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Basic Chemical Graph Theory

  • Mircea Vasile Diudea
Chapter
Part of the Carbon Materials: Chemistry and Physics book series (CMCP, volume 10)

Abstract

Graph Theory applied in Chemistry is called Chemical Graph Theory. This interdisciplinary science takes problems (like isomer enumeration, structure elucidation, etc.) from Chemistry and solve them by Mathematics (using tools from Graph Theory, Set Theory or Combinatorics), thus influencing both Chemistry and Mathematics. This chapter introduces to basic definitions in Graph Theory: graph, walk, path, circuit, planar graph, graph invariant, vertex degree, chemical graph, etc. Then topological matrices are introduced: adjacency, distance, detour, combinatorial matrices, Wiener and Cluj matrices, walk matrix operator (combining three square matrices), reciprocal distance, and layer/shell matrices, on which the centrality indices are defined. Some info about topological symmetry is also presented.

References

  1. Amić D, Trinajstić N (1995) On the detour matrix. Croat Chem Acta 68:53–62Google Scholar
  2. Balasubramanian K (1994) Computer generation of automorphism graphs of weighted graphs. J Chem Inf Comput Sci 34:1146–1150CrossRefGoogle Scholar
  3. Balasubramanian K (1995a) Computer generation of nuclear equivalence classes based on the three-dimensional molecular structure. J Chem Inf Comput Sci 35:243–250CrossRefGoogle Scholar
  4. Balasubramanian K (1995b) Computational strategies for the generation of equivalence classes of Hadamard matrices. J Chem Inf Comput Sci 35:581–589CrossRefGoogle Scholar
  5. Balasubramanian K (1995c) Computer perception of molecular symmetry. J Chem Inf Comput Sci 35:761–770CrossRefGoogle Scholar
  6. Bonchev D, Mekenyan O, Balaban AT (1989) Iterative procedure for the generalized graph center in polycyclic graphs. J Chem Inf Comput Sci 29:91–97CrossRefGoogle Scholar
  7. Ciubotariu D (1987) Structură-reactivitate în clasa derivaţilor acidului carbonic. PhD Thesis, Timisoara, RomaniaGoogle Scholar
  8. Ciubotariu D, Medeleanu M, Vlaia V, Olariu T, Ciubotariu C, Dragos D, Seiman C (2004) Molecular van der Waals space and topological indices from the distance matrix. Molecules 9:1053–1078CrossRefGoogle Scholar
  9. Crippen GM (1977) A novel approach to calculation of conformation: distance geometry. J Comput Phys 24:96–107CrossRefGoogle Scholar
  10. Diudea MV (1994) Layer matrices in molecular graphs. J Chem Inf Comput Sci 34:1064–1071CrossRefGoogle Scholar
  11. Diudea MV (1996) Walk numbers eWM: Wiener-type numbers of higher rank. J Chem Inf Comput Sci 36:535–540CrossRefGoogle Scholar
  12. Diudea MV (1997a) Cluj matrix invariants. J Chem Inf Comput Sci 37:300–305CrossRefGoogle Scholar
  13. Diudea MV (1997b) Cluj matrix, CJu: source of various graph descriptors. MATCH Commun Math Comput Chem 35:169–183Google Scholar
  14. Diudea MV (1997c) Indices of reciprocal property or Harary indices. J Chem Inf Comput Sci 37:292–299CrossRefGoogle Scholar
  15. Diudea MV (1999) Valencies of property. Croat Chem Acta 72:835–851Google Scholar
  16. Diudea MV (2010) Nanomolecules and nanostructures—polynomials and indices, MCM, No. 10. University Kragujevac, SerbiaGoogle Scholar
  17. Diudea MV, Gutman I (1998) Wiener-type topological indices. Croat Chem Acta 71:21–51Google Scholar
  18. Diudea MV, Randić M (1997) Matrix operator, W(M1,M2,M3) and Schultz-type numbers. J Chem Inf Comput Sci 37:1095–1100CrossRefGoogle Scholar
  19. Diudea MV, Ursu O (2003) Layer matrices and distance property descriptors. Indian J Chem 42A:1283–1294Google Scholar
  20. Diudea MV, Topan MI, Graovac A (1994) Layer matrices of walk degrees. J Chem Inf Comput Sci 34:1071–1078Google Scholar
  21. Diudea MV, Parv B, Gutman I (1997a) Detour-Cluj matrix and derived invariants. J Chem Inf Comput Sci 37:1101–1108CrossRefGoogle Scholar
  22. Diudea M, Parv B, Topan MI (1997b) Derived Szeged and Cluj indices. J Serb Chem Soc 62:267–276Google Scholar
  23. Diudea MV, Katona G, Lukovits I, Trinajstić N (1998) Detour and Cluj-detour indices. Croat Chem Acta 71:459–471Google Scholar
  24. Diudea MV, Gutman I, Jäntschi L (2002) Molecular topology. NOVA, New YorkGoogle Scholar
  25. Diudea MV, Florescu MS, Khadikar PV (2006) Molecular topology and its applications. EFICON, BucharestGoogle Scholar
  26. Dobrynin AA (1993) Degeneracy of some matrix invariant and derived topological indices. J Math Chem 14:175–184CrossRefGoogle Scholar
  27. Dobrynin AA, Kochetova AA (1994) Degree distance of a graph: a degree analogue of the Wiener index. J Chem Inf Comput Sci 34:1082–1086CrossRefGoogle Scholar
