benzene.
31. This »novel« description of the aromatic sextet is abstracted from the intriguing book: M. Shroeder, Fractals, Chaos Power laws, W. H. Freeman, New York, 1991, H p. 302; The terms proper and improper sextets seem to have been first introduced H in: N. Ohkami and H. Hosoya, Theor. Chim. Acta 64 (1983) 153, to describe the two resonance structures of benzene.
32. Molecular properties may sometimes be expressed in terms of the various component fragments of a molecule via a formal chemical graph theoretical cluster expansion: D. J. Klein, Int. J. Quantum Chem. S 40 (1986) 153, where a given molecular property X(Γ) is expressed in terms of contributions x(G) for connected subgraphs of graph Γ, i.e. X(Γ) = Σ x(G) (R-1) G ∈ C(Γ) where C (Γ) is a set of connected subgraphs of Γ, Eq. (R - 1) is analogous to Eq. (8) and C(Γ) may be thought to correspond to an equivalence class of Kekulé structures ∈ B. The same approach is applied by : T. G. Schmalz, T. Živković, and D. J. Klein, in R. C. Lacher (Ed.) MATH/CHEM/COMP 1987; Studies in Physical and Theoretical Chemistry 54, Elsevier, Amsterdam, 1988, p. 173. More recently, it was used to »re-visit« the concept of resonance energy of conjugated hydrocarbons: D. Babić and N. Trinajstić, Croat. Chem. Acta 65 (1992) 881.
33. Molecular properties may sometimes be expressed in terms of the various component fragments of a molecule via a formal chemical graph theoretical cluster expansion: D. J. Klein, Int. J. Quantum Chem. S 40 (1986) 153, where a given molecular property X(Γ) is expressed in terms of contributions x(G) for connected subgraphs of graph Γ, i.e. X(Γ) = Σ x(G) (R-1) G ∈ C(Γ) where C (Γ) is a set of connected subgraphs of Γ, Eq. (R - 1) is analogous to Eq. (8) and C(Γ) may be thought to correspond to an equivalence class of Kekulé structures ∈ B. The same approach is applied by : T. G. Schmalz, T. Živković, and D. J. Klein, in R. C. Lacher (Ed.) MATH/CHEM/COMP 1987; Studies in Physical and Theoretical Chemistry 54, Elsevier, Amsterdam, 1988, p. 173. More recently, it was used to »re-visit« the concept of resonance energy of conjugated hydrocarbons: D. Babić and N. Trinajstić, Croat. Chem. Acta 65 (1992) 881.
34. Molecular properties may sometimes be expressed in terms of the various component fragments of a molecule via a formal chemical graph theoretical cluster expansion: D. J. Klein, Int. J. Quantum Chem. S 40 (1986) 153, where a given molecular property X(Γ) is expressed in terms of contributions x(G) for connected subgraphs of graph Γ, i.e. X(Γ) = Σ x(G) (R-1) G ∈ C(Γ) where C (Γ) is a set of connected subgraphs of Γ, Eq. (R - 1) is analogous to Eq. (8) and C(Γ) may be thought to correspond to an equivalence class of Kekulé structures ∈ B. The same approach is applied by : T. G. Schmalz, T. Živković, and D. J. Klein, in R. C. Lacher (Ed.) MATH/CHEM/COMP 1987; Studies in Physical and Theoretical Chemistry 54, Elsevier, Amsterdam, 1988, p. 173. More recently, it was used to »re-visit« the concept of resonance energy of conjugated hydrocarbons: D. Babić and N. Trinajstić, Croat. Chem. Acta 65 (1992) 881.
35. Molecular properties may sometimes be expressed in terms of the various component fragments of a molecule via a formal chemical graph theoretical cluster expansion: D. J. Klein, Int. J. Quantum Chem. S 40 (1986) 153, where a given molecular property X(Γ) is expressed in terms of contributions x(G) for connected subgraphs of graph Γ, i.e. X(Γ) = Σ x(G) (R-1) G ∈ C(Γ) where C (Γ) is a set of connected subgraphs of Γ, Eq. (R - 1) is analogous to Eq. (8) and C(Γ) may be thought to correspond to an equivalence class of Kekulé structures ∈ B. The same approach is applied by : T. G. Schmalz, T. Živković, and D. J. Klein, in R. C. Lacher (Ed.) MATH/CHEM/COMP 1987; Studies in Physical and Theoretical Chemistry 54, Elsevier, Amsterdam, 1988, p. 173. More recently, it was used to »re-visit« the concept of resonance energy of conjugated hydrocarbons: D. Babić and N. Trinajstić, Croat. Chem. Acta 65 (1992) 881.
36. 2 + ..., See, e.g. R. L. Graham, D. E. Knuth and O. Patashink, Concrete Mathematics, Addison - Wesley, Reading (1989) p. 309.
37. H. Hosota, Bull. Chem. Soc. Japan 44 (1971) 2323; Fibonacci Quart. 11 (1973) 255.
38. H. Hosota, Bull. Chem. Soc. Japan 44 (1971) 2323; Fibonacci Quart. 11 (1973) 255.
39. Properties of many families of benzenoid systems are outlined in: E. Clar, The Aromatic Sextet, Wiley, New York, 1972, Polycyclic hydrocarbons, Academic Press, London, 1964 and references therein.
40. Properties of many families of benzenoid systems are outlined in: E. Clar, The Aromatic Sextet, Wiley, New York, 1972, Polycyclic hydrocarbons, Academic Press, London, 1964 and references therein.
41. S. El-Basil, Theor. Chim. Acta 65 (1984), 199, 65 (1984) 191, J. Math. Chem. 2 (1988) 1, S. El-Basil, P. Křivka, and N. Trinajstić, J. Math. Phys. 26 (1985) 2396, I. Gutman and S. El-Basil, MATCH 20 (1986) 81, S. El-Basil, J. Comput. Chem. 8 (1987) 956.
