# The Tutte Polynomial of Some Matroids

Authors: Criel Merino; Marcelino Ramírez-Ibáñez; Guadalupe Rodríguez-Sánchez
Journal: International Journal of Combinatorics
 (2012)

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## Abstract

The Tutte polynomial of a graph or a matroid, named after W. T. Tutte, has the important universal property that essentially any multiplicative graph or network invariant with a deletion and contraction reduction must be an evaluation of it. The deletion and contraction operations are natural reductions for many network models arising from a wide range of problems at the heart of computer science, engineering, optimization, physics, and biology. Even though the invariant is #P-hard to compute in general, there are many occasions when we face the task of computing the Tutte polynomial for some families of
graphs or matroids. In this work, we compile known formulas for the Tutte polynomial of some families of graphs and matroids. Also, we give brief explanations of the techniques that were used to find the formulas. Hopefully, this will be useful for researchers in Combinatorics and elsewhere.


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## Body

## 1. Introduction

Many times, as researchers in Combinatorics, we face the task of computing an evaluation of the Tutte polynomial of a family of graphs or matroids. Sometimes this is not an easy task or at least time consuming. Later, not surprisingly, we find out that a formula was known for a class of graphs or matroids that contains our family. Here we survey some of the best known formulas for some interesting families of graphs and matroids. Our hope is for researchers to have a place to look for a Tutte polynomial before engaging in the search for the Tutte polynomial formula for the considered family.We present along with the formulas, some explanation of the techniques used to compute them. This may also provide tools for computing the Tutte polynomials of new families of graphs or matroids. This survey can also be considered a companion [1]. There, the authors give an introduction of the Tutte polynomial for a general audience of scientists, pointing out relevant relations between different areas of knowledge. But very few explicit calculations are made. Here we consider the practical side of computing the Tutte polynomial.However, we are not presenting evaluations, that are an immense area of research for the Tutte polynomial, nor analysing the complexity of computing the invariant for the different families. For the former, we strongly recommend the book of Welsh [2], for the latter, we recommend Noble’s book chapter [3]. There are many sources for the theory behind this important invariant. The most useful is definitively Brylawski and Oxley book chapter [4]. We already mentioned [1], which also surveys a variety of information about the Tutte polynomial of a graph, some of it new.We also do not address the closely related problem of characterizing families of matroids by their Tutte polynomial, a problem which is generally known as Tutte uniqueness. For this there are also several articles, for example [5–7].We assume knowledge of graph theory as in [8] and matroid theory as in [9]. Further details of many of the concepts treated here can be found in Welsh [2] and Brylawski and Oxley [4]. It is worth noticing that sometimes we switch between graphs and matroids without warning. This is because we really consider the graph G as the graphic matroid M(G).


## 2. Definitions

Some of the richness of the Tutte polynomial is due to its numerous equivalent definitions, which is probably inherited from the vast number of equivalent definitions of the concept of matroid. In this chapter we revise three definitions and we put them to practice by computing the Tutte of some families of matroids, in particular uniform matroids.


### 2.1. The Rank-Nullity Generating Function Definition

One of the simplest definitions, which is often the easiest way to prove properties of the Tutte polynomial, uses the notion of rank.IfM=(E,r) is a matroid, where r is the rank function of M, and A⊆E, we denote r(E)-r(A) by z(A) and |A|-r(A) by n(A), the latter is called the nullity of A.Definition 2.1.
The Tutte polynomial ofM, TM(x,y), is defined as follows:
(2.1)TM(x,y)=∑A⊆E‍(x-1)z(A)(y-1)n(A).Duality
Recall that ifM=(E,r) is a matroid, then M*=(E,r*) is its dual matroid, where r*(A)=|A|-r(E)+r(E∖A). Because zM*(A)=nM(E∖A) and nM*(A)=zM(E∖A), you get the following equality
(2.2)TM(x,y)=TM*(y,x).
This gives the first technique to compute a Tutte polynomial. If you know the Tutte polynomial of M, then, you know the Tutte polynomial of the dual matroid M*.Uniform Matroids
Our first example is the family of uniform matroidsUr,n, where 0≤r≤n. Here the Tutte polynomial can be computed easily using (2.1) because all subsets of size k≤r are independent, so n(·) is zero; all subsets of size k≥r are spanning, so z(·) is zero and if a subset is independent and spanning, then it is a basis of the matroid. Consider the following
(2.3)TUr,n(x,y)=∑i=0r-1(ni)(x-1)r-i+(nr)+∑i=r+1n‍(ni)(y-1)i-r.
Thus, forU2,5 we get
(2.4)TU2,5(x,y)=(x-1)2+5(x-1)+10+10(y-1)+5(y-1)2+(y-1)3=x2+3x+3y+2y2+y3.
AsU3,5=(U2,5)*, we get by using (2.2)
(2.5)TU3,5(x,y)=x3+2x2+3x+3y+y2.Matroid Relaxation
Given a matroidM=(E,r) with a subset X⊆E that is both a circuit and a hyperplane, we can define a new matroid M′ as the matroid with basis ℬ(M′)=ℬ(M)∪{X}. That M' is indeed a matroid is easy to check. For example, F7- is the unique relaxation of F7. The Tutte polynomial of M′ can be computed easily from the Tutte polynomial of M by using (2.1) one has.
(2.6)TM′(x,y)=TM(x,y)-xy+x+y.
Matroid relaxation will be used extensively in the last section.Sparse Paving Matroids
Our second example can be considered a generalization of uniform matroids but it is a much larger and richer family. Apaving matroidM=(E,r) is a matroid whose circuits all have size at least r. Uniform matroids are an example of paving matroids. Paving matroids are closed under minors and the set of excluded minors for the class consists of the matroid U2,2⊕U0,1, see, for example, [10]. The interest about paving matroids goes back to 1976 when Dominic Welsh asked if most matroids were paving, see [9].
Sparse paving matroids were introduced in [11, 12]. A rank-r matroid M is sparse paving if M is paving and for every pair of circuits C1 and C2 of size r we have |C1ΔC2|>2. For example, all uniform matroids are sparse paving matroids.
There is a simple characterization of paving matroids which are sparse in terms of the sizes of their hyperplanes. For a proof, see [10].Theorem 2.2.
LetM be a paving matroid of rank r≥1. Then M is sparse paving if and only if all the hyperplanes of M have size r or r-1.Note that we can say a little more, any circuit of sizer is a hyperplane. Conversely, any proper subset of a hyperplane of size r is independent and so such a hyperplane must be a circuit. Thus, the circuits of size r are precisely the hyperplanes of size r.Many invariants that are usually difficult to compute for a general matroid are easy for sparse paving matroids. For example, observe that ifM is sparse paving, all subsets of size k<r are independent, and all subsets of size k>r are spanning. On the other hand, the subsets of size r are either bases or circuit hyperplanes. Thus, the Tutte polynomial of a rank-r sparse matroid M with n elements and λ circuit-hyperplanes is given by
(2.7)TM(x,y)=∑i=0r-1‍(ni)(x-1)r-i+(nr)+λ(xy-x-y)+∑i=r+1n‍(ni)(y-1)i-r.Clearly, given ar-rank paving matroid M with n elements, we can obtain the uniform matroid Ur,n by a sequence of relaxations from M. If M has λ circuits-hyperplanes, (2.6) also gives (2.7).Free Extension
Another easy formula that we can obtaine from the above definition involves the free extension,M+e, of a matroid M=(E,r) by an element e∉E, which consists of adding the element e to M as independently as possible without increasing the rank. Equivalently, the rank function of M+e is given by the following equations: for X a subset of E,
(2.8)rM+e(X)=rM(X),rM+e(X∪e)={rM(X)+1,ifrM(X)<r(M),r(M),otherwise.
Again, the Tutte polynomial of M+e can be computed easily from the Tutte polynomial of M by using (2.1). Consider
(2.9)TM+e(x,y)=xx-1TM(x,y)+(y-xx-1)TM(1,y).
Here the trick is to notice that TM(1,y)=∑X‍(y-1)|X|-r(X), where the sum is over all subsets of E with r(X)=r(E), that is, spanning subsets of M. The presentation given here is from [13], but the formula can also be found in [14].
For example, the graphic matroidM(K4∖e) has as free extention the matroid Q6. Because M(K4∖e) is sparse paving, by using (2.7), we can compute its Tutte polynomial and obtain
(2.10)TK4∖e(x,y)=x3+2x2+x+2xy+y+y2.
Now, by using (2.9), we get the Tutte polynomial of the free extension:
(2.11)TQ6=x3+3x2+4x+2xy+4y+3y2+y3.
This result can be checked by using (2.7), as Q6 is also sparse paving.


