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The Tutte Polynomial

The Tutte Polynomial. Graph Polynomials 238900 winter 05/06. The Rank Generation Polynomial. Reminder. Number of connected components in G(V,F). The Rank Generation Polynomial. Theorem 2. In addition , S(E n ;x,y) = 1 for G(V,E) where |V|=n and |E|=0.

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The Tutte Polynomial

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  1. The Tutte Polynomial Graph Polynomials 238900 winter 05/06

  2. The Rank Generation Polynomial Reminder Number of connected components in G(V,F)

  3. The Rank Generation Polynomial Theorem 2 In addition, S(En;x,y) = 1 for G(V,E) where |V|=n and |E|=0 Is the graph obtained by omitting edge e Is the graph obtained by contracting edge e

  4. The Rank Generation Polynomial Theorem 2 – Proof Let us define : G’ = G – e G’’ = G / e r’<G> = r<G’> n’<G> = n<G’> r’’<G> = r<G’’> n’’<G> = n<G’’>

  5. The Rank Generation Polynomial Theorem 2 – Proof Let us denote :

  6. The Rank Generation Polynomial Observations

  7. The Rank Generation Polynomial Observations

  8. The Rank Generation Polynomial Observations

  9. The Rank Generation Polynomial Observations 3

  10. The Rank Generation Polynomial Observations 1 2

  11. The Rank Generation Polynomial Reminder

  12. The Rank Generation Polynomial Let us define

  13. The Rank Generation Polynomial 1

  14. The Rank Generation Polynomial

  15. The Rank Generation Polynomial 3 2

  16. The Rank Generation Polynomial

  17. The Rank Generation Polynomial +

  18. The Rank Generation Polynomial

  19. The Rank Generation Polynomial Obvious

  20. The Rank Generation Polynomial Obvious

  21. The Rank Generation Polynomial

  22. The Rank Generation Polynomial Obvious Q.E.D

  23. The Tutte Polynomial

  24. The Tutte Polynomial Reminder 4 5

  25. The Universal Tutte Polynomial

  26. The Universal Tutte Polynomial Theorem 6

  27. The Universal Tutte Polynomial Theorem 6 - Proof 5

  28. The Universal Tutte Polynomial If e is a bridge 4

  29. The Universal Tutte Polynomial If e is a bridge

  30. The Universal Tutte Polynomial If e is a loop 4

  31. The Universal Tutte Polynomial If e is a bridge

  32. The Universal Tutte Polynomial If e is neither a loop nor a bridge

  33. The Universal Tutte Polynomial If e is neither a loop nor a bridge 4

  34. The Universal Tutte Polynomial If e is neither a loop nor a bridge

  35. The Universal Tutte Polynomial If e is neither a loop nor a bridge

  36. The Universal Tutte Polynomial Theorem 6 - Proof Q.E.D

  37. The Universal Tutte Polynomial Theorem 7 then

  38. The Universal Tutte Polynomial Theorem 7 – proof V(G) is dependant entirely on V(G-e) and V(G/e). In addition – Q.E.D

  39. Evaluations of the Tutte Polynomial Proposition 8 Let G be a connected graph. is the number of spanning trees of G Shown in the previous lecture

  40. Evaluations of the Tutte Polynomial Proposition 8 Let G be a connected graph. is the number of connected spanning sub-graphs of G

  41. Evaluations of the Tutte Polynomial Proposition 8 Let G be a connected graph. is the number of connected spanning sub-graphs of G sub-graphs of G connected

  42. Evaluations of the Tutte Polynomial Proposition 8 Let G be a connected graph. is the number of (edge sets forming) spanning forests of G Spanning forests

  43. Evaluations of the Tutte Polynomial Proposition 8 Let G be a connected graph. is the number of spanning sub-graphs of G

  44. The Universal Tutte Polynomial Theorem 9 The chromatic polynomial and the Tutte polynomial are related by the equation :

  45. The Universal Tutte Polynomial Theorem 9 – proof Claim :

  46. The Universal Tutte Polynomial Theorem 9 – proof We must show that :

  47. The Universal Tutte Polynomial Theorem 9 – proof and :

  48. The Universal Tutte Polynomial Theorem 9 – proof Obvious. (empty graphs)

  49. The Universal Tutte Polynomial Theorem 9 – proof Obvious Chromatic polynomial property Chromatic polynomial property

  50. The Universal Tutte Polynomial Theorem 9 – proof Thus :

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