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.

Definition

A graph function f is TG-invariant if: $f(G)=\{\begin{array}{ll}{c}^{|V(G)|}& \text{if G has no edges}\\ xf(G/e)& \text{if }e\text{ is a bridge}\\ yf(G\mathrm{\setminus }e)& \text{if }e\text{ is a loop}\\ af(G/e)+bf(G\mathrm{\setminus }e)& \text{else}\end{array}$ Above G / e denotes edge contraction whereas G \ e denotes deletion. The numbers c, x, y, a, b are parameters.

Generalization to matroids

The matroid function f is TG if:

$$\begin{gathered} \begin{array}{rl}& f(M_1\oplus M_2)=f(M_1)f(M_2)\\ & f(M)=af(M/e)+bf(M\mathrm{\setminus }e) \\ \text{if }e\text{ is not coloop or bridge}\end{array} \end{gathered}$$

It can be shown that f is given by:

$f(M)=a^{|E|-r(E)}{b}^{r(E)}T(M;x/a,y/b)$

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

$T(M;x,y)=\sum _{A\subset E(M)}(x-1{)}^{r(E)-r(A)}(y-1{)}^{|A|-r(A)}$

is the generalization of the Tutte polynomial to matroids.

Grothendieck group

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.