Information-Theoretic Methods in Chemical Graph Theory



During recent years, information theory has been used extensively in chemistry for describing chemical structures and providing good correlations between physicochemical and structural properties. In this chapter, we present a survey on information-theoretic methods which are used in chemical graph theory.


Entropy Information content of molecular graph Information-theoretic methods Molecular graph Shannon relation Topological and information indices 



The research was supported by the RFBR grant 09–01–00244.


  1. As56.
    Ashby, W.R.: An Introduction to Cybernetics. Wiley, New York (1956)MATHGoogle Scholar
  2. AC72.
    Aslangul, C., Constanciel, R., Daudel, R., Kottis, P.: Aspects of the localizability of electrons in atoms and molecules: Loge theory and related methods. Adv. Quant. Chem. 6, 93–141 (1972)CrossRefGoogle Scholar
  3. AC74.
    Aslangul, C., Constanciel, R., Daudel, R., Esnault, L., Ludena, E.: The Loge theory as a starting point for variational calculations. I. General formalism. Int. J. Quant. Chem. 8, 499–522 (1974)CrossRefGoogle Scholar
  4. Ba76.
    Balaban, A.T., (ed.): Chemical Applications of Graph Theory. Academic, New York (1976)Google Scholar
  5. Ba82.
    Balaban, A.T.: Highly discriminating distance-based topological indices. Chem. Phys. Lett. 89, 399–404 (1982)MathSciNetCrossRefGoogle Scholar
  6. Ba83.
    Balaban, A.T.: Topological indices based on topological distances in molecular graphs. Pure Appl. Chem. 55, 199–206 (1983)CrossRefGoogle Scholar
  7. Ba85.
    Balaban, A.T.: Applications of graph theory in chemistry. J. Chem. Inf. Comput. Sci. 25, 334–343 (1985)Google Scholar
  8. BX71.
    Balaban, A.T., Harary, F.: The characteristic polynomial does not uniquely determine the topology of a molecule. J. Chem. Doc. 11, 258–259 (1971)CrossRefGoogle Scholar
  9. BC80.
    Balaban, A.T., Chirac, A., Motoc, I., Simon, Z.: Steric Fit in Quantitative Structure–Activity Relationships. Lecture Notes in Chemistry. Springer, Berlin (1980)Google Scholar
  10. BBC94.
    Balaban, A.T., Basak, S.C., Colburn, T., Grunwald, G.D.: Correlation between structure and normal boiling points of Haloalkanes C1–C4 using neural networks. J. Chem. Inf. Comp. Sci. 34, 1118–1121 (1994)Google Scholar
  11. Ba00.
    Balaban, A.T.: QSAR and computational methods in drug discovery, In: Meyers, R.A. (ed.) Encyclopedia of Analytical Chemistry, vol. 8, pp. 7288–7311. Wiley, Chichester (2000)Google Scholar
  12. BF69.
    Ban, T., Fujita, T.: Mathematical approach to structure-activity study of sympathomimetic amines. J. Med. Chem. 12, 353–356 (1969)CrossRefGoogle Scholar
  13. B99.
    Basak, S.C.: Information theoretic indices of neighborhood complexity and their applications, In: Devillers, J., Balaban, A.T. (eds.) Topological Indices and Related Descriptors in QSAR and QSPR, pp. 563–593. Gordon and Breach Science Publishers, The Netherlands (1999)Google Scholar
  14. BRG80.
    Basak, S.C., Roy, A.B., Ghosh, J.J.: Study of the structure–function relationship of pharmacological and toxicological agents using information theory, In: Avula, X.J.R., Bellman, R., Luke, Y.L., Rigler, A.K. (eds.) Proceeding of the 2nd International Conference on Mathematical Modelling, vol.2, pp. 851–856. University of Missouri–Rolla, Rolla, Missouri (1980)Google Scholar
