Interpretations of the Tutte polynomials of regular matroids

Advances in Applied Mathematics 111 (2019) 101934

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Interpretations of the Tutte polynomials of regular matroids ✩ Martin Kochol MÚ SAV, Štefánikova 49, 814 73 Bratislava 1, Slovakia

a r t i c l e

i n f o

Article history: Received 31 October 2018 Received in revised form 29 April 2019 Accepted 12 August 2019 Available online xxxx MSC: 05C31 05B35

a b s t r a c t A regular chain group N is the set of integral vectors orthogonal with rows of a matrix representing a regular matroid M , i.e., a totally unimodular matrix. N correspond to the set of flows and tensions if M is a graphic and cographic matroid, respectively. We evaluate the Tutte polynomial of M as number of pairs of specified elements of N . © 2019 Elsevier Inc. All rights reserved.

Keywords: Regular matroid Tutte polynomial Regular chain group Totally unimodular matrix Convolution formula

1. Introduction Regular matroids are representable by totally unimodular matrices. Other equivalent characterizations are in [21, Chapter 13] or [5,24,26,29]. A regular chain group N on a finite set E consists of integral vectors (called chains) indexed by E and orthogonal to rows of a totally unimodular matrix (i.e., a representative matrix of a regular ma✩

Partially supported by grant VEGA 2/0024/18. E-mail address: [email protected].

https://doi.org/10.1016/j.aam.2019.101934 0196-8858/© 2019 Elsevier Inc. All rights reserved.

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M. Kochol / Advances in Applied Mathematics 111 (2019) 101934

troid). Suppose that Q(N ; k) denote the number of chains from N with values from {±1, . . . , ±(k−1)}. Moreover, if the coordinates with negative values are indexed by elements of X ⊆ E (resp. are considered mod k), denote this number by Q(N, X; k) (resp. P (N ; k)). Q(N ; k) and P (N ; k) are polynomials in variable k and are sums of Q(N, X; k) where X runs through different subsets of E (determined by an equivalence relation ∼ on the powerset of E). Q(N, X; k) is an Ehrhard polynomial of an integral polytope and P (N ; k) is the characteristic polynomial of dual of the matroid accompanied with N . The Tutte polynomial is a well known invariant that encodes many properties of matroids and has applications in combinatorics, knot theory, statistical physics and coding theory (for surveys see [3,4,6,7,11,14,22]). Evaluating the Tutte polynomial at a point or finding its coefficients is in general a #P -hard problem [1,15], even for planar graphs [28], and evaluations are in general difficult even to approximate [13]. We introduce formulas expressing negative values of the Tutte polynomial of a regular matroid M as sum of products of polynomials Q(N, X; k) of chains associated with minors of M and M ∗ , or as number of pairs of specified chains associated with such minors. In this way we characterize the Tutte polynomial by means of Ehrhart polynomials, thus we can apply a reciprocity law and obtain analogous interpretations for positive values of the Tutte polynomial. Furthermore, we introduce an integral variant of the Tutte polynomial of a regular matroid. 2. Preliminaries In this section we recall some basic properties of regular matroids and regular chain groups presented in [2,18,21,23,25,27,29]. Throughout this paper, E denotes a finite nonempty set. The collection of mappings from E to a set S is denoted by S E . If R is a ring, the elements of RE are considered as vectors indexed by E and we will use notation f + g, −f , and sf for f, g ∈ RE and s ∈ R. A chain on E (over R, or simply an R-chain) is f ∈ RE and the support of f is σ(f ) = {e ∈ E; f (e) = 0}. We say that f is proper if σ(f ) = E. The zero chain (denote by 0) has null support. Given X ⊆ E and f ∈ RE let ρX (f ) ∈ RE be defined so that for each e ∈ E,  [ρX (f )](e) =

−f (e) if e ∈ X, f (e) if e ∈ / X,

and let ρX (Y ) = {ρX (f ), f ∈ Y } for any Y ⊆ RE . Furthermore, let f \X ∈ RE\X be defined so that f \X (e) = f (e) for each e ∈ E \ X. A matroid M on E of rank r(M ) is regular if there exists an r × n (r = r(M ), n = |E|) totally unimodular matrix D (called a representative matrix of M ) such that independent sets of M correspond to independent sets of columns of D. For any basis B of M , D can be transformed to a form (Ir |U ) such that Ir corresponds to B and U is

