Applications of Perfect Matchings in Chemistry

  • Damir Vukičević


Perfect matchings or one factors in mathematics correspond to Kekulé structures in chemistry. In this chapter, we present methods for determination of the existence and enumeration of perfect matchings. The Pfaffian method of enumeration of perfect matchings in planar graphs is presented. The importance of the enumeration of perfect matchings (Kekulé structures) is illustrated with several different chemical applications. A method for coding Kekulé structures which enables efficient storing in the computer is presented. Also, the recently introduced notion of algebraic Kekulé structures is explained and its role in the classification of Kekulé structures according to their significance is discussed. The concept of the resonance graph is presented and its role in the study of fullerene molecules is commented.


Perfect matching Kekulé structure Pfaffian Enumeration Resonance graph Anti-Kekulé number 


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Partial support of the Ministry of Science, Education and Sports of the Republic of Croatia is gratefully acknowledged (grant no. 177–0000000-0884 and grant no. 037-0000000-2779).


  1. 1.
    Lovász L, Plummer MD (1986) Matching theory. North Holland, AmsterdamMATHGoogle Scholar
  2. 2.
    Randić M (2003) Aromaticity of polycyclic conjugated hexagons. Chem Rev 103:3449–3605CrossRefGoogle Scholar
  3. 3.
    Kuhn HW (1955) The Hungarian method for the assignment problem. Nav Res Logist Q 2:83–97CrossRefGoogle Scholar
  4. 4.
    Veljan D (2001) Combinatorial and discrete mathematics, algoritam, zagreb (in Croatian)Google Scholar
  5. 5.
    Valiant L (1979) The complexity of computing the permanent. Theor Comput Sci 8:189–201MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kasteleyn PW (1967) Chapter 2. In: Harary F (ed) Graph theory and theoretical physics. Academic, New YorkGoogle Scholar
  7. 7.
    Jerrum M. Lecture Notes from a Recent Nachdiplomvorlesung at ETH-Zürich “Counting, sampling and integrating: algorithms and complexity” (draft, under construction).\AQPlease update Ref. [7].
  8. 8.
    Gutman I, Cyvin SJ (1999) Introduction to the theory of benzenoid hydrocarbons. Springer, BerlinGoogle Scholar
  9. 9.
    Cyvin SJ, Gutman I (1986) Topological properties of benzenoid systems. Part XXXVI. Algorithm for the number of Kekulé structures in some pericondensed benzenoids. MATCH Commun Math Comput Chem 19:229–242MathSciNetGoogle Scholar
  10. 10.
    Klein DJ, Babić D, Trinajstić N (2002) Enumeration in chemistry. Chem Model Appl Theory 2:56–95Google Scholar
  11. 11.
    Cyvin SJ, Gutman I (1988) Kekulé Structures in benzenoid hydrocarbons. Springer, BerlinGoogle Scholar
  12. 12.
    Morrison R, Boyd R (1992) Organic chemistry. Prentice-Hall, Englewood Cliffs, NJGoogle Scholar
  13. 13.
    Swinborne-Sheldrakem R, Herndon WC, Gutman I (1975) Kekulé structures and resonance energies of benzenoid hydrocarbons. Tetrahedron Lett 16:755–758CrossRefGoogle Scholar
  14. 14.
    Cioslowski J (1986) The generalized McClelland formula. MATCH Commun Math Comput Chem 20:95–101Google Scholar
  15. 15.
    Gutman I, Markovic S, Marinkovic M (1987) Investigation of the Cioslowski formula. MATCH Commun Math Comput Chem 22:277–284Google Scholar
  16. 16.
    Pauling L (1960) The nature of the chemical bond and the structure of molecules and crystals: an introduction to modern structural chemistry. Cornell University Press, Ithaca, NYGoogle Scholar
  17. 17.