  15. BM83.
    Basak, S.C., Magnusson, V.R.: Molecular topology and narcosis: a quantitative structure-activity relationship (QSAR) study of alcohols using complementary informartion content (CIC). Arzneim. Forsch. Drug Res. 33, 501–503 (1983)Google Scholar
  16. BNV91.
    Basak, S.C., Niemi, G.J., Veith, G.D.: Predicting properties of molecules using graph invariants. J. Math. Chem. 7, 243–272 (1991)CrossRefGoogle Scholar
  17. BBG93.
    Basak, S.C., Bertelsen, S., Grunwald, G.D.: Application of graph theoretical parameters in quantifying molecular similarity and structure–activity studies. J. Cem. Inf. Comput. Sci. 34, 270–276 (1993)Google Scholar
  18. BG93.
    Basak, S.C., Grunwald, G.D.: Use of graph invariants, volume and total surface area in predicting boiling point of alkanes. Math. Model. Sci. Comp. 2, 735–740 (1993)Google Scholar
  19. BG95.
    Basak, S.C., Grunwald, G.D.: Molecular similarity and estimation of molecular properties. J. Chem. Inf. Comput. Sci. 35, 366–372 (1995)Google Scholar
  20. BBG00.
    Basak, S.C., Balaban, A.T., Grunwald, G.D., Gute, B.D.: Topological indices: their nature and mutual relatedness. J. Chem. Inf. Comput. Sci. 40, 891–898 (2000)Google Scholar
  21. BGB04.
    Basak, S.C., Gute, B.D., Balaban, A.T. Interrelationship of major topological indices evidenced by clustering. Croat. Chem. Acta CCACAA 77, 331–344 (2004)Google Scholar
  22. BL72.
    Bernstein, R.B., Levine, R.D.: Entropy and chem. change. I. Characterization of product (and reactant) energy distributions in reactive molecular collisions: information and entropy deficiency. J. Chem. Phys. 57, 434–449 (1972)Google Scholar
  23. BLB72.
    Ben-Shaul, A., Levine, R.D., Bernstein, R.B.: Entropy and chem. change. II. Analysis of product energy distributions: temperature and entropy deficiency. J. Chem. Phys. 57, 5427–5447 (1972)Google Scholar
  24. Bon79.
    Bonchev, D.: Information indices for atoms and molecules. MATCH Commun. Math. Comput. Chem. 7, 65–113 (1979)Google Scholar
  25. Bon81.
    Bonchev, D.: Information theory interpretation of the Pauli principle and Hund rule. Int. J. Quant. Chem. 19, 673–679 (1981)CrossRefGoogle Scholar
  26. Bon83.
    Bonchev, D.: Information–theoretic Indices for Characterization of Chemical Structures. Research Studies Press, Chichester (1983)Google Scholar
  27. BKK76.
    Bonchev, D., Kamenski, D., Kamenska, V.: Symmetry and information content of chemical structures. Bull. Math. Biophys. 38, 119–133 (1976)Google Scholar
  28. BT77.
    Bonchev, D., Trinajstić, N.: Information theory, distance matrix, and molecular branching. J. Chem. Phys. 38, 4517–4533 (1977)CrossRefGoogle Scholar
  29. BT78.
    Bonchev, D., Trinajstić, N.: On topological characterization of molecular branching. Int. J. Quant. Chem. S12, 293–303 (1978)Google Scholar
  30. BKT79.
    Bonchev, D., Knop, J.V., Trinajstić, N.: Mathematical models of branching. MATCH Commun. Math. Comput. Chem. 6, 21–47 (1979)Google Scholar
  31. BMT80.
    Bonchev, D., Mekenyan, O., Trinajstić, N.: Topological characterization of cyclic structure. Int. J. Quant. Chem. 17, 845–893 (1980)CrossRefGoogle Scholar
