Tutte polynomial T(G;x,y): the most general graph invariant satisfying deletion-contraction. T encodes: spanning trees (1,1), chromatic P (1-x,0), flow polynomial (0,1-y), reliability (1,p), acyclic orientations (2,0).
How to Use
Select graph:
- T(x,y): Full polynomial
- Evals: Special values
- Specs: Derived polynomials
Specializations
T(1,1) = #spanning trees. T(2,1) = #spanning forests. T(1,2) = #connected spanning subgraphs. T(2,0) = #acyclic orientations. P(G,k) = (-1)^{|V|-c}k^c T(G;1-k,0). Incredibly rich structure.
Computation
Deletion-contraction: exponential time. #P-hard to evaluate at most points (Jaeger-Vertigan-Welsh, 1990). Polynomial only on special curves. For planar graphs: connection to knot theory (Jones polynomial).
Step-by-Step Instructions
- 1Select graph.
- 2Compute T(G;x,y).
- 3Evaluate specials.
- 4Find derived polys.
- 5Compare graphs.