Table of Contents
International Journal of Combinatorics
Volume 2012 (2012), Article ID 430859, 40 pages
http://dx.doi.org/10.1155/2012/430859
Review Article

The Tutte Polynomial of Some Matroids

1Instituto de Matemáticas, Universidad Nacional Autónoma de México, Area de la Investigación Científica, Circuito Exterior, C.U., Coyoácan, 04510 México City, DF, Mexico
2Escuela de Ciencias, Universidad Autónoma Benito Juárez de Oaxaca, 68120 Oaxaca, OAX, Mexico
3Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana, Azcapozalco, Avenue San Pablo No. 180, Col. Reynosa Tamaulipas, Azcapotzalco, 02200 México City, DF, Mexico

Received 23 February 2012; Accepted 10 July 2012

Academic Editor: Cai Heng Li

Copyright © 2012 Criel Merino et al. 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. J. A. Ellis-Monaghan and C. Merino, “Graph polynomials and their applications I: the Tutte polynomial,” in Structural Analysis of Complex Networks: Theory and Applications, M. Dehmer, Ed., Birkhäuser, Boston, Mass, USA, 2011. View at Publisher · View at Google Scholar
  2. D. J. A. Welsh, Complexity: Knots, Colourings and Counting, Cambridge University Press, Cambridge, UK, 1993.
  3. S. D. Noble, “The complexity of graph polynomials,” in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh, G. R. Grimmett and C. J. H. McDiarmid, Eds., Oxford University Press, Oxford, UK, 2007. View at Google Scholar
  4. T. Brylawski and J. Oxley, “The Tutte polynomial and its applications,” in Matroid Applications, Encyclopedia of Mathematics and its Applications, N. White, Ed., Cambridge University Press, Cambridge, UK, 1992. View at Google Scholar
  5. J. Bonin and A. de Mier, “T-uniqueness of some families of k-chordal matroids,” Advances in Applied Mathematics, vol. 32, no. 1-2, pp. 10–30, 2004. View at Publisher · View at Google Scholar
  6. A. de Mier and M. Noy, “On graphs determined by their Tutte polynomials,” Graphs and Combinatorics, vol. 20, no. 1, pp. 105–119, 2004. View at Publisher · View at Google Scholar
  7. I. Sarmiento, “A characterisation of jointless Dowling geometries,” Discrete Mathematics, vol. 197-198, pp. 713–731, 1999. View at Google Scholar
  8. R. Diestel, Graph Theory, Graduate Texts in Mathematics, Springer, New York, NY, USA, 2000.
  9. J. G. Oxley, Matroid Theory, Oxford University Press, New York, NY, USA, 1992.
  10. C. Merino, S. D. Noble, M. Ramírez-Ibáñez, and R. Villarroel, “On the structure of the h-vector of a paving matroid,” European Journal of Combinatorics, vol. 33, no. 8, pp. 1787–1799, 2012. View at Publisher · View at Google Scholar · View at Scopus
  11. M. Jerrum, “Two remarks concerning balanced matroids,” Combinatorica, vol. 26, no. 6, pp. 733–742, 2006. View at Publisher · View at Google Scholar
  12. D. Mayhew and G. F. Royle, “Matroids with nine elements,” Journal of Combinatorial Theory, vol. 98, no. 2, pp. 415–431, 2008. View at Publisher · View at Google Scholar
  13. J. Bonin, A. de Mier, and M. Noy, “Lattice path matroids: enumerative aspects and Tutte polynomials,” Journal of Combinatorial Theory, vol. 104, no. 1, pp. 63–94, 2003. View at Publisher · View at Google Scholar
  14. T. H. Brylawski, “The tutte polynomial, part 1: general theory,” in Matroid Theory and Its Applications. Proceedings of the Third International Mathematical Summer Center (C.I.M.E. 1980), A. Barlotti, Ed., 1982. View at Google Scholar
  15. M. K. Chari, “Matroid inequalities,” Discrete Mathematics, vol. 147, no. 1–3, pp. 283–286, 1995. View at Publisher · View at Google Scholar
  16. B. Jackson, “An inequality for Tutte polynomials,” Combinatorica, vol. 30, no. 1, pp. 69–81, 2010. View at Publisher · View at Google Scholar
  17. N. L. Biggs, R. M. Damerell, and D. A. Sands, “Recursive families of graphs,” Journal of Combinatorial Theory, vol. 12, pp. 123–131, 1972. View at Google Scholar
  18. K. Sekine, H. Imai, and S. Tani, “Computing the Tutte polynomial of a graph of moderate size,” in Lecture Notes in Computer Science, Springer, Berlin, Germany, 1995. View at Publisher · View at Google Scholar
  19. http://www.math.umn.edu/~reiner/Tutte/TUTTE.html.
  20. W. T. Tutte, “A ring in graph theory,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 43, pp. 26–40, 1947. View at Publisher · View at Google Scholar
  21. W. T. Tutte, “A contribution to the theory of chromatic polynomials,” Canadian Journal of Mathematics, vol. 6, pp. 80–91, 1954. View at Google Scholar
  22. W. T. Tutte, “On dichromatic polynomials,” Journal of Combinatorial Theory, vol. 2, no. 3, pp. 301–320, 1967. View at Google Scholar · View at Scopus
  23. A. Björner, “The homology and shellability of matroids and geometric lattices,” in Matroid Applications, Encyclopedia of Mathematics and Its Applications, N. White, Ed., Cambridge University Press, Cambridge, UK, 1992. View at Google Scholar