  32. BMT81.
    Bonchev, D., Mekenyan, O., Trinajstić, N.: Isomer discrimination by topological information approach. J. Comput. Chem. 2, 127–148 (1981)MathSciNetCrossRefGoogle Scholar
  33. BT82.
    Bonchev, D., Trinajstić, N.: Chemical information theory. Structural Aspects. Int. J. Quant. Chem. Symp. 16, 463–480 (1982)Google Scholar
  34. Bril56.
    Brillouin, L.: Science and Information Theory. Academic, New Nork (1956)MATHGoogle Scholar
  35. CS78.
    Chapman, N.B., Shorter J. (eds.): Correlation Analysis in Chemistry. Plenum, New York (1978)Google Scholar
  36. CBF68.
    Crum-Brown, A., Fraser, T.R.: Trans Royal Soc. Edinburgh 25, 151–203, 257–274, 693–739 (1868-1869)Google Scholar
  37. DB74.
    Daudel, R., Bader, R.F., Stephens, M.E., Borett, D.S.: The electron pair in chemistry. Can. J. Chem. 52, 1310–1320 (1974)CrossRefGoogle Scholar
  38. DBE08.
    Dehmer, M., Borgert, S., Emmert–Streib, F.: Entropy bounds for hierarchical molecular networks. PLoS ONE 3, e3079 (2008)Google Scholar
  39. DE08.
    Dehmer, M., Emmert–Streib, F.: Structural information content of networks: graph entropy based on local vertex functionals. Comput. Biol. Chem. 32, 131–138 (2008)Google Scholar
  40. D96.
    Devillers, J. (ed.): Genetic Algorithms in Molecular Modeling (Principles of QSAR and Drug Design). Academic, London (1996)Google Scholar
  41. DB99.
    Devillers, J., Balaban, A.T. (eds.).: Topological Indices and Related Descriptors in QSAR and QSPR. Gordon and Breach, Amsterdam, Netherlands (1999)Google Scholar
  42. DG98.
    Diudea, M.V., Gutman, I.: Wiener–type topological indices. Croat. Chem. Acta CCACAA 71, 21–51 (1998)Google Scholar
  43. Do98.
    Dobrynin, A.A.: Discriminating power of the Schultz index for cata–condensed benzenoid graphs. MATCH Commun. Math. Comput. Chem. 38, 19–32 (1998)MathSciNetMATHGoogle Scholar
  44. DG77.
    Doyle, J.K., Graver, J.E.: Mean distance in graph. Discrete Math. 17, 147–154 (1977)MathSciNetMATHGoogle Scholar
  45. DLA73.
    Dubois,J.E., Laurent, D., Aranda, A.: Perturbation of environments which are limited, concentric and ordered. J. Claim. Phys. 11, 1608–1616 (1973)Google Scholar
  46. DLA74.
    Dubois, J.E.: DARC system in chemistry, In: Wipke, W.T., Heller, S., Fellmann, R., Hyde, E. (eds.) Computer Representation and Manipulation of Chemical Information, pp. 239–263. Wiley, New York (1974)Google Scholar
  47. En76.
    Entringer, R.C., Jackson, D.E., Snyder, D.A.: Distance in graphs. Czechoslovak Math. J. 2, 283–297 (1976)MathSciNetGoogle Scholar
  48. En97.
    Entringer, R.C.: Distance in graphs: trees. J. Combin. Math. Combin. Comput. 24, 65–84 (1997)MathSciNetMATHGoogle Scholar
  49. F94.
    Fisher, E.: Einfluss der Configuration auf die Wirkung der Enzyme. Chem. Ber. 74, 70–77 (1986)Google Scholar
  50. FBE80.
    Fratev, F., Bonchev, D., Enchev, V.: A theoretical information approach to ring and total aromaticity in ground and excited states. Croat. Chem. Acta CCACAA 53, 545–554 (1980)Google Scholar