### 2.2. Deletion and Contraction

In the second (equivalent) definition of the Tutte polynomial we use a linear recursion relation given by deleting and contracting elements that are neither loops nor coloops. This is by far the most used method to compute the Tutte polynomial.Definition 2.3.
IfM is a matroid, and e is an element that is neither a loop nor a coloop, then
(2.12)TM(x,y)=TM∖e(x,y)+TM/e(x,y).
If e is a coloop, then
(2.13)TM(x,y)=xTM∖e(x,y).
If e is a loop, then
(2.14)TM(x,y)=yTM/e(x,y).The proof that Definitions2.1 and 2.3 are equivalent can be found in [4]. From this it is clear that you just need the Tutte polynomial of the matroid without loops and coloops. Also, if you know the Tutte polynomial of M∖e and M/e, then you know the Tutte polynomial of the matroid M. This way of computing the Tutte polynomial leads naturally to linear recursions. We present two examples where these linear recursions give formulas.The CycleCn
As a first example of this subsection we consider the graphic matroidM(Cn). Here, by deleting an edge e from Cn we obtained a path whose Tutte polynomial is xn-1, while if we contract e we get a smaller cycle Cn-1. Thus, as the Tutte polynomial of the 2-cycle is x+y, we obtain
(2.15)TCn(x,y)=∑i=1n-1‍xi+y.
By duality, the Tutte polynomial of the graphCn* with two vertices and n parallel edges between them is
(2.16)TCn*(x,y)=∑i=1n-1‍yi+x.Parallel and Series Classes
As an almost trivial application of deletion and contraction, we look into the common case when you have a graph or a matroid with parallel elements. Let us define aparallel class in a matroid M as maximal subset X of E(M) such that any two distinct members of X are parallel and no member of X is a loop, see [9]. Then the following result is well known and has been found many times, see [15, 16].Lemma 2.4.
LetX be a parallel class of a matroid M with |X|=p+1. If X is not a cocircuit, then
(2.17)TM(x,y)=TM∖X(x,y)+(yp+yp-1+⋯+1)TM/X(x,y).The proof is by induction onp with the case p=0 being (2.12). The rest of the proof follows easily from the fact that each loop introduces a multiplicative factor of y in the Tutte polynomial.Aseries class in M is just a parallel class in M* and by duality we get the following.Lemma 2.5.
LetX be a series class of a matroid M with |X|=p+1. If X is not a circuit, then
(2.18)TM(x,y)=(xp+xp-1+⋯+1)TM∖X(x,y)+TM/X(x,y).The Rectangular LatticeLm,n
The rectangular lattice, that we will define in a moment, is our first example where computing its Tutte polynomial is really far from trivial and no complete answer is known. The interest resides probably in the importance of computing the Potts partition function, which is equivalent to the Tutte polynomial, of the square lattice. However, even a complete resolution of this problem will be just a small step towards the resolution of the really important problem of computing the Tutte polynomial of the cubic lattice.
Letm and n be integers, m,n>1. The grid or rectangular latticeLm,n is a connected planar graph with mn vertices, such that its faces are squares, except one face that is a polygon with 2(m+n)-4 edges. The grid graph Lm,n can be represented as in Figure 1.Figure 1
The grid graphLm,n.The vertices ofLm,n are denoted as ordered pairs, such that the vertex in the intersection of the row i and the column j is denoted by ij.As an example, we show a recursive formula for the Tutte polynomials of grid graphsLm,n when m=2. For m>3, the formulas are very complicated to use in practical calculations.The graphsL2,1, L2,2, and L2,3 are shown in Figure 2, with the labels that correspond to their vertices.Figure 2
The first three terms are from the sequence{L2,n} and the last three are from {Q2,n}.For every positive integern. The graph Q2,n is defined from the graph L2,n by contraction of the edge {11,21} of L2,n. In particular, the graph Q2,1 is one isolated vertex, see Figure 2.The initial conditions to construct the recurrence relations for the Tutte polynomial of the grid graphsL2,n are.(i)
TQ2,1(x,y)=1;(ii)
TL2,1(x,y)=x;(iii)
TL2,2(x,y)=x3+x2+x+y.The third condition is true because the graphL2,2 is isomorphic to the cycle C4, so we can use (2.15). The formula for the recurrence relation to calculate the Tutte polynomial of the graphs L2,n is
(2.19)TL2,n(x,y)=(x2+x+1)TL2,n-1(x,y)+yTQ2,n-1(x,y),TQ2,n(x,y)=(x+1)TL2,n-1(x,y)+yTQ2,n-1(x,y).From this, we get a linear recurrence of order two,(2.20)TL2,n-(x2+x+1+y)TL2,n-1+x2yTL2,n-2=0,
that is easy to solve. Thus, the general formula is
(2.21)TL2,n(x,y)=a1λ1+a2λ1-λ2λ1n-2+a1λ2+a2λ2-λ1λ2n-2,
for n≥2, where a1=y+x+x2+x3, a2=-yx3, and λ1 and λ2 are
(2.22)12[(1+y+x+x2)±(y2+2y(1+x-x2)+(1+x+x2)2)1/2].Arecursive family of graphsGm is a sequence of graphs with the property that the Tutte polynomials TGm(x,y) satisfy a linear homogeneous recursion relation in which the coefficients are polynomials in x and y with integral coefficients, independent of m, see [17]. So, the sequence {L2,n} is such a family and in general, {Lk,n}, for a fixed k, is a recursive family.Graphic and Representable Matroids
Probably the most common class of matroids, when evaluations of the Tutte polynomial are considered, is graphic matroids. IfM is the graphic matroid of a graph G, then deleting an element e from M amounts to deleting the corresponding edge from G, and contracting e to contracting the edge. This is easy to do in a small graph and by using Definition 2.3 you can compute the Tutte polynomial. An example is shown in Figure 3.Figure 3
An example of computing the Tutte polynomial of a graphic matroid using deletion and contraction.As we mentioned, computing the Tutte polynomial is not in general computationally tractable. However, for graphic matroids, there are some resources to compute it for reasonably sized graphs of about 100 edges. These include Sekine et al. [18], which provide an algorithm to implement the recursive Definition 2.3. Common computer algebra systems such as Maple and Mathematica will compute the Tutte polynomial for very small graphs, and there are also some implementations freely available on the Web, such as http://homepages.mcs.vuw.ac.nz/~djp/tutte/  by Haggard and Pearce.Similarly, given a matroidM=(E,r), representable over a field 𝔽, you can compute its Tutte polynomial using deletion and contraction. For any element e∈E, the matroids M∖e and M/e are easily computed from the representation of M, see [9]. This can be automatized and there are already computer programs that compute the Tutte polynomial of a representable matroid. For our calculations we have used the one given by Michael Barany, for more information about this program and how to use it, see [19]. Some of the calculations made in the Section 4 were computed using this program.