  24. D. Mayhew, M. Newman, D. Welsh, and G. Whittle, “On the asymptotic proportion of connected matroids,” European Journal of Combinatorics, vol. 32, no. 6, pp. 882–890, 2011. View at Publisher · View at Google Scholar
  25. T. H. Brylawski, “A decomposition for combinatorial geometries,” Transactions of the American Mathematical Society, vol. 171, pp. 235–282, 1972. View at Google Scholar
  26. T. H. Brylawski, “Intersection theory for embeddings of matroids into uniform geometries,” Studies in Applied Mathematics, vol. 61, no. 3, pp. 211–244, 1979. View at Google Scholar
  27. J. E. Bonin and H. Qin, “Tutte polynomials of q-cones,” Discrete Mathematics, vol. 232, no. 1–3, pp. 95–103, 2001. View at Publisher · View at Google Scholar
  28. J. P. S. Kung, “Critical problems,” in Matroid Theory, J. Bonin, J. G. Oxley, and B. Servatius, Eds., American Mathematical Society, Providence, RI, USA, 1996. View at Google Scholar
  29. R. P. Stanley, Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge, UK, 1999. View at Publisher · View at Google Scholar
  30. D. J. A. Welsh, “Counting colourings and flows in random graphs,” in Combinatorics, Paul Erdös is Eighty, D. Miklós, V. T. Sós, and T. Szönyi, Eds., pp. 491–505, János Bolyai Mathematical Society, Budapest, Hungary, 1996. View at Google Scholar
  31. J. L. Martin and V. Reiner, “Cyclotomic and simplicial matroids,” Israel Journal of Mathematics, vol. 150, pp. 229–240, 2005. View at Publisher · View at Google Scholar
  32. E. G. Mphako, “Tutte polynomials of perfect matroid designs,” Combinatorics, Probability and Computing, vol. 9, no. 4, pp. 363–367, 2000. View at Publisher · View at Google Scholar
  33. M. Barany and V. Reiner, “The Tutte polynomial of a finite projective space,” preprint.
  34. R. P. Stanley, Enumerative Combinatorics, vol. 1, Cambridge University Press, Cambridge, UK, 1997.
  35. N. Calkin, C. Merino, S. Noble, and M. Noy, “Improved bounds for the number of forests and acyclic orientations in the square lattice,” The Electronic Journal of Combinatorics, vol. 10, no. 1, 2003. View at Google Scholar
  36. S.-C. Chang and R. Shrock, “Zeros of Jones polynomials for families of knots and links,” Physica A, vol. 301, no. 1–4, pp. 196–218, 2001. View at Publisher · View at Google Scholar
  37. R. Shrock, “Exact Potts model partition functions on ladder graphs,” Physica A, vol. 283, no. 3-4, pp. 388–446, 2000. View at Publisher · View at Google Scholar
  38. D. J. A. Welsh and C. Merino, “The Potts model and the Tutte polynomial,” Journal of Mathematical Physics, vol. 41, no. 3, pp. 1127–1152, 2000. View at Publisher · View at Google Scholar
  39. L. Beaudin, J. Ellis-Monaghan, G. Pangborn, and R. Shrock, “A little statistical mechanics for the graph theorist,” Discrete Mathematics, vol. 310, no. 13-14, pp. 2037–2053, 2010. View at Publisher · View at Google Scholar
  40. S.-C. Chang and R. Shrock, “Exact Potts model partition function on strips of the triangular lattice,” Physica A, vol. 286, no. 1-2, pp. 189–238, 2000. View at Publisher · View at Google Scholar
  41. S.-C. Chang and R. Shrock, “Exact Potts model partition functions on wider arbitrary-length strips of the square lattice,” Physica A, vol. 296, no. 1-2, pp. 234–288, 2001. View at Publisher · View at Google Scholar
  42. S.-C. Chang and R. Shrock, “Tutte polynomials and related asymptotic limiting functions for recursive families of graphs,” Advances in Applied Mathematics, vol. 32, no. 1-2, pp. 44–87, 2004. View at Publisher · View at Google Scholar
  43. K. Truemper, Matroid Decomposition, Academic Press, 1992.
  44. A. Andrzejak, “Splitting formulas for Tutte polynomials,” Journal of Combinatorial Theory, vol. 70, no. 2, pp. 346–366, 1997. View at Publisher · View at Google Scholar
  45. T. H. Brylawski, “A combinatorial model for series-parallel networks,” Transactions of the American Mathematical Society, vol. 154, pp. 1–22, 1971. View at Google Scholar
  46. J. Bonin and A. de Mier, “Tutte polynomials of generalized parallel connections,” Advances in Applied Mathematics, vol. 32, no. 1-2, pp. 31–43, 2004. View at Publisher · View at Google Scholar
  47. F. Jaeger, D. L. Vertigan, and D. J. A. Welsh, “On the computational complexity of the Jones and Tutte polynomials,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 108, no. 1, pp. 35–53, 1990. View at Publisher · View at Google Scholar
  48. J. E. Bonin, “Strongly inequivalent representations and Tutte polynomials of matroids,” Algebra Universalis, vol. 49, no. 3, pp. 289–303, 2003. View at Publisher · View at Google Scholar
  49. D. J. A. Welsh, Matroid Theory, Dover publications, New York, NY, USA, 2010.