  51. FEP82.
    Fratev, F., Enchev, V., Polansky, O.E., Bonchev, D.: A theoretical-information study on the electron delocalization (aromaticity) of annulenes with and without bond alternation. THEOCHEM 88, 105–118 (1982)CrossRefGoogle Scholar
  52. FW64.
    Free, S.M., Wilson, I.W.: A mathematical contribution to structure–activity studies. J. Med. Chem. 7, 395–399 (1964)CrossRefGoogle Scholar
  53. FB71.
    Fujita, T., Ban, T.: Structure-activity relation 3. Structure-activity study of phenethylamines as substrates of biosynthetic enzymes of sympathetic transmitters. J. Med. Chem. 14, 148–152 (1971)Google Scholar
  54. Gu94.
    Gutman, I.: Selected properties of the Schultz molecular index. J. Chem. Inf. Comput. Sci. 34, 1087–1089 (1994)Google Scholar
  55. GC89.
    Gutman, I., Cyvin, S.J.: Introduction to the Theory of Benzenoid Hydrocarbons. Springer, Berlin (1989)Google Scholar
  56. GC90.
    Gutman, I., Cyvin, S.J. (eds): Advances in the Theory of Benzenoid Hydrocarbons. Springer, Berlin (1990)Google Scholar
  57. HSK72.
    Hansch, C., Schaeffer, J., Kerley, R. Alcohol dehydrogenase structure–activity relationships. J. Biol. Chem. 247, 4703–4710 (1972)Google Scholar
  58. H77.
    Hansch, C.: On the predictive value of QSAR. In: Buisman, K. (ed.) Biological Activity and Chemical Structure, pp.47–61. Elsevier, Amsterdam (1977)Google Scholar
  59. H80.
    Hansch, C.: Use of quantitative structure-activity relationships (QSAR) in drug design (review). Pharm. Chem. J. 14, 678–691 (1980)CrossRefGoogle Scholar
  60. H81.
    Hansch, C.: The physicochemical approach to drug design and discovery (QSAR). Drug Dev. Res. 1, 267–309 (1981)CrossRefGoogle Scholar
  61. HL95.
    Hansch C., Leo A.: Exploring QSAR Fundamentals and Applications in Chemistry and Biology. ACS, Washington DC, USA (1995)Google Scholar
  62. Ha69.
    Harary, F.: Graph Theory. Addison–Wesley, Reading, MA (1969)Google Scholar
  63. HGL90.
    Hernandez-Gallegos, Z., Lehmann, F.P.A. : A Free–Wilson/Fujita–Ban analysis and prediction of the analgesic potency of some 3-hydroxy- and 3-methoxy-N-alkylmorphinan-6-one opioids. J. Med. Chem. 33, 2813–2817 (1990)CrossRefGoogle Scholar
  64. Ho71.
    Hosoya, H.: Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of hydrocarbons. Bull. Chem. Soc. Jpn. 44, 2332–2339 (1971)Google Scholar
  65. JBM88.
    Johnson, M., Basak, S.C., Maggiora, G.: A characterization of molecular similarity methods for property prediction. Math. Comput. Modell. 11, 630–6634 (1988)MathSciNetCrossRefGoogle Scholar
  66. JM90.
    Johnson, M., Maggiora, G.: Concepts and Applications of Molecular Similarity. Wiley, New York (1990)Google Scholar
  67. K00.
    Karelson, M.: Molecular descriptors in QSAR/QSPR. Wiley, New York (2000)Google Scholar
  68. KL94.
    Karlitzky, A.R., Lobanov, V.S.: CODESSA, Version 5.3, University of Florida, Gainesville (1994)Google Scholar
  69. KH76.
    Kier, L.B., Hall, L.H.: Molecular Connectivity in Chemistry and Drug Research. Academic, New York (1976)Google Scholar
  70. KH81.
    Kier, L.B., Hall, L.H.: Derivation and significance of valence molecular connectivity. J. Pharm. Sci. 70, 583–589 (1981)CrossRefGoogle Scholar
