Table of Contents
International Journal of Combinatorics
Volume 2013 (2013), Article ID 347613, 14 pages
http://dx.doi.org/10.1155/2013/347613
Research Article

An Algebraic Representation of Graphs and Applications to Graph Enumeration

Centro de Estruturas Lineares e Combinatórias, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal

Received 23 July 2012; Accepted 25 September 2012

Academic Editor: Xueliang Li

Copyright © 2013 Ângela Mestre. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. C. Read, “A survey of graph generation techniques,” in Combinatorial Mathematics 8, vol. 884 of Lecture Notes in Math., pp. 77–89, Springer, Berlin, Germany, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, New York, NY, USA, 1973. View at Zentralblatt MATH
  3. F. Bergeron, G. Labelle, and P. Leroux, Combinatorial Species and Tree-Like Structures, vol. 67, Cambridge University Press, Cambridge, UK, 1998. View at Zentralblatt MATH
  4. C. Itzykson and J. B. Zuber, Quantum Field Theory, McGraw-Hill, New York, NY, USA, 1980.
  5. C. Jordan, “Sur les assemblages de lignes,” Journal für die Reine und Angewandte Mathematik, vol. 70, pp. 185–190, 1869. View at Google Scholar
  6. R. C. Read, “Every one a winner or how to avoid isomorphism search when cataloguing combinatorial configurations. Algorithmic aspects of combinatorics (Conference in Vancouver Island, BC, Canada, 1976),” Annals of Discrete Mathematics, vol. 2, pp. 107–120, 1978. View at Google Scholar · View at Zentralblatt MATH
  7. Â. Mestre and R. Oeckl, “Combinatorics of n-point functions via Hopf algebra in quantum field theory,” Journal of Mathematical Physics, vol. 47, no. 5, p. 052301, 16, 2006. View at Publisher · View at Google Scholar
  8. Â. Mestre and R. Oeckl, “Generating loop graphs via Hopf algebra in quantum field theory,” Journal of Mathematical Physics, vol. 47, no. 12, Article ID 122302, p. 14, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. Â. Mestre, “Combinatorics of 1-particle irreducible n-point functions via coalgebra in quantum field theory,” Journal of Mathematical Physics, vol. 51, no. 8, Article ID 082302, 2010. View at Publisher · View at Google Scholar
  10. O. Dziobek, “Eine formel der substitutionstheorie,” Sitzungsberichte der Berliner Mathematischen Gesellschaft, vol. 17, pp. 64–67, 1947. View at Google Scholar
  11. A. Cayley, “A theorem on trees,” Quarterly Journal of Pure and Applied Mathematics, vol. 23, pp. 376–378, 1889. View at Google Scholar
  12. G. Pólya, “Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen,” Acta Mathematica, vol. 68, pp. 145–254, 1937. View at Google Scholar
  13. R. Tarjan, “Depth-first search and linear graph algorithms,” SIAM Journal on Computing, vol. 1, no. 2, pp. 146–160, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. N. Abel, “Beweis eines Ausdruckes, von welchem die Binomial-Formel ein einzelner Fall ist,” Journal für die Reine und Angewandte Mathematik, vol. 1, pp. 159–160, 1826. View at Google Scholar
  15. L. Székely, “Abel’s binomial theorem,” http://www.math.sc.edu/~szekely/abel.pdf.
  16. R. Diestel, Graph Theory, vol. 173, Springer, Berlin, Germany, 5rd edition, 2005. View at Zentralblatt MATH
  17. M. J. Atallah and S. Fox, Algorithms and Theory of Computation Handbook, CRC Press, Boca Raton, Fla, USA, 1998.
  18. G. Kirchhoff, “Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird,” Annual Review of Physical Chemistry, vol. 72, pp. 497–508, 1847. View at Google Scholar
  19. Â. Mestre, “Generating connected and 2-edge connected graphs,” Journal of Graph Algorithms and Applications, vol. 13, no. 2, pp. 251–281, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. J. Cuntz, R. Meyer, and J. M. Rosenberg, Topological and Bivariant K-Theory, vol. 36, Birkhäuser, Basel, Switzerland, 2007. View at Zentralblatt MATH
  21. C. Kassel, Quantum Groups, vol. 155, Springer, New York, NY, USA, 1995. View at Publisher · View at Google Scholar
  22. J.-L. Loday, Cyclic Homology, vol. 301, Springer, Berlin, Germany, 1992.
  23. M. Livernet, “A rigidity theorem for pre-Lie algebras,” Journal of Pure and Applied Algebra, vol. 207, no. 1, pp. 1–18, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. J. W. Moon, “Various proofs of Cayley's formula for counting trees,” in A Seminar on Graph Theory, pp. 70–78, Holt, Rinehart & Winston, New York, NY, USA, 1967. View at Google Scholar
  25. K. Husimi, “Note on Mayers' theory of cluster integrals,” The Journal of Chemical Physics, vol. 18, pp. 682–684, 1950. View at Publisher · View at Google Scholar
  26. J. Mayer, “Equilibrium statistical mechanics,” in The International Encyclopedia of Physical Chemistry and Chemical Physics, Pergamon Press, Oxford, UK, 1968. View at Google Scholar
  27. P. Leroux, “Enumerative problems inspired by Mayer's theory of cluster integrals,” Electronic Journal of Combinatorics, vol. 11, no. 1, research paper 32, p. 28, 2004. View at Google Scholar · View at Zentralblatt MATH