### 2.3. Internal and External Activity

Being a polynomial, it is natural to ask for the coefficients of the Tutte polynomial. Tutte’s original definition gives its homonymous polynomial in terms of its coefficients by means of a combinatorial interpretation, but before we give the third definition of the Tutte polynomial, we introduce the relevant notions.Let us fix an ordering≺ on the elements of M, say E={e1,…,em}, where ei≺ej if i<j. Given a fixed basis S, an element e is called internally active if e∈S and it is the smallest edge with respect to ≺ in the only cocircuit disjoint from S∖{e}. Dually, an element f is externally active if f∉S and it is the smallest element in the only circuit contained in S∪{f}. We define tij to be the number of bases with i internally active elements and j externally active elements. In [20–22] Tutte defined TM using these concepts. A proof of the equivalence with Definition 2.1 can be found in [23].Definition 2.6.
IfM=(E,r) is a matroid with a total order on its ground set, then
(2.23)TM(x,y)=∑i,j‍tijxiyj.
In particular, the coefficients tij are independent of the total order used on the ground set.Uniform Matroids Again
Our first use of (2.23) is to get a slightly different expression for the Tutte polynomial of uniform matroids. This time we get
(2.24)TUr,n(x,y)=∑j=1n-r(n-j-1r-1)yj+∑i=1r(n-i-1n-r-1)xi,
when 0<r<n, while TUn,n(x,y)=xn and TU0,n(x,y)=yn. This can also be established by expanding (2.3).Paving Matroids
Paving matroids were defined above. The importance of paving matroids is its abundance and meaning, most matroids of upto 9 elements are paving. This was checked in [12] and it has been conjecture in [24] that this is indeed true for all matroids, that is, when n is large, the probability that you choose a paving matroid uniformly at random among all matroids with up to n elements is approaching 1.
Now, in order to get a formula for the Tutte polynomial of paving matroids, we need the following definition from [9]. Given integers k>1 and m>0, a collection 𝒯={T1,…,Tk} of subsets of a set E, such that each member of 𝒯 has at least m elements and each m-element subset of E is contained in a unique member of 𝒯 and is called an m-partition of E. The elements of the partition are called blocks. The following proposition is also from [9].Proposition 2.7.
If𝒯 is an m-partition of E, then 𝒯 is the set of hyperplanes of a paving matroid of rank m+1 on E. Moreover, for r≥2, the set of hyperplanes of every rank-r paving matroid on E is an (r-1)-partition of E.Brylawski gives the following proposition in [25], but we mentioned that his proof does not use activities.Proposition 2.8.
LetM be a rank-r matroid with n elements and Tutte polynomial ∑i,j‍tijxiyj. Then, M is paving if and only if tij=0 for all (i,j)≥(2,1). In addition if the (r-1)-partition of E has bk blocks of cardinality k, for k=r-1,…n, then
(2.25)ti0=(n-i-1r-i)∀i≥2,t10=∑k=0∞(r-2+kr-2)bk+r-1+(n-2r-1)(nr-1),
and for all j>0,
(2.26)t1j=∑k=0∞(r-2+kr-2)bk+j+r-1,t0j=(n-j-1r-1)-∑k=0∞(r-1+kr-1)bk+j+r-1.Note that whenM is the uniform matroid Ur,n, M is a paving matroid with (r-1)-partition all subsets of size r-1. Thus the above formula gives us (2.24).As an example, consider the matroidR9 with geometric representation given in Figure 4. This matroid is paving but not sparse paving. The blocks correspond to lines in the representation. There are 7 blocks of size 3, 2 blocks of size 4, and 3 blocks of size 2, corresponding to trivial lines that do not appear in Figure 4. Using the above formulas we get that the Tutte polynomial of R9 is
(2.27)TR9(x,y)=x3+6x2+8x+11xy+2xy2+8y+13y2+10y3+6y4+3y5+y6.Figure 4
The matroidR9.Catalan Matroids
Our final example in this subsection is a family of matroids that although simple, it has a surprisingly natural combinatorial definition together with a nice interpretation of the internal and external activity.
Consider an alphabet constituted by the letters{N,E}. The length of a word w=w1w2⋯wn is n, the number of letters in w. Every word can be associated with a path on the plane ℤ2, with an initial point A=(m1,n1) and a final point B=(m2,n2), m1<m2 and n1<n2. The letter N is identified with a north step and the letter E with an east step, such that the first step of A=(0,0) to B will be N=(0,1) or E=(1,0).
LetA and B be a couple of fixed points of ℤ2. A lattice path is a path in ℤ2 from A to B using only steps N or E. The lattice paths from A to B have m steps E and n steps N. Let P and Q be two lattice paths, with initial points (x0,yP) and (x0,yQ) of P and Q, respectively. If yP<yQ for every x0 in [m1,m2]; then the set of lattice paths in this work is lattice paths from A to B bounded between P and Q, that we called PQ-bounded lattice paths.
The lattice pathsP and Q can be substituted by lines. If P is the line y=0, Q the line y=x and m=n, then the number of PQ-bounded lattice paths from (0,0) to (n,n) is the nth Catalan number:
(2.28)cn=1n+1(2nn).
Denote by[n] the set {1,2,…,n}. Consider the word w=w1w2⋯wm+n that corresponds to a PQ-bounded lattice path. We can associate to w a subset Xw of [m+n] by defining that i∈Xw if wi=N. Xw is the support set of the steps N in w. The family of support sets Xw of PQ-bounded lattice paths is the set of bases of a transversal matroid, denoted M[P,Q] with ground set [m+n], see [13].
If the bounds of the lattice paths are the linesy=0 and y=x, the matroid is denoted by Mn and is named Catalan matroid. Note that Mn has a loop that corresponds to the label 1 and a coloop that corresponds to the label 2n, and that Mn is autodual.
The fundamental result to compute the Tutte polynomial of Catalan matroids is the following, see [13], where here, for a basis B, we denote by i(B) the number of internally active elements and by e(B) the externally active elements.Proposition 2.9.
LetB∈ℬ be a basis of M[P,Q] and let wB be the PQ-bounded lattice path associated with B. Then i(B) is the number of times wB meets the upper path Q in a north step and e(B) is the number of times wB meets the lower path P in an east step.Then the Tutte polynomial of the Catalan matroidsMn for n>1, is
(2.29)TMn(x,y)=∑i,j>0‍i+j-2n-1(2n-i-j-1n-i-j+1)xiyj.Note that the coefficient ofxiyj in the Tutte polynomial of the matroid Mn depends only on n and the sum i+j.In the Figure5 are shown the bases of M3 with its internal and external activities. Thus the Tutte polynomial of M3 is
(2.30)TM3(x,y)=x3y+x2y+x2y2+xy2+xy3.Figure 5
Bases ofM3 with their internal and external activity.