42. S. El-Basil, Theor. Chim. Acta 65 (1984), 199, 65 (1984) 191, J. Math. Chem. 2 (1988) 1, S. El-Basil, P. Křivka, and N. Trinajstić, J. Math. Phys. 26 (1985) 2396, I. Gutman and S. El-Basil, MATCH 20 (1986) 81, S. El-Basil, J. Comput. Chem. 8 (1987) 956.
43. S. El-Basil, Theor. Chim. Acta 65 (1984), 199, 65 (1984) 191, J. Math. Chem. 2 (1988) 1, S. El-Basil, P. Křivka, and N. Trinajstić, J. Math. Phys. 26 (1985) 2396, I. Gutman and S. El-Basil, MATCH 20 (1986) 81, S. El-Basil, J. Comput. Chem. 8 (1987) 956.
44. S. El-Basil, Theor. Chim. Acta 65 (1984), 199, 65 (1984) 191, J. Math. Chem. 2 (1988) 1, S. El-Basil, P. Křivka, and N. Trinajstić, J. Math. Phys. 26 (1985) 2396, I. Gutman and S. El-Basil, MATCH 20 (1986) 81, S. El-Basil, J. Comput. Chem. 8 (1987) 956.
45. S. El-Basil, Theor. Chim. Acta 65 (1984), 199, 65 (1984) 191, J. Math. Chem. 2 (1988) 1, S. El-Basil, P. Křivka, and N. Trinajstić, J. Math. Phys. 26 (1985) 2396, I. Gutman and S. El-Basil, MATCH 20 (1986) 81, S. El-Basil, J. Comput. Chem. 8 (1987) 956.
46. S. El-Basil, Theor. Chim. Acta 65 (1984), 199, 65 (1984) 191, J. Math. Chem. 2 (1988) 1, S. El-Basil, P. Křivka, and N. Trinajstić, J. Math. Phys. 26 (1985) 2396, I. Gutman and S. El-Basil, MATCH 20 (1986) 81, S. El-Basil, J. Comput. Chem. 8 (1987) 956.
47. R. P. Grimaldi, Discrete and Combinatorial Mathematics, Addison-Wesley, Reading, 1985, ch. 11.
48. I. Gutman, Theor. Chim. Acta 45 (1977) 309.
49. S. El-Basil, J. Math. Chem. 1 (1987) 153.
50. S. El-Basil and M. Randić, Advan. Quantum Chem. 24 (1992) 239.
51. Several recent papers deal with benzenoid systems. See e.g. M. Randić, D. J. Klein, H. Zhu, N. Trinajstić, and T. Živković, Fizika A 3 (1994) 61, A. T. Balaban, X. Liu, S. J. Cyvin, and D. J. Klein, J. Chem. Inf. Comput. Sci. 33 (1993) 429, D. J. Klein, Chem. Phys. Lett. 217 (1994) 261, D. J. Klein, J. Chem. Inf. Comput. Sci. 34 (1994) 453, M. Randić, Y. Tsukano, and H. Hosoya, Nat. Rep. Ochan. Univ. 45 (1994) 101.
52. Several recent papers deal with benzenoid systems. See e.g. M. Randić, D. J. Klein, H. Zhu, N. Trinajstić, and T. Živković, Fizika A 3 (1994) 61, A. T. Balaban, X. Liu, S. J. Cyvin, and D. J. Klein, J. Chem. Inf. Comput. Sci. 33 (1993) 429, D. J. Klein, Chem. Phys. Lett. 217 (1994) 261, D. J. Klein, J. Chem. Inf. Comput. Sci. 34 (1994) 453, M. Randić, Y. Tsukano, and H. Hosoya, Nat. Rep. Ochan. Univ. 45 (1994) 101.
53. Several recent papers deal with benzenoid systems. See e.g. M. Randić, D. J. Klein, H. Zhu, N. Trinajstić, and T. Živković, Fizika A 3 (1994) 61, A. T. Balaban, X. Liu, S. J. Cyvin, and D. J. Klein, J. Chem. Inf. Comput. Sci. 33 (1993) 429, D. J. Klein, Chem. Phys. Lett. 217 (1994) 261, D. J. Klein, J. Chem. Inf. Comput. Sci. 34 (1994) 453, M. Randić, Y. Tsukano, and H. Hosoya, Nat. Rep. Ochan. Univ. 45 (1994) 101.
54. Several recent papers deal with benzenoid systems. See e.g. M. Randić, D. J. Klein, H. Zhu, N. Trinajstić, and T. Živković, Fizika A 3 (1994) 61, A. T. Balaban, X. Liu, S. J. Cyvin, and D. J. Klein, J. Chem. Inf. Comput. Sci. 33 (1993) 429, D. J. Klein, Chem. Phys. Lett. 217 (1994) 261, D. J. Klein, J. Chem. Inf. Comput. Sci. 34 (1994) 453, M. Randić, Y. Tsukano, and H. Hosoya, Nat. Rep. Ochan. Univ. 45 (1994) 101.
55. Several recent papers deal with benzenoid systems. See e.g. M. Randić, D. J. Klein, H. Zhu, N. Trinajstić, and T. Živković, Fizika A 3 (1994) 61, A. T. Balaban, X. Liu, S. J. Cyvin, and D. J. Klein, J. Chem. Inf. Comput. Sci. 33 (1993) 429, D. J. Klein, Chem. Phys. Lett. 217 (1994) 261, D. J. Klein, J. Chem. Inf. Comput. Sci. 34 (1994) 453, M. Randić, Y. Tsukano, and H. Hosoya, Nat. Rep. Ochan. Univ. 45 (1994) 101.