  71. KH86.
    Kier, L.B., Hall, L.H.: Molecular Connectivity in Structure–Activity Analysis. Research Studies Press, Letchworth (1986)Google Scholar
  72. KR87.
    King, R.B., Rouvray, D.H. (eds.): Graph Theory and Topology in Chemistry. Elsevier, Amsterdam (1987)MATHGoogle Scholar
  73. Kol69.
    Kolmogorov, A.N.: On logic basis of information theory. Probl. Peredachi Inf. 5, 3–7 (1969)MathSciNetMATHGoogle Scholar
  74. KP90.
    Konstantinova, E.V., Paleev, A.A.: Sensitivity of topological indices of polycyclic graphs. Vychisl. Sistemy 136, 38–48 (1990)MathSciNetMATHGoogle Scholar
  75. Kon96.
    Konstantinova, E.V.: The discrimination ability of some topological and information distance indices for graphs of unbranched hexagonal systems. J. Chem. Inf. Comput. Sci. 36, 54–57 (1996)Google Scholar
  76. KD00.
    Konstantinova, E.V., Diudea, M.V.: The Wiener polynomial derivatives and other topological indices in chemical research. Croat. Chem. Acta CCACAA 73, 383–403 (2000)Google Scholar
  77. KV03.
    Konstantinova, E.V., Vidyuk, M.V.: Discriminating tests of information and topological indices. Animals and trees. J. Chem. Inf. Comp. Sci. 43, 1860–1871 (2003)Google Scholar
  78. K88.
    Kubinyi, H.: Free–Wilson Analysis. Theory, applications and its relationship to Hansch analysis. Quant. Struct. Act. Relat. 7, 121–133 (1988)Google Scholar
  79. K93.
    Kubinyi, H.: QSAR: Hansch Analysis and Related Approaches. In: Mannhold, R., Kroogsgard-Larsen, P., Timmerman, H. (eds.) Methods and Principles in Medicinal Chemstry. VCH, Weinheim (1993)Google Scholar
  80. L90.
    Lajiness, M.S.: Molecular similarity–based methods for selecting compounds for screening. In: Rouvray, D.H. (ed.) Computational Chemical Graph Theory. Nova Science Publishers, New York (1990)Google Scholar
  81. MH83.
    Magnuson, V.R., Harris, D.K., Basak, S.C.: Topological indices based on neighborhood symmetry: chemical and biological application. In: King, R.B. (ed.) Chemical Applications of Topology and Graph Theory, pp. 178–191. Elsevier, Amsterdam (1983)Google Scholar
  82. Mar71.
    Marshall, C.W.: Applied Graph Theory. Wiley-Interscience, New York (1971)MATHGoogle Scholar
  83. MB86.
    Mekenyan, O., Bonchev, D.: OASIS method for predicting biological activity of chemical copounds. Acta Pharm. Jugosl. 36, 225–237 (1986)Google Scholar
  84. MMB93.
    Mekenyan, O., Mercier, C., Bonchev, D., Dubois, J.E.: Comparative study of DARC/PELCO and OASIS methods. II. Modelling PNMT inhibitory potency of benzylamines and amphetamines. Eur. J. Med. Chem. 28, 811–819 (1993)Google Scholar
  85. MC03.
    MolConnZ, Ver.4.05, Hall Ass. Consult., Quincy, MA (2003)Google Scholar
  86. Mo65.
    Morgan, H.L.: The generation of a unique machine description of chemical structures – a technique developed at chemical abstracts service. J. Chem. Doc. 5, 107–113 (1965)CrossRefGoogle Scholar
  87. Mor55.
    Morovitz, H.: Some order–disorder considerations in living systems. Bull. Math. Biophys. 17, 81–86 (1955)CrossRefGoogle Scholar