## 2.1. The Rank-Nullity Generating Function Definition

One of the simplest definitions, which is often the easiest way to prove properties of the Tutte polynomial, uses the notion of rank.IfM=(E,r) is a matroid, where r is the rank function of M, and A⊆E, we denote r(E)-r(A) by z(A) and |A|-r(A) by n(A), the latter is called the nullity of A.Definition 2.1.
The Tutte polynomial ofM, TM(x,y), is defined as follows:
(2.1)TM(x,y)=∑A⊆E‍(x-1)z(A)(y-1)n(A).Duality
Recall that ifM=(E,r) is a matroid, then M*=(E,r*) is its dual matroid, where r*(A)=|A|-r(E)+r(E∖A). Because zM*(A)=nM(E∖A) and nM*(A)=zM(E∖A), you get the following equality
(2.2)TM(x,y)=TM*(y,x).
This gives the first technique to compute a Tutte polynomial. If you know the Tutte polynomial of M, then, you know the Tutte polynomial of the dual matroid M*.Uniform Matroids
Our first example is the family of uniform matroidsUr,n, where 0≤r≤n. Here the Tutte polynomial can be computed easily using (2.1) because all subsets of size k≤r are independent, so n(·) is zero; all subsets of size k≥r are spanning, so z(·) is zero and if a subset is independent and spanning, then it is a basis of the matroid. Consider the following
(2.3)TUr,n(x,y)=∑i=0r-1(ni)(x-1)r-i+(nr)+∑i=r+1n‍(ni)(y-1)i-r.
Thus, forU2,5 we get
(2.4)TU2,5(x,y)=(x-1)2+5(x-1)+10+10(y-1)+5(y-1)2+(y-1)3=x2+3x+3y+2y2+y3.
AsU3,5=(U2,5)*, we get by using (2.2)
(2.5)TU3,5(x,y)=x3+2x2+3x+3y+y2.Matroid Relaxation
Given a matroidM=(E,r) with a subset X⊆E that is both a circuit and a hyperplane, we can define a new matroid M′ as the matroid with basis ℬ(M′)=ℬ(M)∪{X}. That M' is indeed a matroid is easy to check. For example, F7- is the unique relaxation of F7. The Tutte polynomial of M′ can be computed easily from the Tutte polynomial of M by using (2.1) one has.
(2.6)TM′(x,y)=TM(x,y)-xy+x+y.
Matroid relaxation will be used extensively in the last section.Sparse Paving Matroids
Our second example can be considered a generalization of uniform matroids but it is a much larger and richer family. Apaving matroidM=(E,r) is a matroid whose circuits all have size at least r. Uniform matroids are an example of paving matroids. Paving matroids are closed under minors and the set of excluded minors for the class consists of the matroid U2,2⊕U0,1, see, for example, [10]. The interest about paving matroids goes back to 1976 when Dominic Welsh asked if most matroids were paving, see [9].
Sparse paving matroids were introduced in [11, 12]. A rank-r matroid M is sparse paving if M is paving and for every pair of circuits C1 and C2 of size r we have |C1ΔC2|>2. For example, all uniform matroids are sparse paving matroids.
There is a simple characterization of paving matroids which are sparse in terms of the sizes of their hyperplanes. For a proof, see [10].Theorem 2.2.
LetM be a paving matroid of rank r≥1. Then M is sparse paving if and only if all the hyperplanes of M have size r or r-1.Note that we can say a little more, any circuit of sizer is a hyperplane. Conversely, any proper subset of a hyperplane of size r is independent and so such a hyperplane must be a circuit. Thus, the circuits of size r are precisely the hyperplanes of size r.Many invariants that are usually difficult to compute for a general matroid are easy for sparse paving matroids. For example, observe that ifM is sparse paving, all subsets of size k<r are independent, and all subsets of size k>r are spanning. On the other hand, the subsets of size r are either bases or circuit hyperplanes. Thus, the Tutte polynomial of a rank-r sparse matroid M with n elements and λ circuit-hyperplanes is given by
(2.7)TM(x,y)=∑i=0r-1‍(ni)(x-1)r-i+(nr)+λ(xy-x-y)+∑i=r+1n‍(ni)(y-1)i-r.Clearly, given ar-rank paving matroid M with n elements, we can obtain the uniform matroid Ur,n by a sequence of relaxations from M. If M has λ circuits-hyperplanes, (2.6) also gives (2.7).Free Extension
Another easy formula that we can obtaine from the above definition involves the free extension,M+e, of a matroid M=(E,r) by an element e∉E, which consists of adding the element e to M as independently as possible without increasing the rank. Equivalently, the rank function of M+e is given by the following equations: for X a subset of E,
(2.8)rM+e(X)=rM(X),rM+e(X∪e)={rM(X)+1,ifrM(X)<r(M),r(M),otherwise.
Again, the Tutte polynomial of M+e can be computed easily from the Tutte polynomial of M by using (2.1). Consider
(2.9)TM+e(x,y)=xx-1TM(x,y)+(y-xx-1)TM(1,y).
Here the trick is to notice that TM(1,y)=∑X‍(y-1)|X|-r(X), where the sum is over all subsets of E with r(X)=r(E), that is, spanning subsets of M. The presentation given here is from [13], but the formula can also be found in [14].
For example, the graphic matroidM(K4∖e) has as free extention the matroid Q6. Because M(K4∖e) is sparse paving, by using (2.7), we can compute its Tutte polynomial and obtain
(2.10)TK4∖e(x,y)=x3+2x2+x+2xy+y+y2.
Now, by using (2.9), we get the Tutte polynomial of the free extension:
(2.11)TQ6=x3+3x2+4x+2xy+4y+3y2+y3.
This result can be checked by using (2.7), as Q6 is also sparse paving.