M. Kochol / Advances in Applied Mathematics 111 (2019) 101934

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totally unimodular. The dual of M is regular matroid M ∗ with a representative matrix (−U T |In−r ) (where In−r corresponds to E \ B). By a regular chain group N on E (associated with D) we mean the set of chains on E over Z that are orthogonal to each row of D (i.e., are integral combinations of rows of a representative matrix of M ∗ ). The set of chains orthogonal to every chain of N is a chain group called orthogonal to N and denoted by N ⊥ (clearly, N ⊥ is the set of integral combinations of rows of D). By rank of N we mean r(N ) = n − r(M ) = r∗ (M ). Then r(N ⊥ ) = n − r(N ) = r(M ). Throughout this paper, we always assume that a regular chain group N is associated with a matrix D = D(N ) representing a matroid M = M (N ). For any X ⊆ E, define by   N −X = f \X ; f ∈ N, σ(f ) ∩ X = ∅ ,   N/X = f \X ; f ∈ N .

(1)

Clearly, M (N −X) = M − X and D(N −X) arises from D(N ) after deleting the columns corresponding to X. Furthermore (N −X)⊥ = N ⊥ /X, (N/X)⊥ = N ⊥ −X, and M (N/X) = M/X. Let A be an Abelian group with additive notation. We shall consider A as a (right) Z-module such that the scalar multiplication a · z of a ∈ A by z ∈ Z is equal to 0 if z −z z = 0, 1 a if z > 0 and 1 (−a) if z < 0. Similarly if a ∈ A and f ∈ ZE then define a · f ∈ AE so that (a · f )(e) = a · f (e) for each e ∈ E. If N is a regular chain group on E, define by m A(N ) = { i=1 ai · fi ; ai ∈ A, fi ∈ N, m ≥ 1} , A[N ] = {f ∈ A(N ); σ(f ) = E} . Notice that A(N ) = N if A = Z. By [2, Proposition 1], g ∈ AE is from A(N ) if and only if for each f ∈ N ⊥ ,  e∈E g(e) · f (e) = 0.

(2)

Let P (N, A) = |A[N ]| and denote by P (N ; k) = P (N, Zk ) = |Zk [N ]|. We will denote by Z+ the set of positive integers. Define by Nk = {f ∈ N ; 1 ≤ |f (e)| ≤ k − 1 for each e ∈ E}, Nk (X) = {f ∈ Nk ; ρX (f ) ∈ ZE + },

X ⊆ E.

Let Q(N ; k) = |Nk | and Q(N, X; k) = |Nk (X)|, X ⊆ E. Clearly, Nk is equal to the (disjoint) union of Nk (X) where X runs through all subsets of E. We denote by P(E) the set of subsets of E. For any X ⊆ E denote by χX ∈ ZE such that χX (e) = 1 (resp. χX (e) = 0) for each e ∈ E (resp. e ∈ E \ X).

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M. Kochol / Advances in Applied Mathematics 111 (2019) 101934

Define the equivalence relation ∼ on P(E) by: X, X  ∈ P(E) satisfy to X ∼ X  if and only if χX − χX  ∈ N . The set of the equivalence classes will be denoted by P(E)/ ∼. By [2, Proposition 3(a)], for each X ∈ P(E)/ ∼ and X, X  ∈ X , Q(N, X; k) = Q(N, X  ; k).

(3)

Thus we can define Q(N, X ; k) to be equal Q(N, X; k) for some X ∈ X . We say that X ⊆ E is positive if ρX (N ) ∩ ZE + = ∅. X ∈ P(E)/ ∼ will be said to be positive whenever some element of X is positive (because by (3) then every element of X will be positive). We shall denote the set of positive elements of P(E) (resp. of P(E)/ ∼) by O(N )+ (resp. O(N )+ ). A chain f of N we shall call k-chain if 0 ≤ f (e) ≤ k−1 for each e ∈ E. For any X ⊆ E, denote by N k (X) the set of all k-chains of ρX (N ) and denote by Q(N, X; k) = |N k (X)|. Furthermore, Q(N, X; k) = Q(N, X  ; k) if X, X  ∈ X ∈ P(E)/ ∼, and we can define Q(N, X ; k) = Q(N, X; k) for some X ∈ X . Let H ⊆ E and  be a labeling of elements of E by pairwise different integers. For any f ∈ N denote by ef ∈ E such that (ef ) = min{(e); e ∈ σ(f )} and we say that f is (H, )-compatible if f (ef ) and (χH − χE\H )(ef ) have the same sign. Denote by OH, (N ) the subset of O(N )+ consisting of sets X such that each c ∈ N 2 (X) is (H, )-compatible. By Propositions 3, 7, 10, and 15 from [2] and [18, Lemma 2] (see also [17, Theorems 1, 2] for similar results about graphic and cographic matroids) we have P (N ; k) and Q(N ; k) are polynomials in k of degree r(N ), if X ∈ O(N )+ , then Q(N, X; k) is a polynomial in k of degree r(N ), |OH, (N ) ∩ X | = 1 for each X ∈ O(N )+ , |O(N )+ | = (−1)r(N ) P (N ; −1),