  88. Mo68a.
    Mowshovitz, A.: The information content of digraphs and infinite graphs. Bull. Math. Biophys. 30, 225–240 (1968)MathSciNetCrossRefGoogle Scholar
  89. Mo68b.
    Mowshovitz, A.: An index of the relative complexity of a graph. Bull. Math. Biophys. 30, 175–204 (1968)MathSciNetCrossRefGoogle Scholar
  90. Mo68c.
    Mowshovitz, A.: Graphs with prescribed information content. Bull. Math. Biophys. 30, 387–414 (1968)MathSciNetCrossRefGoogle Scholar
  91. Mo68d.
    Mowshovitz, A.: Entropy measures and graphical structure. Bull. Math. Biophys. 30, 533–546 (1968)MathSciNetCrossRefGoogle Scholar
  92. NST93.
    Nekrasov, Yu.S., Tepfer, E.E., Sukharev, Yu.N.: On the relationship between the mass–spectral and structural indices of arylsilanes. Russ. Chem. Bull. 42, 343–346 (1993)CrossRefGoogle Scholar
  93. NS93.
    Nekrasov, Yu.S., Sukharev, Yu.N., Molgacheva, N.S., Tepfer, E.E.: Generalized characteristics of mass–spectra of aromatic compounds and their correlation with the constants of substituents. Russ. Chem. Bull. 42, 1986–1990 (1993)CrossRefGoogle Scholar
  94. NS96.
    Nekrasov, Yu.S., Sukharev, Yu.N., Tepfer, E.E., Molgacheva, N.S.: Establishment of correlations between the structure and reactivity of molecules in the gas phase based on information theory. Russ. Chem. Bull. 45, 2542–2546 (1996)CrossRefGoogle Scholar
  95. NS02.
    Nekrasov, Yu.S., Sukharev, Yu.N., Tepfer, E.E., Yakushin, S.: Electron impact mass spectra data processing for evaluation of gas–phase reactivity of cymantrene (tricarbonyl η5–cyclopentadienylmanganese) derivatives. Eur. J. Mass Spectrom. 8, 247–251 (2002)CrossRefGoogle Scholar
  96. NST05.
    Nekrasov, Yu.S., Sukharev, Yu.N., Tepfer, E.E.: Determination of spectrum–structure correlations based on integral parameters of mass–spectra. J. Analyt. Chem. 20, 1035–1037 (2005)CrossRefGoogle Scholar
  97. NTM95.
    Nikolić, S., Trinajstić, N., Mihalić, Z.: The Wiener index: developments and applications. Croat. Chem. Acta CCACAA 68, 105–129 (1995)Google Scholar
  98. P52.
    Platt, J.R.: Prediction of isomeric differences in paraffin properties. J. Phys. Chem. 56, 328–336 (1952)CrossRefGoogle Scholar
  99. PBC73.
    Purcell, W.P., Bass, G.E., Clayton, J.M.: Strategy in Drug Design. A Molecular Guide to Biological Activity. Wiley–lnterscience, New York (1973)Google Scholar
  100. QS81.
    Quintas, L.V., Slater, P.J.: Pairs of non–isomorphic graphs having the same path degree sequence. MATCH Commun. Math. Comput. Chem. 12, 75–86 (1981)MathSciNetMATHGoogle Scholar
  101. Ra75.
    Randić, M.: On characterization of molecular branching. J. Am. Chem. Soc. 69, 6609–6615 (1975)CrossRefGoogle Scholar
  102. Ra79.
    Randić, M.: Characterization of atoms, molecules and classes of molecules based on paths enumerations. Proc. Bremen Konferenz zur Chemie Univ. Bremen. 2, 5–64 (1979)Google Scholar
  103. Ra84.
    Randić, M.: On molecular identification numbers. J. Chem. Inf. Comput. Sci. 24, 164–175 (1984)Google Scholar
  104. Ra91.
    Randić, M.: Generalized molecular descriptors. J. Math. Chem. 7, 155–168 (1991)Google Scholar
