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An Introduction of Tutte Polynomial

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An Introduction of Tutte Polynomial An Introduction of Tutte Polynomial Bo Lin December 12, 2013 Abstract Tutte polynomial, defined for matroids and graphs, has the important property that any multiplicative graph invariant with a deletion and con- traction property is an evaluation of it. In the article we introduce the three definitions of Tutte polynomial, show their equivalence, and present some examples of matroids and graphs to show how to compute their Tutte polynomials. This article is based on [3]. 1 Contents 1 Definitions3 1.1 Rank-Nullity.............................3 1.2 Deletion-Contraction.........................3 1.3 Internal and External activity....................5 2 Equivalence of Definitions5 2.1 First and Second Definition.....................5 2.2 First and Third Definition......................7 3 Examples 10 3.1 Dual Matroids............................ 10 3.2 Uniform Matroids.......................... 11 3.3 An example of graphic matroids.................. 12 2 1 Definitions Tutte polynomial is named after W. T. Tutte. It is defined on both matroids and graphs. In this paper, we focus on the Tutte polynomial of a matroid. We introduce three equivalent definitions in Section1, and then show their equivalence in Section2. Throughout this section, Let M = (E; r) be a matroid, where r is the rank function of M. 1.1 Rank-Nullity Definition 1.1. For all A ⊆ E, we denote z(A) = r(E) − r(A) and n(A) = jAj − r(A): n(A) is called the nullity of A. For example, if A is a basis of M, then z(A) = n(A) = 0: Then Tutte polynomial is defined by Definition 1.2. X z(A) n(A) TM (x; y) = (x − 1) (y − 1) : A⊆E 1.2 Deletion-Contraction We can also define Tutte polynomial in a recursive way. We need the notion of deletion and contraction. Definition 1.3. Let T be a subset of E. The deletion of M with respect to T , denoted as MnT , is a matroid with ground set E − T , and independent sets I(MnT ) = fI \ (E − T )jI 2 I(M)g: If T = feg, we also write Mne for MnT . Definition 1.4. Let T be a subset of E. The restriction of M with respect to T , denoted as M(T ), is the matroid Mn(E − T ): If T = feg, we also write M(e) for M(T ). Definition 1.5. M ∗ is a matroid, called the dual of M, with ground set E, and basis B(M ∗) = fE − BjB 2 B(M)g: 3 Definition 1.6. Let T be a subset of E. The contraction of M with respect to T , denoted as M=T , is the matroid (M ∗nT )∗: If T = feg, we also write M=e for M=T . Definition 1.7. Let e 2 E, call e a loop of M if feg is not an independent set of M; call e a coloop (or isthmus) if every basis of M contains e. It's trivial to show Proposition 1.8. Let e 2 E. If e is either a loop or coloop of M, then M=e = Mne: With these notions, now we can give another definition of Tutte polynomial. We define TM (x; y) recursively as follows: Definition 1.9. If M is the empty matroid(E is ;), then define TM (x; y) = 1: If e 2 E is a loop of M, then define TM (x; y) = yTM=e(x; y): If e 2 E is a coloop of M, then define TM (x; y) = xTM=e(x; y): Finally if e 2 E is neither a loop nor a coloop of M, then define TM (x; y) = TM=e(x; y) + TMne(x; y): This definition is recursive because if e 2 E then the ground sets of both M=e and Mne are one element less than E. Definition 1.10. Let f be a function defined on all matroids and e 2 E. If f(M) = f(M(e))f(M=e) if e is a loop or coloop and f(M) = f(Mne) + f(M=e) if e is neither a loop nor coloop ; then we call f a Tutte-Grothendieck invariant, T-G invariant for short. From this deletion-contraction definition we can also see that the Tutte poly- nomial is a T-G invariant. Actually we can show that any T-G invariant is an evaluation of the Tutte polynomial.