  28. Estrada E, Rodriguez L (1997) Matrix algebric manipulation of molecular graphs, Harary- and MTI-like molecular descriptors. MATCH Commun Math Comput Chem 35:157–167Google Scholar
  29. Estrada E, Rodriguez L, Gutierrez A (1997) Matrix algebric manipulation of molecular graphs, distance and vertex-adjacency matrices. MATCH Commun Math Chem 35:145–156Google Scholar
  30. Euler L (1752–1753) Elementa doctrinae solidorum. Novi Comm Acad Scient Imp Petrop 4:109–160Google Scholar
  31. Graovac A, Babić D (1990) The evaluation of quantum chemical indices by the method of moments. Int J Quantum Chem Quantum Chem Symp 24:251–262CrossRefGoogle Scholar
  32. Gutman I (1994) A formula for the Wiener number of trees and its extension to graphs containing cycles. Graph Theory Notes NY 27:9–15Google Scholar
  33. Gutman I, Polansky OE (1986) Mathematical concepts in organic chemistry. Springer, Berlin, pp 108–116CrossRefGoogle Scholar
  34. Halberstam FY, Quintas LV (1982) Distance and path degree sequences for cubic graphs. Pace University, New York. A note on table of distance and path degree sequences for cubic graphs, Pace University, New YorkGoogle Scholar
  35. Harary F (1969) Graph theory. Addison-Wesley, ReadingCrossRefGoogle Scholar
  36. Hargittai M, Hargittai I (2010) Symmetry through the eyes of a chemist. Springer, DordrechtGoogle Scholar
  37. Horn RA, Johnson CR (1985) Matrix analysis. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  38. Hungerford TW (1974) Algebra. Graduate texts in mathematics, vol 73. Springer, New York (Reprint of the original 1980)Google Scholar
  39. Ionescu T (1973) Graphs-applications (in Romanian), Ed. Ped., BucharestGoogle Scholar
  40. Ivanciuc O, Balaban TS, Balaban AT (1993) Reciprocal distance matrix, related local vertex invariants and topological indices. J Math Chem 12:309–318CrossRefGoogle Scholar
  41. Janežič D, Miličević A, Nikolić S, Trinajstić N (2007) Graph theoretical matrices in chemistry. Mathematical chemistry monographs. University Kragujevac, KragujevacGoogle Scholar
  42. Klein DJ, Lukovits I, Gutman I (1995) On the definition of the hyper-Wiener index for cycle-containing structures. J Chem Inf Comput Sci 35:50–52CrossRefGoogle Scholar
  43. Kuratowski K (1930) Sur le Problème des Courbes Gauches en Topologie. Fund Math 15:271–283CrossRefGoogle Scholar
  44. Lukovits I (1996) The detour index. Croat Chem Acta 69:873–882Google Scholar
  45. Mirman R (1999) Point groups, space groups, crystals, molecules. World Scientific, River EdgeCrossRefGoogle Scholar
  46. Petitjean M (2007) A definition of symmetry. Symmetry Cult Sci 18:99–119Google Scholar
  47. Plavšić D, Nikolić S, Trinajstić N, Mihalić Z (1993) On the Harary index for the characterization of chemical graphs. J Math Chem 12:235–250CrossRefGoogle Scholar
  48. Randić M (1991) Generalized molecular descriptors. J Math Chem 7:155–168CrossRefGoogle Scholar
  49. Randić M (1993) Novel molecular description for structure-property studies. Chem Phys Lett 211:478–483CrossRefGoogle Scholar
  50. Randić M (1995) Restricted random walks on graphs. Theor Chim Acta 92:97–106CrossRefGoogle Scholar
  51. Randić M, Guo X, Oxley T, Krishnapriyan H (1993) Wiener matrix: source of novel graph invariants. J Chem Inf Comput Sci 33:700–716Google Scholar
  52. Randić M, Guo X, Oxley T, Krishnapriyan H, Naylor L (1994) Wiener matrix invariants. J Chem Inf Comput Sci 34:361–367CrossRefGoogle Scholar
  53. Razinger M, Balasubramanian K, Munk ME (1993) Graph automorphism perception algorithms in computer-enhanced structure elucidation. J Chem Inf Comput Sci 33:197–201CrossRefGoogle Scholar
  54. Skorobogatov AV, Dobrynin AA (1988) Metric analysis of graphs. MATCH Commun Math Comput Chem 23:105–151Google Scholar
  55. Sylvester JJ (1874) 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 1:64–90CrossRefGoogle Scholar
  56. Tratch SS, Stankevich MI, Zefirov NS (1990) Combinatorial models and algorithms in chemistry. The expanded Wiener number—a novel topological index. J Comput Chem 11:899–908CrossRefGoogle Scholar
  57. Trinajstić N (1983) Chemical graph theory. CRC Press, Boca RatonGoogle Scholar
  58. Ursu O, Diudea MV (2005) TOPOCLUJ software program. Babes-Bolyai University, ClujGoogle Scholar
  59. Wiener H (1947) Structural determination of paraffin boiling points. J Am Chem Soc 69:17–20CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Mircea Vasile Diudea
    • 1
  1. 1.Department of Chemistry, Faculty of Chemistry and Chemical EngineeringBabes-Bolyai UniversityCluj-NapocaRomania

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