Download The Tutte and Chromatic Polynomials: Relationships and Computations and more Study notes Mathematics in PDF only on Docsity!
Wednesday 2/
The Tutte Polynomial
Definition 1. Let M be a matroid with ground set E and let e ∈ E. The Tutte polynomial T (M ) =
T (M ; x, y) is computed recursively as follows:
(T1) If E = ∅, then T (M ) = 1.
(T2a) If e ∈ E is a loop, then T (M ) = y · T (M/e).
(T2b) If e ∈ E is a coloop, then T (M ) = x · T (M − e).
(T3) If e ∈ E is neither a loop nor a coloop, then T (M ) = T (M − e) + T (M/e).
We prove that T (M ) is well-defined by giving a closed formula for it, the corank-nullity
∗ generating function.
Theorem 1. Let r be the rank function of the matroid M. Then
(1) T (M ; x, y) =
A⊆E
(x − 1)
r(E)−r(A) (y − 1)
|A|−r(A) .
Proof. Let
T (M ) =
T (M ; x, y) denote the generating function on the right-hand side of (1). We will prove
by induction on n = |E| that
T (M ) obeys the recurrence of Definition 1 for every ground set element e,
hence equals T (M ). Let r
′ and r
′′ denote the rank functions on M − e and M/e respectively.
For (T1), if E = ∅, then (1) gives
T (M ) = 1 = T (M ).
For (T2a), let e be a loop. Then r
′ (A) = r(A) = r(A ∪ e) for every A ⊂ E \ e, so
T (M ) =
A⊆E
(x − 1)
r(E)−r(A) (y − 1)
|A|−r(A)
A⊆E
e 6 ∈A
(x − 1)
r(E)−r(A) (y − 1)
|A|−r(A)
B⊆E
e∈B
(x − 1)
r(E)−r(B) (y − 1)
|A|−r(B)
A⊆E\e
(x − 1)
r
′ (E\e)−r
′ (A) (y − 1)
|A|−r
′ (A)
A⊆E\e
(x − 1)
r
′ (E\e)−r
′ (A) (y − 1)
|A|+1−r
′ (A)
= (1 + (y − 1))
A⊆E\e
(x − 1)
r
′ (E\e)−r
′ (A) (y − 1)
|A|−r
′ (A)
= y
T (M − e).
For (T2b), let e be a coloop. Then r
′′ (A) = r(A) = r(A ∪ e) − 1 for every A ⊂ E \ e, so
T (M ) =
A⊆E
(x − 1)
r(E)−r(A)
(y − 1)
|A|−r(A)
e 6 ∈A⊆E
(x − 1)
r(E)−r(A) (y − 1)
|A|−r(A)
e∈B⊆E
(x − 1)
r(E)−r(B) (y − 1)
|A|−r(B)
A⊆E\e
(x − 1)
(r
′′ (E\e)+1)−r
′′ (A) (y − 1)
|A|−r
′′ (A)
A⊆E\e
(x − 1)
(r
′′ (E\e)+1)−(r
′′ (A)+1)
(y − 1)
|A|+1−(r
′′ (A)+1)
∗ The quantity r(E) − r(A) is the corank of A; it is the minimum number of elements one needs to add to A to obtain a
spanning set of M. Meanwhile, |A| − r(A) is the nullity of A: the minimum number of elements one needs to remove from A
to obtain an acyclic set.
A⊆E\e
(x − 1)
r
′′ (E\e)+1−r
′′ (A) (y − 1)
|A|−r
′′ (A)
A⊆E\e
(x − 1)
r
′′ (E\e)−r
′′ (A) (y − 1)
|A|−r
′′ (A)
= ((x − 1) + 1)
A⊆E\e
(x − 1)
r
′′ (E\e)−r
′′ (A) (y − 1)
|A|−r
′′ (A)
= x
T (M/e).
Finally, suppose that e is neither a loop nor a coloop. Then
r
′ (A) = r(A) and r
′′ (A) = r(A ∪ e) − 1
so
T (M ) =
A⊆E
(x − 1)
r(E)−r(A)
(y − 1)
|A|−r(A)
A⊆E\e
[
(x − 1)
r(E)−r(A) (y − 1)
|A|−r(A)
]
[
(x − 1)
r(E)−r(A∪e) (y − 1)
|A∪e|−r(A∪e)
]
A⊆E\e
[
(x − 1)
r
′ (E\e)−r
′ (A) (y − 1)
|A|−r
′ (A)
]
[
(x − 1)
(r
′′ (E)+1)−(r
′′ (A)+1) (y − 1)
|A|+1−(r
′′ (A)−1)
]
A⊆E\e
(x − 1)
r
′ (E\e)−r
′ (A) (y − 1)
|A|−r
′ (A)
A⊆E\e
(x − 1)
r
′′ (E\e)−r
′′ (A) (y − 1)
|A|−r
′′ (A)
T (M − e) +
T (M/e)
which is (T3).
As a consequence, we can obtain several invariants of a matroid easily from its Tutte polynomial.
Corollary 2. For every matroid M , we have
(1) T (M ; 1, 1) = number of bases of M ;
(2) T (M ; 2, 2) = |E|;
(3) T (M ; 2, 1) = number of independent sets of M ;
(4) T (M ; 1, 2) = number of spanning sets of M.
Proof. We’ve already proved (1) and (2), but they also follow from the corank-nullity generating function.
Plugging in x = 2, y = 2 will change every summand to 1. Plugging in x = 1 and y = 1 will change every
summand to 0, except for those sets A that have corank and nullity both equal to 0 — that is, those sets
that are both spanning and independent. The verifications of (3) and (4) are analogous.
A little more generally, we can use the Tutte polynomial to enumerate independent and spanning sets by
their cardinality:
A⊆E independent
q
|A| = q
r(M ) (2) T (1/q + 1, 1);
A⊆E spanning
q
|A| = q
r(M ) (3) T (1, 1 /q + 1).
Another easy fact is that T (M ) is multiplicative on direct sums:
T (M
1
⊕ M
2
) = T (M
1
)T (M
2