(4)

P (N ; 0), |O(N ) | = |OH, (N )| = (−1)   P (N ; k) = Q(N, X ; k) = Q(N, X ; k). +

r(N )

X ∈P(E)/∼

X ∈O(N )+

By Propositions 9 and 12–14 from [2] and [18, Lemma 2] we have Q(N, X; −k) = (−1)r(N ) Q(N, X; k + 1), for each X ∈ O(N )+ , |X | = Q(N, X ; 2), for each X ∈ O(N )+ ,  Q(N, X; k)  P (N ; k) = = Q(N, X; k), Q(N, X; 2) X∈O (N ) X∈O(N )+ H,

P (N ; −k) = (−1)

r(N )

= (−1)r(N )

 X ∈O(N )+

 X∈O(N )+

(5)

Q(N, X; k + 1) Q(N, X; 2)

Q(N, X ; k + 1) = (−1)r(N )

 X∈OH, (N )

Q(N, X; k + 1).

M. Kochol / Advances in Applied Mathematics 111 (2019) 101934

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The reciprocity law expressed in the first row of (5) follows from the fact that Q(N, X; k) is an Ehrhart polynomial of an integral polytope (for more details see [2, Section III.2] and [10, Theorem 5.1, Corollary B.1, p. 23]). From the definition of Q(N ; k), (3), and (5) we have 

Q(N ; k) =

Q(N, X; k) =

X∈O(N )+

Q(N, X ; k)Q(N, X ; 2),

X ∈O(N )+



Q(N ; −k) = (−1)r(N )



Q(N, X; k + 1) =

(6)

X∈O(N )+



(−1)r(N )

Q(N, X ; k + 1)Q(N, X ; 2).

X ∈O(N )+

3. The Tutte polynomial Let M be a matroid on E with rank function r. The Tutte polynomial of M is 

T (M ; x, y) =

(x − 1)r(E)−r(X) (y − 1)r

∗

(E)−r ∗ (E\X)

.

X⊆E

The duality formula is T (M ; x, y) = T (M ∗ ; y, x) and the convolution formula of Kook, Reiner, and Stanton [19] (see also [12,20]) is 

T (M ; x, y) =

T (M/X; x, 0) T (M |X; 0, y)

X⊆E

where M |X = M − (E\X). Let N be a regular chain group on E. By [7, Corollary 6.2.6] and [18, Lemma 1], P (N ; k) = (−1)r(N ) T (M (N ); 0, 1 − k), thus by duality and convolution formulas we get T (M (N ); x, y) = 



T (M (N )∗ −X; 0, x) T (M (N )|X; 0, y) =

X⊆E

(−1)r(N

⊥

−X)

P (N ⊥ −X; 1−x) (−1)r(N |X) P (N |X; 1−y)

X⊆E

where N |X = N − (E\X). By definition of N ⊥ and (1) we have r(N ⊥ −X) = r(E\X), r(N |X) = r∗ (X) = |X| − r(E) + r(E\X), and thus

M. Kochol / Advances in Applied Mathematics 111 (2019) 101934

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r(N ⊥ −X) + r(N |X) = r(E) + |X| − 2(r(E) − r(E\X)),

(7)

whence 

T (M (N ); x, y) =

(−1)r(E)+|X| P (N ⊥ −X; 1−x) P (N |X; 1−y)

(8)

X⊆E

and applying (4) and (5) we have T (M (N ); x, y) = ⎞ ⎛   (−1)r(E)+|X| ⎝ Q(N ⊥ −X, Y; 1−x)⎠ X⊆E

Y∈O(N ⊥ −X)+

⎛



⎝

(9)

