Tutte–Grothendieck invariant

In mathematics, a Tutte–Grothendieck (TG) invariant is a type of graph invariant that satisfies a generalized deletion–contraction formula. Any evaluation of the Tutte polynomial would be an example of a TG invariant.[1][2]

Definition

edit

A graph function f is TG-invariant if:[2]

 

Above G / e denotes edge contraction whereas G \ e denotes deletion. The numbers c, x, y, a, b are parameters.

Generalization to matroids

edit

The matroid function f is TG if:[1]

 

It can be shown that f is given by:

 

where E is the edge set of M; r is the rank function; and

 

is the generalization of the Tutte polynomial to matroids.

Grothendieck group

edit

The invariant is named after Alexander Grothendieck because of a similar construction of the Grothendieck group used in the Riemann–Roch theorem. For more details see:

  • Tutte, W. T. (2008). "A ring in graph theory". Mathematical Proceedings of the Cambridge Philosophical Society. 43 (1): 26–40. doi:10.1017/S0305004100023173. ISSN 0305-0041. MR 0018406.
  • Brylawski, T. H. (1972). "The Tutte-Grothendieck ring". Algebra Universalis. 2 (1): 375–388. doi:10.1007/BF02945050. ISSN 0002-5240. MR 0330004.

References

edit
  1. ^ a b Welsh. Complexity, Knots, Colourings and Counting.
  2. ^ a b Goodall, Andrew (2008). "Graph polynomials and Tutte-Grothendieck invariants: an application of elementary finite Fourier analysis". arXiv:0806.4848 [math.CO].