## 2.2. Deletion and Contraction

In the second (equivalent) definition of the Tutte polynomial we use a linear recursion relation given by deleting and contracting elements that are neither loops nor coloops. This is by far the most used method to compute the Tutte polynomial.Definition 2.3.
IfM is a matroid, and e is an element that is neither a loop nor a coloop, then
(2.12)TM(x,y)=TM∖e(x,y)+TM/e(x,y).
If e is a coloop, then
(2.13)TM(x,y)=xTM∖e(x,y).
If e is a loop, then
(2.14)TM(x,y)=yTM/e(x,y).The proof that Definitions2.1 and 2.3 are equivalent can be found in [4]. From this it is clear that you just need the Tutte polynomial of the matroid without loops and coloops. Also, if you know the Tutte polynomial of M∖e and M/e, then you know the Tutte polynomial of the matroid M. This way of computing the Tutte polynomial leads naturally to linear recursions. We present two examples where these linear recursions give formulas.The CycleCn
As a first example of this subsection we consider the graphic matroidM(Cn). Here, by deleting an edge e from Cn we obtained a path whose Tutte polynomial is xn-1, while if we contract e we get a smaller cycle Cn-1. Thus, as the Tutte polynomial of the 2-cycle is x+y, we obtain
(2.15)TCn(x,y)=∑i=1n-1‍xi+y.
By duality, the Tutte polynomial of the graphCn* with two vertices and n parallel edges between them is
(2.16)TCn*(x,y)=∑i=1n-1‍yi+x.Parallel and Series Classes
As an almost trivial application of deletion and contraction, we look into the common case when you have a graph or a matroid with parallel elements. Let us define aparallel class in a matroid M as maximal subset X of E(M) such that any two distinct members of X are parallel and no member of X is a loop, see [9]. Then the following result is well known and has been found many times, see [15, 16].Lemma 2.4.
LetX be a parallel class of a matroid M with |X|=p+1. If X is not a cocircuit, then
(2.17)TM(x,y)=TM∖X(x,y)+(yp+yp-1+⋯+1)TM/X(x,y).The proof is by induction onp with the case p=0 being (2.12). The rest of the proof follows easily from the fact that each loop introduces a multiplicative factor of y in the Tutte polynomial.Aseries class in M is just a parallel class in M* and by duality we get the following.Lemma 2.5.
LetX be a series class of a matroid M with |X|=p+1. If X is not a circuit, then
(2.18)TM(x,y)=(xp+xp-1+⋯+1)TM∖X(x,y)+TM/X(x,y).The Rectangular LatticeLm,n
The rectangular lattice, that we will define in a moment, is our first example where computing its Tutte polynomial is really far from trivial and no complete answer is known. The interest resides probably in the importance of computing the Potts partition function, which is equivalent to the Tutte polynomial, of the square lattice. However, even a complete resolution of this problem will be just a small step towards the resolution of the really important problem of computing the Tutte polynomial of the cubic lattice.
Letm and n be integers, m,n>1. The grid or rectangular latticeLm,n is a connected planar graph with mn vertices, such that its faces are squares, except one face that is a polygon with 2(m+n)-4 edges. The grid graph Lm,n can be represented as in Figure 1.Figure 1
The grid graphLm,n.The vertices ofLm,n are denoted as ordered pairs, such that the vertex in the intersection of the row i and the column j is denoted by ij.As an example, we show a recursive formula for the Tutte polynomials of grid graphsLm,n when m=2. For m>3, the formulas are very complicated to use in practical calculations.The graphsL2,1, L2,2, and L2,3 are shown in Figure 2, with the labels that correspond to their vertices.Figure 2
The first three terms are from the sequence{L2,n} and the last three are from {Q2,n}.For every positive integern. The graph Q2,n is defined from the graph L2,n by contraction of the edge {11,21} of L2,n. In particular, the graph Q2,1 is one isolated vertex, see Figure 2.The initial conditions to construct the recurrence relations for the Tutte polynomial of the grid graphsL2,n are.(i)
TQ2,1(x,y)=1;(ii)
TL2,1(x,y)=x;(iii)
TL2,2(x,y)=x3+x2+x+y.The third condition is true because the graphL2,2 is isomorphic to the cycle C4, so we can use (2.15). The formula for the recurrence relation to calculate the Tutte polynomial of the graphs L2,n is
(2.19)TL2,n(x,y)=(x2+x+1)TL2,n-1(x,y)+yTQ2,n-1(x,y),TQ2,n(x,y)=(x+1)TL2,n-1(x,y)+yTQ2,n-1(x,y).From this, we get a linear recurrence of order two,(2.20)TL2,n-(x2+x+1+y)TL2,n-1+x2yTL2,n-2=0,
that is easy to solve. Thus, the general formula is
(2.21)TL2,n(x,y)=a1λ1+a2λ1-λ2λ1n-2+a1λ2+a2λ2-λ1λ2n-2,
for n≥2, where a1=y+x+x2+x3, a2=-yx3, and λ1 and λ2 are
(2.22)12[(1+y+x+x2)±(y2+2y(1+x-x2)+(1+x+x2)2)1/2].Arecursive family of graphsGm is a sequence of graphs with the property that the Tutte polynomials TGm(x,y) satisfy a linear homogeneous recursion relation in which the coefficients are polynomials in x and y with integral coefficients, independent of m, see [17]. So, the sequence {L2,n} is such a family and in general, {Lk,n}, for a fixed k, is a recursive family.Graphic and Representable Matroids
Probably the most common class of matroids, when evaluations of the Tutte polynomial are considered, is graphic matroids. IfM is the graphic matroid of a graph G, then deleting an element e from M amounts to deleting the corresponding edge from G, and contracting e to contracting the edge. This is easy to do in a small graph and by using Definition 2.3 you can compute the Tutte polynomial. An example is shown in Figure 3.Figure 3
An example of computing the Tutte polynomial of a graphic matroid using deletion and contraction.As we mentioned, computing the Tutte polynomial is not in general computationally tractable. However, for graphic matroids, there are some resources to compute it for reasonably sized graphs of about 100 edges. These include Sekine et al. [18], which provide an algorithm to implement the recursive Definition 2.3. Common computer algebra systems such as Maple and Mathematica will compute the Tutte polynomial for very small graphs, and there are also some implementations freely available on the Web, such as http://homepages.mcs.vuw.ac.nz/~djp/tutte/  by Haggard and Pearce.Similarly, given a matroidM=(E,r), representable over a field 𝔽, you can compute its Tutte polynomial using deletion and contraction. For any element e∈E, the matroids M∖e and M/e are easily computed from the representation of M, see [9]. This can be automatized and there are already computer programs that compute the Tutte polynomial of a representable matroid. For our calculations we have used the one given by Michael Barany, for more information about this program and how to use it, see [19]. Some of the calculations made in the Section 4 were computed using this program.


