Tutte–Grothendieck invariant
Appearance
This article needs additional citations for verification. (August 2019) |
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.