[2][6.2.6] 4 1.3 Internal and External activity This alternative definition is Tutte's original definition. It explicitly gives the coefficients of the polynomial. We introduce two relevant notions. In this sub- section let's fix an order ≺ on the elements of E, say E = fe1; ··· ; eng, where ei ≺ ej if and only if i < j. Definition 1.11. Let B be a basis of M. An element e 2 B is called internally active if e is the smallest element with respect to order ≺ in the unique cocir- cuit disjoint from B − feg, otherwise e is called internally passive. Dually an element f 2 E − B is called externally active if f is the smallest element in the unique circuit contained in B [ffg, otherwise f is called externally passive. Let IA(B);IP (B); EA(B); EP (B) be the set of internally active, internally passive, externally active and externally passive elements of B respectively. Remark 1.12. We need the following lemma for the existence and uniqueness of the circuit contained in B [ ffg. Lemma 1.13. Let B be a basis of M and f 2 E −B, then there exists a unique circuit C contained in B [ ffg and f 2 C. Proof. Since B is a basis, so it is a maximal independent set. So B [ ffg is not an independent set. By the definition of circuit there exists a circuit C contained in B [ ffg. Of course C is not contained in B, so we have f 2 C. Suppose there is another distinct circuit C0 contained in B [ ffg, then f 2 C0 as well. But by the third axiom of circuits we have another circuit C2 such that 0 C2 ⊆ C [ C − ffg, then C2 ⊆ B, a contradiction. So C is the unique circuit contained in B [ ffg. Now we could carry out our third definition of Tutte polynomials. Definition 1.14. For any nonnegative integers i; j let ti;j be the number of basis of M with exactly i internally active elements and j externally active elements. Then X i j TM (x; y) = ti;jx y : i;j≥0 Remark 1.15. The equivalence of these definitions shows that actually ti;j is independent to the choice of order ≺. 2 Equivalence of Definitions In this section we prove the equivalence of the three definitions. 2.1 First and Second Definition In order to show that first definition is equivalent to the second, it's enough to show that X (x − 1)z(A)(y − 1)n(A) A⊆E 5 satisfies the equalities in Definiton 1.9. Actually the equality for empty matroid is trivial. If E = feg, then e is either a loop or coloop. If e is a loop, then the sum is (x − 1)z(;)(y − 1)n(;) + (x − 1)z(feg)(y − 1)n(feg) = 1 + (y − 1) = y: If e is a coloop, then the sum is (x − 1)z(;)(y − 1)n(;) + (x − 1)z(feg)(y − 1)n(feg) = 1 + (x − 1) = x: It's enough to verify the deletion-contraction relation for an element e that is neither a loop nor a coloop. In this case let z0 and r0 be the counterparts of z and r for M=e. Since e is neither a loop nor a coloop, we have r(E − e) = r(E) and r0(E − e) = r(E) − 1: For A ⊆ E − feg, we have r0(A) = jAj − r∗(E − e) + r∗(E − e − A) = jAj − (jE − ej − r(E) + r(e)) + (jE − e − Aj − r(E) + r(A [ e)) = jAj − jEj + 1 + r(E) − r(e) + jEj − 1 − jAj − r(E) + r(A [ e) = r(A [ e) − r(e) = r(A [ e) − 1: Then we have X z(A) n(A) TM (x; y) = (x − 1) (y − 1) A⊆E X = (x − 1)z(A)(y − 1)n(A) A⊆E−{eg X + (x − 1)z(A[feg)(y − 1)n(A[feg) A⊆E−{eg X = (x − 1)r(E−e)−r(A)(y − 1)n(A) A⊆E−{eg X + (x − 1)r(E)−r(A[feg)(y − 1)jAj+1−r(A[feg) A⊆E−{eg X r0(E−e)−r0(A) jA|−r0(A) = TMne(x; y) + (x − 1) (y − 1) A⊆E−{eg = TMne(x; y) + TM=e(x; y): So this equivalence is proved. Remark 2.1. The deletion-contraction definition enables us to calculate the Tutte polynomial of a matroid, though this matroid is not efficient when the ground set is very large. We will discuss this later in section3. 6 2.2 First and Third Definition In this subsection we show the following equivalence: Theorem 2.2. Fix an order ≺ on the ground set E, we have X i j X z(A) n(A) ti;jx y = (x − 1) (y − 1) : i;j≥0 A⊆E We present the proof given in [1]. We need to introduce the notion of shelling. The definitions of simplicial complex, facet, and pure are from [1] as well. Definition 2.3. Let ∆ be a pure simplicial complex. A shelling of ∆ is a linear sequence F1;F2; ··· ;Ft of the facets of ∆ such that for all 1 ≤ i < j ≤ t there exists a facet Fk satisfying 1 ≤ k < j and an element x 2 Fj such that Fi \ Fj ⊆ Fk \ Fj = Fj − fxg: A complex is called shellable if it is pure and admits a shelling.
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