## 2.3. Internal and External Activity

Being a polynomial, it is natural to ask for the coefficients of the Tutte polynomial. Tutte’s original definition gives its homonymous polynomial in terms of its coefficients by means of a combinatorial interpretation, but before we give the third definition of the Tutte polynomial, we introduce the relevant notions.Let us fix an ordering≺ on the elements of M, say E={e1,…,em}, where ei≺ej if i<j. Given a fixed basis S, an element e is called internally active if e∈S and it is the smallest edge with respect to ≺ in the only cocircuit disjoint from S∖{e}. Dually, an element f is externally active if f∉S and it is the smallest element in the only circuit contained in S∪{f}. We define tij to be the number of bases with i internally active elements and j externally active elements. In [20–22] Tutte defined TM using these concepts. A proof of the equivalence with Definition 2.1 can be found in [23].Definition 2.6.
IfM=(E,r) is a matroid with a total order on its ground set, then
(2.23)TM(x,y)=∑i,j‍tijxiyj.
In particular, the coefficients tij are independent of the total order used on the ground set.Uniform Matroids Again
Our first use of (2.23) is to get a slightly different expression for the Tutte polynomial of uniform matroids. This time we get
(2.24)TUr,n(x,y)=∑j=1n-r(n-j-1r-1)yj+∑i=1r(n-i-1n-r-1)xi,
when 0<r<n, while TUn,n(x,y)=xn and TU0,n(x,y)=yn. This can also be established by expanding (2.3).Paving Matroids
Paving matroids were defined above. The importance of paving matroids is its abundance and meaning, most matroids of upto 9 elements are paving. This was checked in [12] and it has been conjecture in [24] that this is indeed true for all matroids, that is, when n is large, the probability that you choose a paving matroid uniformly at random among all matroids with up to n elements is approaching 1.
Now, in order to get a formula for the Tutte polynomial of paving matroids, we need the following definition from [9]. Given integers k>1 and m>0, a collection 𝒯={T1,…,Tk} of subsets of a set E, such that each member of 𝒯 has at least m elements and each m-element subset of E is contained in a unique member of 𝒯 and is called an m-partition of E. The elements of the partition are called blocks. The following proposition is also from [9].Proposition 2.7.
If𝒯 is an m-partition of E, then 𝒯 is the set of hyperplanes of a paving matroid of rank m+1 on E. Moreover, for r≥2, the set of hyperplanes of every rank-r paving matroid on E is an (r-1)-partition of E.Brylawski gives the following proposition in [25], but we mentioned that his proof does not use activities.Proposition 2.8.
LetM be a rank-r matroid with n elements and Tutte polynomial ∑i,j‍tijxiyj. Then, M is paving if and only if tij=0 for all (i,j)≥(2,1). In addition if the (r-1)-partition of E has bk blocks of cardinality k, for k=r-1,…n, then
(2.25)ti0=(n-i-1r-i)∀i≥2,t10=∑k=0∞(r-2+kr-2)bk+r-1+(n-2r-1)(nr-1),
and for all j>0,
(2.26)t1j=∑k=0∞(r-2+kr-2)bk+j+r-1,t0j=(n-j-1r-1)-∑k=0∞(r-1+kr-1)bk+j+r-1.Note that whenM is the uniform matroid Ur,n, M is a paving matroid with (r-1)-partition all subsets of size r-1. Thus the above formula gives us (2.24).As an example, consider the matroidR9 with geometric representation given in Figure 4. This matroid is paving but not sparse paving. The blocks correspond to lines in the representation. There are 7 blocks of size 3, 2 blocks of size 4, and 3 blocks of size 2, corresponding to trivial lines that do not appear in Figure 4. Using the above formulas we get that the Tutte polynomial of R9 is
(2.27)TR9(x,y)=x3+6x2+8x+11xy+2xy2+8y+13y2+10y3+6y4+3y5+y6.Figure 4
The matroidR9.Catalan Matroids
Our final example in this subsection is a family of matroids that although simple, it has a surprisingly natural combinatorial definition together with a nice interpretation of the internal and external activity.
Consider an alphabet constituted by the letters{N,E}. The length of a word w=w1w2⋯wn is n, the number of letters in w. Every word can be associated with a path on the plane ℤ2, with an initial point A=(m1,n1) and a final point B=(m2,n2), m1<m2 and n1<n2. The letter N is identified with a north step and the letter E with an east step, such that the first step of A=(0,0) to B will be N=(0,1) or E=(1,0).
LetA and B be a couple of fixed points of ℤ2. A lattice path is a path in ℤ2 from A to B using only steps N or E. The lattice paths from A to B have m steps E and n steps N. Let P and Q be two lattice paths, with initial points (x0,yP) and (x0,yQ) of P and Q, respectively. If yP<yQ for every x0 in [m1,m2]; then the set of lattice paths in this work is lattice paths from A to B bounded between P and Q, that we called PQ-bounded lattice paths.
The lattice pathsP and Q can be substituted by lines. If P is the line y=0, Q the line y=x and m=n, then the number of PQ-bounded lattice paths from (0,0) to (n,n) is the nth Catalan number:
(2.28)cn=1n+1(2nn).
Denote by[n] the set {1,2,…,n}. Consider the word w=w1w2⋯wm+n that corresponds to a PQ-bounded lattice path. We can associate to w a subset Xw of [m+n] by defining that i∈Xw if wi=N. Xw is the support set of the steps N in w. The family of support sets Xw of PQ-bounded lattice paths is the set of bases of a transversal matroid, denoted M[P,Q] with ground set [m+n], see [13].
If the bounds of the lattice paths are the linesy=0 and y=x, the matroid is denoted by Mn and is named Catalan matroid. Note that Mn has a loop that corresponds to the label 1 and a coloop that corresponds to the label 2n, and that Mn is autodual.
The fundamental result to compute the Tutte polynomial of Catalan matroids is the following, see [13], where here, for a basis B, we denote by i(B) the number of internally active elements and by e(B) the externally active elements.Proposition 2.9.
LetB∈ℬ be a basis of M[P,Q] and let wB be the PQ-bounded lattice path associated with B. Then i(B) is the number of times wB meets the upper path Q in a north step and e(B) is the number of times wB meets the lower path P in an east step.Then the Tutte polynomial of the Catalan matroidsMn for n>1, is
(2.29)TMn(x,y)=∑i,j>0‍i+j-2n-1(2n-i-j-1n-i-j+1)xiyj.Note that the coefficient ofxiyj in the Tutte polynomial of the matroid Mn depends only on n and the sum i+j.In the Figure5 are shown the bases of M3 with its internal and external activities. Thus the Tutte polynomial of M3 is
(2.30)TM3(x,y)=x3y+x2y+x2y2+xy2+xy3.Figure 5
Bases ofM3 with their internal and external activity.


## 3. Techniques

We move now to present three techniques that are sometimes useful to compute the Tutte polynomial, when the initial trial with the above definitions is not successful.


