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Symmetry properties of chemical graphs VIII. On complementarity of isomerization modes

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Abstract

Isomerization mode defines the process of interconversion of one isomer into another. Several mechanisms are conceivable for degenerate rearrangements and, in general, lead to a distinctive network of relations between participating isomers. Here we consider selected modes which are complementary in the sense that if mode 1 transforms an isomer A into B, C, D etc., then mode 2 transforms the same isomer A into X, Y, Z, etc., which includes all isomers not comprised by the first mode. Physico-chemical complementarity can be translated into mathematical complementarity of associated chemical graphs. This allows us to use the tool of Graph Theory. One example of graph theoretical use is the theorem that graph G and its complement G have the same automorphism group (i.e., the same symmetry). We have shown that a close examination of a graph and its complement and their components allows us to recognize the automorphism group in some complex cases without resorting to canonical numbering or other involved procedures, and even allows us to determine isomorphism of different processes.

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References

  1. Balaban, A. T., Farcasiu, D., Banica, R.: Rev. Roum. Chim. 11, 1205 (1966).

    Google Scholar 

  2. For a concise explanation of graph theoretical terms see: Essam, J. W., Fisher, M. E.: Rev. Mod. Phys., 42, 272 (1970)

    Google Scholar 

  3. For an introduction to graph theoretical concepts see: Wilson, R. J.: Introduction to Graph Theory, London: Longman, 1972

    Google Scholar 

  4. Harary, F.: Graph Theory, p. 165. Reading, Mass.: Addison-Wesley, 1969

    Google Scholar 

  5. Balaban, A. T.: Rev. Roum. Chim. 18, 811 (1973)

    Google Scholar 

  6. Gielen, M.: Bull. Soc. Chim. Belges, 78, 351 (1969)

    Google Scholar 

  7. Muetterties, E. L.: J. Amer. Chem. Soc. 91, 4115 (1969)

    Google Scholar 

  8. Ray, P., Dutt, N. K.: J. Indian Chem. Soc. 20, 81 (1943)

    Google Scholar 

  9. Randić, M., Davis, M. I.: Part VI of this series, Intnl J. Quant. Chem. 26, 69 (1984).

    Google Scholar 

  10. Lloyd, E. K..: MATCH, 7, 255 (1979)

    Google Scholar 

  11. Frucht, R. (Univ. Tecnica Federico Sta Maria, Valparaiso, Chile): private information, 1982

  12. Davidson, R. A.: private correspondence

  13. Randić, M.: J. Chem. Phys. 60, 3920 (1974); (this paper suggested two unrelated schemes to derive the canonical labels, one of which was subsequently recognized as deficient by Mackay, A. L.: J. Chem. Phys. 62, 308 (1975)). The other scheme has been demonstrated practical, even in case of rather complex graphs: Randić, M.: J. Chem. Inf. & Computer Sci. 17, 171 (1977). For computer implementation consult: Randi/'c, M., Brissey, G. M., Wilkins, C. L.: J. Chem. Inf. & Computer Sci. 21, 52 (1981)

    Google Scholar 

  14. Davis, M. I., Ellzey, Jr., M. L.: J. Comput. Chem. 4, 267 (1983)

    Google Scholar 

  15. Dunitz, J. D., Prelog, V.: Agnew. Chem. 80, 700 (1968)

    Google Scholar 

  16. Tutte, W. T.: The Connectivity of Graphs, Toronto Univ. Press, 1967

  17. Sommerville, D. M. Y.: An Introduction to the Geometry of N Dimensions, p. 108, New York: Dover 1958

    Google Scholar 

  18. Randić, M.: J. Comput. Chem. 4, 73 (1983); Part V of this series

    Google Scholar 

  19. Randić, M.: Croat. Chem. Acta 49, 643 (1977); Part I of this series

    Google Scholar 

  20. Randić, M.: Chem. Phys. Lett. 42, 283 (1976)

    Google Scholar 

  21. Hasselbarth, W., Ruch, E.: Theoret. Chim. Acta (Berl.) 29, 259 (1973)

    Google Scholar 

  22. Klein, D. J., Cowley, A. H.: J. Amer. Chem. Soc. 97, 1633 (1975)

    Google Scholar 

  23. Balasubramanian, K.: Theoret. Chim. Acta (Berl.) 53, 129 (1979); Balasubramanian, K.: Int. J. Quant. Chem., 22, 385 (1982); Balasubramanian, K.: Int. J. Quant. Chem., 21, 411 (1982); Balasubramanian, K.: Int. J. Quant. Chem., 22, 1013 (1982); Balasubramanian, K.: Theoret. Chim. Acta (Berl.), 51, 37 (1979); Balasubramanian, K.: J. Chem. Phys., 72, 665 (1980); Balasubramanian, K.: J. Chem. Phys., 75, 4572 (1981); Balasubramanian, K.: “Symmetries and Properties of Non-Rigid Molecules: A Comprehensive Survey” (J. Maruani and J. Serre, Eds.), Proceedings of an International Symposium, Paris (July 1982) (Studies in Physical and Theoretical Chemistry, 23, 149), Amsterdam: Elsevier Publ. 1983

    Google Scholar 

  24. Bauder, A., Meyer, R., Günthard, Hs. H.: Mol. Phys., 28, 1305 (1974); Frei, H., Meyer, R., Bauder, A., Günthard, Hs. H.: Mol. Phys. 32, 43 (1976); Frei, H., Groner, P., Bauder, A., Günthard, Hs. H.: Mol. Phys. 36, 1469 (1978)