⎞

Q(N |X, Y  ; 1−y)⎠ ,

Y  ∈O(N |X)+

and T (M (N ); x, y) = ⎞ ⎛ ⊥   Q(N −X, Y ; 1−x) ⎠ (−1)r(E)+|X| ⎝ Q(N ⊥ −X, Y ; 2) X⊆E Y ∈O(N ⊥ −X)+ ⎛



⎝

Y  ∈O(N |X)+

(10)

⎞

Q(N |X, Y  ; 1−y) ⎠ . Q(N |X, Y  ; 2)

For any X ⊆ E, let (X) be a labeling of elements from X by pairwise different integers and H(X) ⊆ X. By (5) and (9), we have



⎛

T (M (N ); x, y) = 

(−1)r(E)+|X| ⎝

Y ∈OH(E\X),(E\X)

X⊆E

⎛



⎝ Y

⎞ Q(N ⊥ −X, Y ; 1−x)⎠

(N ⊥ −X)

(11) ⎞

Q(N |X, Y  ; 1−y)⎠ .

 ∈O H(X),(X) (N |X)

Theorem 1. Let N be a regular chain group on E, (X) be a labeling of elements from X by pairwise different integers and H(X) ⊆ X for any X ⊆ E. For each x, y < 0, T (M (N ); x, y) = s1 − s2 where s1 (resp. s2 ) is the number of all couples of chains (f, f  ) such that f ∈ (N ⊥ −X)1−x (Y ), f  ∈ (N |X)1−y (Y  ), Y ∈ OH(E\X),(E\X) (N ⊥ −X),

M. Kochol / Advances in Applied Mathematics 111 (2019) 101934

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Y  ∈ OH(X),(X) (N |X), and X ⊆ E where |X| has the same (resp. different) parity as r(M (N )). Proof. Follows directly from (11) and the fact that (−1)r(E)+|X| equals 1 (resp. −1) if and only if |X| has the same (resp. different) parity as r(M (N )). 2 If  is a labeling of elements of E by numbers 1, . . . , |E| and H ⊆ E, we can set H(X) = H ∩ X and (X) to be the restriction of  to X (X ⊆ E). Hence by (11) and Theorem 1,

 X⊆E

⎛

T (M (N ); x, y) = 

(−1)r(E)+|X| ⎝

⎞

Q(N ⊥ −X, Y ; 1−x)⎠

Y ∈OH\X, (N ⊥ −X)

⎛



⎝

(12)

⎞

Q(N |X, Y  ; 1−y)⎠ .

Y  ∈OH∩X, (N |X)

Corollary 1. Let N be a regular chain group on E,  be a labeling of elements of E by numbers 1, . . . , |E|, and H ⊆ E. For each x, y < 0, T (M (N ); x, y) = s1 − s2 where s1 (resp. s2 ) is the number of all couples of chains (f, f  ) such that f ∈ (N ⊥ −X)1−x (Y ), f  ∈ (N |X)1−y (Y  ), Y ∈ OH\X, (N ⊥ −X), Y  ∈ OH∩X, (N |X), and X ⊆ E where |X| has the same (resp. different) parity as r(M (N )). Furthermore, by (5), (6), (7), and (8), we have

T (M (N ); x, y) =



⎛ ⎝

X⊆E

⎛

Q(N ⊥ −X, Y; x)⎠

Y∈O(N ⊥ −X)+



⎝

⎞



⎞

(13)

Q(N |X, Y  ; y)⎠ =

Y  ∈O(N |X)+



⎛ ⎝

X⊆E

⎛ ⎝



Y ∈O(N ⊥ −X)+

 Y  ∈O(N |X)+

⎞ Q(N ⊥ −X, Y ; x) ⎠ Q(N ⊥ −X, Y ; 2) 

⎞

Q(N |X, Y ; y) ⎠ = Q(N |X, Y  ; 2)

(14)

M. Kochol / Advances in Applied Mathematics 111 (2019) 101934

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 X⊆E

⎛

⎞



⎝

Q(N ⊥ −X, Y ; x)⎠

Y ∈OH(E\X),(E\X) (N ⊥ −X)

⎛



⎝

⎞

(15)

Q(N |X, Y  ; y)⎠ ,

 ∈O H(X),(X) (N |X)