    Randic M (2004) Algebraic Kekulé formulas for benzenoid hydrocarbons. J Chem Inform Comput Sci 44:365–372Google Scholar
  18. 18.
    Fowler PW, Manolopoulos DE (1995) An atlas of fullerenes. Clarendon Press, OxfordGoogle Scholar
  19. 19.
    Gutman I, Vukičević D, Graovac A, Randić M (2004) Algebraic kekulé structures of benzenoid hydrocarbons. J Chem Inform Comput Sci 44:296–299Google Scholar
  20. 20.
    Vukičević D, Žigert P (2008) Binary coding of algebraic kekulé structures of catacondensed benzenoid graphs. Appl Math Lett 21(7):712–716MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Vukičević D, Sedlar J (2004) Total forcing number of the triangular grid, Math Commun 9:169–179MATHMathSciNetGoogle Scholar
  22. 22.
    Vukičević D, Došlić T (2007) Global forcing number of grid graphs. Australas J Combinator 38:47–62MATHGoogle Scholar
  23. 23.
    Došlić T (2007) Global forcing number of benzenoid graphs. J Math Chem 41:217–229MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Wang H, Ye D, Zhang H, Wang H (2008) The forcing number of toroidal polyhexes. J Math Chem 43:457–475MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Vukičević D, Randić M (2005) On kekulé structures of buckminsterfullerene. Chem Phys Lett 401:446–450CrossRefGoogle Scholar
  26. 26.
    Vukičević D, Gutman I, Randić M (2006) On instability of fullerene C72. Croat Chem Acta 79:429–436Google Scholar
  27. 27.
    Kroto HW, Heath JR, O’Brien SC, Curl RF, Smalley RE (1985) C60: Buckminsterfullerene. Nature 318:162–163CrossRefGoogle Scholar
  28. 28.
    Randić M, Kroto H, Vukičević D. Kekulé structures of buckminsterfullerene, Adv Quantum Chem (submitted)Google Scholar
  29. 29.
    Vukičević D, Kroto HW, Randić M (2005) Atlas of Kekulé valence structures of buckminsterfullerene. Croat Chem Acta 78:223–234Google Scholar
  30. 30.
    El-Basil S (1993) Kekulé structures as graph generators. J Math Chem 14:305–318CrossRefMathSciNetGoogle Scholar
  31. 31.
    Gründler W (1982) Signinkante elektronenstrukturen fur benzenoide kohlenwasserstoffe. Wiss Z Univ Halle 31:97–116Google Scholar
  32. 32.
    Randić M (1997) Resonance in catacondensed benzenoid hydrocarbons. Int J Quantum Chem 63:585–600CrossRefGoogle Scholar
  33. 33.
    Zhang F, Guo X, Chen R (1988) Z-transformation graphs of perfect matchings of hexagonal systems. Discrete Math 72:405–415MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Chen R, Zhang F (1997) Hamilton paths in Z-transformation graphs of perfect matchings of hexagonal systems. Discrete Appl Math 74:191–196MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Klavžar S, Žigert P. Resonance graphs of catacondensed benzenoid graphs are median, ManuscriptGoogle Scholar
  36. 36.
    Klavžar S, Žigert P, Brinkmann G (2002) Resonance graphs of catacondensed even ring systems and medians. Discrete Math 253:35–43MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Flocke N, Schmalz TG, Klein DJ (1998) Variational resonance valence bond study on the ground state of C 60 using the Heisenberg model. J Chem Phys 109:873–880CrossRefGoogle Scholar
  38. 38.
    Randić M, Vukičević D (2006) Kekulé structures of Fullerene C70. Croat Chem Acta 79:471–481Google Scholar
  39. 39.
    Vukičević D. Total forcing number and Anti-forcing number of C 20, preprintGoogle Scholar
  40. 40.
    Kutnar K, Sedlar J, Vukičević D (2009) On the Anti-Kekulé number of Leapfrog Fullerenes. J Math Chem 45:406–416MATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Faculty of Mathematics and Natural SciencesUniversity of SplitSplitCroatia

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