    Google Scholar 

  25. Ruch, E., Klein, D. J.: Theoret. Chim Acta (Berl.) 63, 447 (1983)

    Google Scholar 

  26. Ezra, G. S.: “Symmetry Properties of Molecules”, Lecture Notes in Chemistry #28, Berlin: Springer-Verlag, 1982. We would like to add few references not included in the above monograph: Musher, J. I.: Inorg. Chem., 11, 2335 (1972); Klemperer, W. G.: J. Amer. Chem. Soc., 94, 6940 (1972); Eaton, S. S., Eaton, G. R.: J. Amer. Chem. Soc., 95, 1825 (1973); Bouman, T. D., Trindle, C.: Theoret. Chim. Acta (Berl.) 37, 217 (1975); King, R. B.: Theoret. Chim. Acta (Berl.), 56, 269 (1980); Nourse, J. G.: J. Amer. Chem. Soc. 101, 1210 (1979); Nourse, J. G.: J. Amer. Chem. Soc., 102, 4883 (1980)

    Google Scholar 

  27. Balaban, A. T.: Rev. Roum. Chim., 23, 733 (1978)

    Google Scholar 

  28. Randić, M., Katović, V.: Int. J. Quant. Chem. 21, 647 (1982); (Part IV of this series); Randić, M., Katović, V., Trinajstić, N.: “Symmetries and Properties of Non-Rigid Molecules: A Comprehensive Survey” (Maruani, J., Serre, J. Eds.), Proceedings of an International Symposium, Paris (July 1982) (Studies in Physical and Theoretical Chemistry, 23, 399) Amsterdam: Elsevier, 1983

    Google Scholar 

  29. Randić, M.: SIAM J. on Algebraic and Descrite Methods (in press)

  30. Orwell, G.: Animal Farm: “All animals are equal but some animals are more equal than others”

  31. Ordering of structures, which may result in complete or a partial order, appears to have been neglected, yet it offers important new insights into the structures and their properties. For a selection of diverse applications consult: Randić, M.: J. Amer. Chem. Soc. 97, 6609 (1975); Randić, M.: Acta Cryst. A34, 275 (1978); Randić, M.: Chem. Phys. Lett. 55, 547 (1978); Randić, M.: Int. J. Quant. Chem: Quant. Biol. Symp. 5, 245 (1978); Randić, M., Wilkins, C. L.: Chem. Phys. Lett. 63, 332 (1979); Randić, M., Wilkins, C. L.: J. Phys. Chem. 83, 1525 (1979); Randić, M., Wilkins, C. L.: Int. J. Quant. Chem: Quant. Biol. Symp. 6, 55 (1979); Wilkins, C. L., Randić, M.: Theoret. Chim. Acta (Berl.), 58, 45 (1980); Randić, M.: J. Magn. Res., 39, 431 (1980); Randić, M., Gerstein, B. C.: J. Magn. Res. 43, 207 (1981); Wilkins, C. L., Randić, M., Schuster, S. M., Markin, R. S., Steiner, S., Dorgan, L.: Anal. Chim. Acta 133, 637 (1981); Randić, M.: Int. J. Quant. Chem. 23, 1707 (1983); Randić, M., Trinajstić, N.: MATCH 13, 271 (1982); Randić, M. Džonova Jerman-Blažić, B.: J.C.S. Faraday Trans. II (submitted); DeBruin, K. E., Naumann, K. Zon, G., Mislow, K.: J. Amer. Chem. Soc. 91, 7031 (1969); Mislow, K.: Acc. Chem. Res. 3, 381 (1970); Gust, D., Finocchiaro, P., Mislow, K.: Proc. Nat. Acad. Sci., 70, 3445 (1973); Hutchings, M. G., Nourse, J. G., Mislow, K.: Tetrahedron 30, 1535 (1974); Mislow, K.: Acc. Chem. Res. 9, 26 (1976); El-Basil, S., Shalabi, A. S.: Math. Chem. (MATCH), 14, 191 (1983); El-Basil, S.: Int. J. Quant. Chem. (in press)

    Google Scholar 

  32. Randić, M.: Int. J. Quant. Chem. 15, 663 (1979); Part II; Randić, M.: Int. J. Quant. Chem. (Symp.) 14, 557 (1980)

    Google Scholar 

  33. Brandt, J., von Scholley, A.: Comp. & Chem., 7, 51 (1983)

    Google Scholar 

  34. Uchino, M.: J. Chem. Inf. Comput. Sci. 20, 116 (1980) and subsequent papers; Balaban, A. T., Mekenyan, C., Bonchev, D.: J. Chem. Info, and Comput. Sci. (submitted)

    Google Scholar 

  35. Hendrickson, J. B., Toczko, A. G.: J. Chem. Info. and Comput. Sci., 23, 171 (1983)

    Google Scholar 

  36. Randić, M., Woodworth, W. L., Kleiner, A. F.: (work in progress)

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Drake University, 50311, Des Moines, IA

    Milan Randić

  2. Ames Laboratory, Iowa State University, 50011, Ames, IA, USA

    Milan Randić

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  1. Milan Randić
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Dedicated to Professor Kurt Mislow of Princeton University

Operated for the U.S. Department of Energy by Iowa State University under contract W-7405-Eng-82. Supported in part by the Office of the Director.

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Randić, M. Symmetry properties of chemical graphs VIII. On complementarity of isomerization modes. Theoret. Chim. Acta 67, 137–155 (1985). https://doi.org/10.1007/BF00547899

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