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

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

${\displaystyle f(G)={\begin{cases}c^{|V(G)|}&{\text{if G has no edges}}\\xf(G/e)&{\text{if }}e{\text{ is a bridge}}\\yf(G\backslash e)&{\text{if }}e{\text{ is a loop}}\\af(G/e)+bf(G\backslash e)&{\text{else}}\end{cases}}}$

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:^[1]

${\displaystyle {\begin{aligned}&f(M_{1}\oplus M_{2})=f(M_{1})f(M_{2})\\&f(M)=af(M\backslash e)+bf(M/e)\ \ \ {\text{if }}e{\text{ is not coloop or bridge}}\end{aligned}}}$

It can be shown that f is given by:

${\displaystyle f(M)=a^{|E|-r(E)}b^{r(E)}T(M;x_{0}/b,y_{0}/a)}$

where ⁠${\displaystyle x_{0}}$⁠ is the value f takes on coloops, ⁠${\displaystyle y_{0}}$⁠ is the value f takes on loops, E is the edge set of M; r is the rank function; and

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

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.