The Tutte polynomial is an invariant of a matroid that is universal with respect to satisfying a deletion-contraction recurrence. Indeed, one way to define the Tutte polynomial of a matroid is: if M is a matroid consisting of a loops and b coloops, then TM(x, y) = xayb, and if e ∈ M is neither a loop nor a coloop, then TM(x, y) := TM/e(x, y) + TM\e(x, y), where M\e is the deletion of M with respect to {e}, and M/e is the contraction of M with respect to {e}. Many invariants of a matroid can be determined by substituting values into its Tutte polynomial - cf. tutteEvaluate.
i1 : tuttePolynomial matroid completeGraph 4 3 3 2 2 o1 = x + y + 3x + 4x*y + 3y + 2x + 2y o1 : ZZ[x, y] |
i2 : tuttePolynomial specificMatroid "nonpappus" 6 5 4 3 3 2 2 o2 = y + 3y + 6y + x + 10y + 6x + 7x*y + 15y + 14x + 14y o2 : ZZ[x, y] |