Y

where (X) is a labeling of elements from X by pairwise different integers and H(X) ⊆ X for any X ⊆ E. This implies the following statement. Theorem 2. Let N be a regular chain group on E, (X) be a labeling of elements from X by pairwise different integers and H(X) ⊆ X for any X ⊆ E. For each x, y ≥ 2, T (M (N ); x, y) is equal to the number of all couples of chains (f, f  ) such that f ∈ [N ⊥ −X]x (Y ), f  ∈ [N |X]y (Y  ), X ⊆ E, Y ∈ OH(E\X),(E\X) (N ⊥ −X), Y  ∈ OH(X),(X) (N |X), and X ⊆ E. If  is a labeling of elements of E by numbers 1, . . . , |E| and H ⊆ E, we can set H(X) = H ∩ X and (X) to be the restriction of  to X (X ⊆ E). Hence by (15) and Theorem 2,



T (M (N ); x, y) =

⎛ ⎝

X⊆E

⎛ ⎝



Y ∈OH\X, (N ⊥ −X)



⎞ Q(N ⊥ −X, Y ; x)⎠ ⎞

(16)

Q(N |X, Y  ; y)⎠ .

Y  ∈OH∩X, (N |X)

Corollary 2. Let N be a regular chain group on E,  be a labeling of elements of E by numbers 1, . . . , |E|, and H ⊆ E. For each x, y ≥ 2, T (M (N ); x, y) is equal to the number of all couples of chains (f, f  ) such that f ∈ [N ⊥ −X]x (Y ), f  ∈ [N |X]y (Y  ), X ⊆ E, Y ∈ OH\X, (N ⊥ −X), Y  ∈ OH∩X, (N |X), and X ⊆ E. In fact, Q(N ; k) is an integral variant of P (N ; k) and with respect to (8), it is natural to define an integral variant of the Tutte polynomial so that

I(M (N ); x, y) =

 X⊆E

whence by (6),

(−1)r(E)+|X| Q(N ⊥ −X; 1−x) Q(N |X; 1−y),

(17)

M. Kochol / Advances in Applied Mathematics 111 (2019) 101934

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I(M (N ); x, y) = ⎞ ⎛   (−1)r(E)+|X| ⎝ Q(N ⊥ −X, Y ; 1−x)⎠ X⊆E

Y ∈O(N ⊥ −X)+

⎛



⎝

(18) ⎞

Q(N |X, Y  ; 1−y)⎠ ,

 ∈O(N |X)+

Y

and the following statement holds true. Theorem 3. If N is a regular chain group on E, then for each x, y < 0, I(M (N ); x, y) = s1 − s2 where s1 (resp. s2 ) is the number of all couples of chains (f, f  ) such that f ∈ [N ⊥ −X]1−x (Y ), f  ∈ [N |X]1−y (Y  ), X ⊆ E, Y ∈ O(N ⊥ −X)+ , Y  ∈ O(N |X)+ , and X ⊆ E where |X| has the same (resp. different) parity as r(M (N )). Furthermore by (7), (18), and (5),

I(M (N ); x, y) = ⎛ ⎝

 X⊆E

⎛



⎝ Y

∈O(N ⊥ −X)+



⎞ Q(N ⊥ −X, Y ; x)⎠ ⎞

(19)

Q(N |X, Y  ; y)⎠ ,

Y  ∈O(N |X)+

whence we get the following. Theorem 4. If N is a regular chain group on E, then for each x, y ≥ 2, I(M (N ); x, y) is equal to the number of all couples of chains (f, f  ) such that f ∈ [N ⊥ −X]x (Y ), f  ∈ [N |X]y (Y  ), Y ∈ O(N ⊥ −X)+ , and Y  ∈ O(N |X)+ , and X ⊆ E. As pointed out before, Q(N, X; k) are Ehrhart polynomials, thus Q(N, X; k) are sums of Ehrhart polynomials. Therefore formulas (11), (15), (18), and (19) present T (M (N ); x, y) and I(M (N ); x, y) as sums of products of Ehrhart polynomials. Notice that analogues of formulas (8)–(10), (13), (14), (17)–(19) were studied for graphic matroids in [8,9,16]. References [1] J.D. Annan, The complexities of the coefficients of the Tutte polynomial, Discrete Appl. Math. 57 (1995) 93–103. [2] D.K. Arrowsmith, F. Jaeger, On the enumeration of chains in regular chain-groups, J. Combin. Theory Ser. B 32 (1982) 75–89. [3] L. Beaudin, J. Ellis-Monaghan, G. Pangborn, R. Shrock, A little statistical mechanics for the graph theorist, Discrete Math. 310 (2010) 2037–2053. [4] N. Biggs, Algebraic Graph Theory, second ed., Cambridge University Press, Cambridge, 1993.

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