  105. RT88.
    Randić, M., Trinajstić, N.: Composition as a method for data redustion: Application to carbon-13 NMR chemical shifts. Theor. Chim. Acta 73, 233–246 (1988)CrossRefGoogle Scholar
  106. Rash55.
    Rashevsky, N.: Life, information theory and topology. Bull. Math. Biophys. 17, 229–235 (1955)MathSciNetCrossRefGoogle Scholar
  107. Rash60.
    Rashevsky, N.: Life, information theory, probability and physics. Bull. Math. Biophys. 22, 351–364 (1960)MathSciNetCrossRefGoogle Scholar
  108. RR84.
    Raychaudhary, C., Ray, S.K., Ghosh, J.J., Roy, A.B., Basak, S.C.: Discrimination of isomeric structures using information theoretic topological indices. J. Comput. Chem. 5, 581–588 (1984)CrossRefGoogle Scholar
  109. RCD85.
    Razinger, M., Chretien, J.R., Dubois, J.K.: Structural selectivity of topological indices in alkane series. J. Chem. Inf. Comput. Sci. 25, 23–27 (1985)Google Scholar
  110. RBT88.
    Romero, D.L., Busso, M., Tan, C.K., Reusser, F., Palmer, J.R., Poppe, S.M., Aristoff, P.A., Downey, K.M., So, A.G., Resnick, L., Tarpley, W.G.: Nonnucleoside reverse transcriptase inhibitors that potently and specifically block human immunodeficiency virus type I replication. Proc. Natl. Acad. Sci. USA. 88, 8806–8810 (1991)CrossRefGoogle Scholar
  111. Rou83.
    Rouvray, D.H.: Should we have designs on topological indices? In: Chemical applications of topology and graph theory. In: King, R.B. (ed.) Studies in Physical and Theoretical Chemistry. Elsevier, Amsterdam (1983)Google Scholar
  112. Rou89.
    Rouvray, D.H.: The limits of applicability of topological indices. J. Mol. Struc. (Theochem) 185, 187–201 (1989)Google Scholar
  113. RB79.
    Rouvray, D.H., Balaban, A.T.: Chemical applications of graph theory. In: Wilson, R.J., Beineke, L.W. (eds.) Applications of Graph Theory, pp. 177–221. Academic, New York (1979)Google Scholar
  114. RB91.
    Rouvray, D.H., Bonchev, D.: Chemical Graph Theory: Introduction and Fundamentals. Abacus Press, Tunbridge Wells, Kent (1991)MATHGoogle Scholar
  115. SHP81.
    Schaad, L.J., Hess (Jr.) B.A., Purcell, W.P., Cammarata, A., Franke, R., Kubinyi, H.: Compatibility of the Free–Wilson and Hansch quantitative structure–activity relations. J. Med. Chem. 24(7), 900–901 (1981)Google Scholar
  116. S03.
    Selassie, C.D.: History of Quantitative Structure–Activity Relationships, In: Abraham, D.J. (ed.) Burger’s Medicinal Chemistry and Drug Discovery, pp. 1–48. Wiley, New York (2003)Google Scholar
  117. Sh49.
    Shannon, C., Weaver, W.: Mathematical Theory of Communications. University of Illinois, Urbana (1949)Google Scholar
  118. SK91.
    Skorobogatov, V.A., Konstantinova, E.V., Nekrasov, Yu.S., Sukharev, Yu.N., Tepfer, E.E.: On the correlation between the molecular information topological and mass–spectra indices of organometallic compounds. MATCH Commun. Math. Comput. Chem. 26, 215–228 (1991)Google Scholar
  119. Sl82.
    Slater, P.J.: Counterexamples to Randić’s conjecture on distance degree sequences for trees. J. Graph Theory 6, 89–92 (1982)MathSciNetMATHCrossRefGoogle Scholar
