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Matroids :: tuttePolynomial(Matroid)

tuttePolynomial(Matroid) -- Tutte polynomial of a matroid

Synopsis

Description

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 T_M(x, y) = x^ay^b, and if $e \in M$ is neither a loop nor a coloop, then T_M(x, y) := T_{M/e}(x, y) + T_{M\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]

See also