### 3.1. Equivalent Polynomials: The Coboundary Polynomial

For some families of matroids and graphs it is easier to compute polynomials that are equivalent to the Tutte polynomial. One such polynomial is Crapo’s coboundary polynomial, see also [26]. Consider
(3.1)χ-M(λ,t)=∑X∈ℱM‍t|X|χM/X(λ),
where ℱM is the set of flats of M and χN(λ) is the characteristic polynomial of the matroid N. The characteristic polynomial of a matroid M is defined by
(3.2)χM(λ)=∑A⊆E‍(-1)|X|λz(A).The Tutte and the coboundary polynomial are related as follows:(3.3)TM(x,y)=1(y-1)r(E)χ-M((x-1)(y-1),y),(3.4)χ-M(λ,t)=(t-1)r(E)TM(λ+t-1t-1,t).WhenM is a graphic matroid M(G) of a connected graph G, the characteristic polynomial is equivalent to the chromatic polynomial, that we denote χG:
(3.5)χG(λ)=λχM(G)(λ).And also, in this case, the coboundary is the bad-colouring polynomial. Thebad-colouring polynomial is the generating function
(3.6)BG(λ,t)=∑j‍bj(G;λ)tj,
where bj(G;λ) is the number of λ-colourings of G with exactly j bad edges. Note that when we set t=0 we obtain the chromatic polynomial. Then, the polynomials are related as follows:
(3.7)BG(λ,t)=λχ-M(G)(λ,t).Theq-Cone
We base this section on the work of Bonin and Qin, see [27].Definition 3.1.
LetM be a rank-r simple matroid representable over GF(q). A matroid N is a q-cone of M with base S and apex a if(1)
the restrictionPG(r,q)∣S of PG(r,q) to the subset S is isomorphic to M,(2)
the pointa∈PG(r,q) is not contained in the closure of S, CLPG(r,q)(S), in PG(r,q), and(3)
the matroidN is the restriction of PG(r,q) to the set ⋃s∈S‍ClPG(r,q)({a,s}).That is, one representsM as a set S of points in PG(r,q) and constructs N by restricting PG(r,q) to the set of points on the lines joining the points of S to the fixed point a outside the hyperplane of PG(r,q) spanned by S. The basic result is by Kung who proved it in [28].Theorem 3.2.
For everyq-cone N of a rank-r simple matroid M, one has
(3.8)χN(λ)=(λ-1)qrχM(λq).To extend this result toχ-, the authors in [27] identify the flats F of N and the contractions N/F of N by these flats. They prove the following formula
(3.9)χ-N(λ,t)=tχ-M(λ,tq)+qr(λ-1)χ-M(λq,t).
By using (3.3) we get a formula for the Tutte polynomial of N in terms of the Tutte polynomial of M.Theorem 3.3.
IfM is a rank-r matroid representable over GF(q) and N is a q-cone of M, then
(3.10)TN(x,y)=y(yq-1)r(y-1)r+1TM((x-1)(y-1)yq-1+1,yq)+qr(xy-x-y)y-1TM(x+1q+1,y).Thus, for example, the2-cone of the 3-point line is the matroid PG(2,2). The Tutte polynomial of the former matroid is just x2+x+y, then by using the above formula we get the Tutte polynomial of PG(2,2) to be
(3.11)x3+4x2+3x+7xy+3y+6y2+3y3+y4.
As the q-cone of PG(r-1,q) is PG(r,q), this method can be used to compute the Tutte polynomial of any PG(r,q), but we do this later using the coboundary polynomial directly.Complete Graphs
Given the apparent simplicity of many of the formulas for invariants in complete graphs, like the number of spanning trees or acyclic orientations, it is not so straightforward to compute the whole Tutte polynomial of complete graphs; many researches, however, have tried with different amounts of success. The amount of frustration after failing to compute this seemingly innocuous invariant of many of our colleagues was our original motivation for writing this paper.
Using the exponential formula, see Stanley [29], we can give an exponential generating function for the Tutte polynomial of the complete graphs. Let us denote the vertices of Kn by V and by Bn its bad colouring. To compute Bn, observe that any λ-colouring partitions the vertices V into λ color classes of subsets Vi of vertices each of cardinality ni, for i=1,…,λ. So, we have that n1+⋯+nλ=n. The number of bad edges with both ends in the set Vi is t(ni2). Thus, by the exponential formula we get the following formula
(3.12)(∑n∈ℕ‍t(n2)unn!)λ=1+∑n≥1‍Bn(λ,t)unn!.
Now, we can use (3.7) to get
(3.13)(∑n∈ℕ‍t(n2)unn!)λ=1+λ∑n≥1‍χ-n(λ,t)unn!.
LetTn(x,y) be the Tutte polynomial of Kn. Tutte in [22] and Welsh in [30] give the following exponential generating function for Tn(x,y) that follows from the previous equation and (3.3)
(3.14)(∑n≥0‍y(n2)unn!)(x-1)(y-1)=1+(x-1)∑n≥1‍(y-1)nTn(x,y)unn!.Even that the previous formulas seem difficult to handle, (3.13) is quite easy to use in Maple or Mathematica. For example, the following program in Maple can compute T30(x,y) in no time:Coboundary:= proc(n,q,v) local i,x,g;g:=x->(add(v^(i*(i-1)/2)*x^i/i!,i=0..n))^q;simplify(eval(diff(g(x),x∖$n),x=0)/q); end proc;T:= proc(n,x,y)simplify((1/((y-1)^(n−1)))*subs({q=(x−1)*(y−1),v=y},Coboundary(n,q,v))) end proc;By computingT5(x,y) we obtain
(3.15)T5(x,y)=y6+4y5+x4+5xy3+10y4+6x3+10x2y+15xy2+15y3+11x2+20xy+15y2+6x+6y.Complete Bipartite Graphs
It is just natural to use this technique for complete bipartite graphsKn,m with similar results as above. This time let the vertex set be V1∪V2 and denote by Bn,m its bad-colouring polynomial. To compute Bn,m, observe that any λ-colouring partitioned the vertices in subsets V1(i)∪V2(i) each of cardinality nimi. The number of the bad edges with both ends colour i is nimi. Thus, by the exponential formula we get the following:
(3.16)(∑(n,m)∈ℕ2‍tnmunn!umm!)λ=1+∑(n,m)∈ℕ2(n,m)≠(0,0)Bn,m(λ,t)unn!vmm!=1+λ∑(n,m)∈ℕ2(n,m)≠(0,0)χ-n,m(λ,t)unn!vmm!.
Thus, a formula for the Tutte polynomial of the bipartite complete graph can be found in a similar way as before. LetTn,m(x,y) be the Tutte polynomial of Kn,m. The following formula is from Stanley’s book [29], see also [31]
(3.17)(∑(n,m)∈ℕ2‍ynmunn!vmm!)(x-1)(y-1)=1+(x-1)∑(n,m)∈ℕ2(n,m)≠(0,0)(y-1)nmTn,m(x,y)unn!vmm!.
As before, (3.16) is quite easy to use in Maple to compute Tn,m(x,y) for small values of n and m. In this way we get the following:
(3.18)T3,3(x,y)=x5+4x4+10x3+9x2y+11x2+6xy2+15xy+5x+y4+5y3+9y2+5y.Projective Geometries and Affine Geometries
The role played by complete graphs in graphic matroids is taken by projective geometries in representable matroids. Herein lies the importance of projective geometries. Even though a formula for their Tutte polynomial has been known for a while and has been discovered at least twice, not much work has been done in the actual combinatorial interpretations for the value of the different evaluations of the Tutte polynomial.
For this part we follow Mphako [32], see also [33]. For all nonnegative integers m and k we define the Gaussian coefficients as
(3.19)[mk]q=(qm-1)(qm-q)⋯(qm-qk-1)(qk-1)(qk-q)⋯(qk-qk-1).
Note that [m0]q=1 since by convention an empty product is 1 and [0k]q=0.LetPG(r-1,q) be the (r-1) dimensional projective geometry over GF(q). Then, as a matroid, it has rank r and [r1]q elements. Also, every rank-k flat X is isomorphic to PG(k-1,q) and the simplification of M/X is isomorphic to PG(r-k-1,q). The number of rank-k flats is [rk]q. The characteristic polynomial of PG(r-1,q) is known to be, see [4],
(3.20)χPG(r-1,q)(λ)=∏i=0r-1‍(λ-qi).
Thus, using (3.1) we get
(3.21)χ-PG(r-1,q)(λ,t)=∑k=0r‍t[k1]q[rk]q∏i=0r-k-1‍(λ-qi).
The matroid PG(2,3) has a geometric representation given in Figure 6, and it is isomorphic to the unique Steiner system S(2,4,13). The matroid is paving but not sparse paving and has the following representation over GF(3):
(3.22)124|35678910111213[100|1221100112010|1110111011001|0121212210].
To compute its Tutte polynomial we could use the program in [19] or use the above formula to get
(3.23)χ-PG(2,3)(λ,t)=(λ-1)(λ-3)(λ-9)+13t(λ-1)(λ-3)+13t4(λ-1)+t13.Figure 6
The projective planePG(2,3) with a marked hyperplane.Finally the Tutte polynomial ofPG(2,3) is obtained from the coboundary by a substitution λ=(x-1)(y-1) and t=y and by multiplying by the factor 1/(y-1)3:
(3.24)TPG(2,3)(x,y)=x3+10x2+13xy2+26xy+16x+16y+32y2+36y3+28y4+21y5+15y6+10y7+6y8+3y9+y10.Every time we have a projective geometryPG(r-1,q) we can get an affine geometry, AG(r-1,q), simply by deleting all the points in a hyperplane of PG(r-1,q). For example, by deleting all the points in the red line from PG(2,3) in Figure 6 we get AG(2,3) with geometric representation in Figure 7.Figure 7
Affine planeAG(2,3).The situation to compute the Tutte polynomial is very similar for ther-dimensional affine geometry over GF(q), AG(r,q). It has rank r+1 and qr elements. Any flat of X rank k+1 is isomorphic to AG(k,q) and the simplification of AG(r,q)/X is isomorphic to PG(r-k-1,q). The number of rank-k flats is qr-k[rk]q. The characteristic polynomial of AG(r,q) is known to be, see [4],
(3.25)χAG(r,q)(λ)=(λ-1)∑k=0r‍(-1)kλr-k∏i=0k-1‍(qr-i).
Now, using the result of Mphako we can compute the coboundary polynomial of AG(r,q):
(3.26)χ-AG(r,q)(λ,t)=χAG(r,q)(λ)+∑k=0r‍tqkqr-k[rk]q∏i=0r-k-1‍(λ-qi).The affine planeAG(2,3) is isomorphic to the unique Steiner triple system S(2,3,9), as every line contains exactly 3 points and every pair of points is in exactly one line. S(2,3,9) is a sparse paving matroid with 12 circuit hyperplanes so by using (2.7) we could compute its Tutte polynomial. However, we use the above formula to obtain the same result consider the following.
(3.27)χ-AG(2,3)(λ,t)=(λ-1)(λ2-8λ+16)+9t(λ-1)(λ-3)+12t3(λ-1)+t9.Again, the Tutte polynomial ofAG(2,3) is obtained from the coboundary by a substitution λ=(x-1)(y-1) and t=y and by multiplying by the factor 1/(y-1)3(3.28)TAG(2,3)(x,y)=x3+6x2+12xy+9x+9y+15y2+10y3+6y4+3y5+y6.