  120. Spi63.
    Spialter, L.: The atom connectivity matrix (ACM) and its charactereistic polynimial (ACMCP): a new computer–oriented chemical nomenclature. J. Am. Chem. Soc. 85, 2012–2013 (1963)CrossRefGoogle Scholar
  121. Spi64a.
    Spialter, L.: The atom connectivity matrix (ACM) and its charactereistic polynimial (ACMCP). J. Chem. Doc. 4, 261–269 (1964)CrossRefGoogle Scholar
  122. Spi64b.
    Spialter, L.: The atom connectivity matrix characteristic polynimial (ACMCP) and its physico–geometric (topological) significance. J. Chem. Doc. 4, 269–274 (1964)CrossRefGoogle Scholar
  123. SBJ79.
    Stuper, A., Brugger, W., Jurs, P.: Computer Assisted Studies of Chemical Structure and Biological Function. Wiley, New York (1979)Google Scholar
  124. SN93.
    Sukharev, Yu.N., Nekrasov, Yu.S., Molgacheva, N.S., Tepfer, E.E.: Computer processing and interpretation of mass–spectral information. Part IX - Generalized characteristics of mass–spectra. Org. Mass Spectrom. 28, 1555–1561 (1993)Google Scholar
  125. Syl78.
    Sylvester, J.J.: On an application of the new atomic theory to the graphical representation of the invariants and covariants of binary quantics, with three appendices. Am. J. Math. 1, 64–125 (1878)MathSciNetCrossRefGoogle Scholar
  126. TC00.
    Todeschini, R., Consonni, V.: Handbook of Moleculat Descriptors. Wiley, Weinheim, Germany (2000)CrossRefGoogle Scholar
  127. TCM06.
    Todeschini, R., Consonni, V., Mauri, A., Pavan, M.: DRAGON – Software for the calculation of molecular descriptors. Ver.5.4 for Windows, Talete srl, Milan, Italy (2006)Google Scholar
  128. Tri92.
    Trinajstić, N.: Chemical Graph Theory, 2nd edn. (revised). CRC Press, Boca Raton, FL (1992)Google Scholar
  129. Tr56a.
    Trucco, E.: A note of the information content of graphs. Bull. Math. Biophys. 17, 129–135 (1956)MathSciNetCrossRefGoogle Scholar
  130. Tr56b.
    Trucco, E.: On the informational content of graphs-compound symbols. Different states for each point. Bull. Math. Biophys. 18, 237–245 (1956)MathSciNetGoogle Scholar
  131. Val63.
    Valentinuzzi, M., Valentinuzzi, M.E.: Information content of chemical structures and some possible biological applications. Bull. Math. Biophys. 25, 11–27 (1963)CrossRefGoogle Scholar
  132. Wi47.
    Wiener, H.: Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69, 17–20 (1947)CrossRefGoogle Scholar
  133. Wi48a.
    Wiener, H.: Vapor pressure–temperature relationships among the branched paraffin hydrocarbons. J. Phys. Chem. 52, 425–430 (1948)CrossRefGoogle Scholar
  134. Wi48b.
    Wiener, H.: Relation of the physical properties of the isomeric alkanes to molecular structure. J. Phys. Chem. 52, 1082–1089 (1948)CrossRefGoogle Scholar
  135. YAK08.
    Yousefi-Azari, H., Ashrafi, A.R., Khalifeh, M.H.: Topological indices of nanotubes, nanotori and nanostars. Dig. J. Nanomater. Bios. 3, 251–255 (2008)Google Scholar
  136. ZT08.
    Zhou, Bo; Trinajstic, N.: Bounds on the Balaban index. Croatia Chemica Acta CCACCA 81, 319–323 (2008)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsYeungnam UniversityGyeongsanSouth Korea
  2. 2.Sobolev Institute of MathematicsSiberian Branch of Russian Academy of SciencesNovosibirskRussia

Personalised recommendations