### 3.2. Transfer-Matrix Method

Using formula (2.1) quickly becomes prohibitive as the number of states grows exponentially with the size of the matroid. You can get around this problem when you have a family of graphs that are constructed using a simple graph that you repeat in a path-like fashion; the bookkeeping of the contribution of each state can be done with a matrix, the update can be done by matrix multiplication after the graph grows a little more. This is the essence of our second method.The theoretical background of the transfer-matrix method, taken from [34], is described below.Adirected graph or digraphG→ is a triple (V,E,ϕ), where V={v1,…,vp} is a set of vertices, E is a finite set of directed edges or arcs, and ϕ is a map from E to V×V. If ϕ(e)=(u,v), then e is called an edge from u to v, with initial vertex u and final vertex v. A directed walkΓ in G→ of lengthn from u to v is a sequence e1,…,en of n edges such that the final vertex of ei is the initial vertex of ei+1, for 1≤i≤n-1.Now letw:E→R be a weight function on E with values in some commutative ring R. If Γ=e1,…,en is a walk, then the weight of Γ is defined by w(Γ)=w(e1)⋯w(en). For 1≤i,j≤p and n∈ℕ, we define
(3.29)Ai,j(n)=∑Γ‍w(Γ),
where the sum is over all walks Γ in G→ of length n from vi to vj. In particular, Ai,j(0)=δij. The fundamental problem treated by the transfer-matrix method is the evaluation of Ai,j(n). The idea is to interpret Ai,j(n) as an entry in a certain matrix. Define a p×p matrix D=(Di,j) by
(3.30)Di,j=∑e‍w(e),
where the sum is over all edges e satisfying that its initial vertex is vi and its final vertex is vj. In other words, Di,j=Ai,j(1). The matrix D is called the adjacency matrix of G→, with respect to the weight function w.Theorem 3.4.
Letn∈ℕ. Then the (i,j)-entry of Dn is equal to Ai,j(n). Here one defines D0=Ip even if D is not invertible, where Ip is the identity matrix.Proof.
See [34].Rectangular Lattice Again
The transfer-matrix method gives us another way to computeTLm,n(x,y) for a fixed width m at point (x,y) which is described in Calkin et al. [35]. In this case we have the same restriction as before, a fixed width m for small values of m, but it has the advantage of being easily automatized.Theorem 3.5 (see, Calkin et. al. [35]).
For indeterminatesx and y and integers n,m≥2, m fixed, one has
(3.31)TLm,n(x+1,y+1)=xnm-1Xmt·(Λm)n-1·1→,
where Xm, a vector of length cm, and Λm, a cm×cm matrix, depend on x,y, and m but not n. And 1→ is the vector of length cm with all entries equal to 1.The quantitycm is the mth Catalan number, so the method is just practical for small values of m. Computing the vectors Xm and the matrix Λm can be easily done in a computer.For example, forL2,n we get that TL2,n(x+1,y+1) equals
(3.32)x2n-1(x-1,1)(x-1+3x-2+yx-21+2x-1x-1+2x-2+x-31+2x-1+x-2)n-1(11).Wheels and Whirls
The transfer-matrix method takes a nice turn when combined with the physics idea of boundary conditions. In this case more lineal algebra is required but the method is still suitable to use in a computer algebra package.A well-known family of self-dual graphs are wheel graphs,Wn. The graph Wn has n+1 vertices and 2n edges, see Figure 8. The rim of the wheel graph Wn is a circuit hyperplane of the corresponding graphic matroid, the relaxation of this circuit-hyperplane gives the matroid Wn, the whirl matroid on n elements. In [36], Chang and Shrock using results from [37] compute the Tutte polynomial of Wn as follows:
(3.33)TWn(x,y)=12n[(1+x+y)+((1+x+y)2-4xy)1/2]n+12n[(1+x+y)-((1+x+y)2-4xy)1/2]n+xy-x-y-1.Figure 8
Then-wheel.From this expression, it is easy to compute an expression for the Tutte polynomial of whirls using (2.6). Consider
(3.34)TWn(x,y)=12n[(1+x+y)+((1+x+y)2-4xy)1/2]n+12n[(1+x+y)+((1+x+y)2-4xy)1/2]n-1.Now, the way Chang and Shrock computeTWn(x,y) is by using the Potts model partition function together with the transfer matrix method. Remarkably, the Tutte polynomial and the Potts model partition function are equivalent. But rather than defining the Potts model we invite the reader to check the surveys in [38, 39]. Here we show how to do the computation using the coboundary polynomial and the transfer method.Let us compute the bad colouring polynomial forWn when we have 3 colours. For this we define the 3×3 matrix D3(3.35)(t211tt1t1t).
The idea is that the entry ij of the matrix (D3)n=D3n will contain all the contributions of bad edges to the bad colouring polynomial for all the colourings σ, with σ(h)=1, σ(1)=i, and σ(n)=j for the fan graph Fn, obtained from Wn by deleting one edge from the rim. Then, to get the bad-colouring polynomial of Fn we just add all the entries in D3n and multiply by λ=3. To get the bad-colouring polynomial of Wn we take the trace of D3n, and multiply by λ=3. This works because, by taking the trace you are considering just colourings with the same initial and final configuration, that is you are placing periodic boundary conditions.Thus, for example, the trace inD33 is t6+8t3+6t2+12t, so BW3(3,t)=3t6+24t3+18t2+36t. By computing the eigenvalues of D3, we obtain the bad colouring polynomial of Wn with λ=3(3.36)BWn(3,t)=32n[(t2+t+1)+(t4-2t3-t2+10t+1)1/2]n+32n[(t2+t+1)+(t4-2t3-t2+10t+1)1/2]n+3(t-1)n.For arbitraryλ we need to compute the eigenvalues of the matrix of λ×λ given below
(3.37)(t211⋯1tt1⋯1t1t⋯1⋮⋮⋮⋱⋮t11⋯t).The eigenvalues of the matrix are(3.38)e1,2=12[(t2+t+λ-2)±(t4-2t3-(2λ-5)t2+(6λ-8)t+(λ-2)2)1/2],
each with multiplicity 1 and e3=(t-1) with multiplicity λ-2. Thus the bad colouring polynomial of Wn is λ(e1n+e2n+(λ-2)e3n). To get formula (3.33) we just need to make the corresponding change of variables as seen at the beginning of the previous section.Other examples that were computed by using this technique include Möbius strips, cycle strips, and homogeneous clan graphs. For the corresponding formulas you can check [36, 37, 40–42].


### 3.3. Splitting the Problem: 1-Sum, 2-Sum and 3-Sum

Our last technique is based on the recurrent idea of splitting a problem that otherwise may be too big. The notion of connectedness in matroid theory offers a natural setting for our purposes, but we do not explain this theory here and we direct the reader to the book of Oxley [9] for such a subject. Also, for 1-sum, 2-sum and 3-sum of binary matroids the book of Truemper [43] is the best reference.However, we would like to explain a little of the relation of 1-sum, 2-sum and 3-sum and the splitting of a matroid. For a matroidM=(E,r), a partition (X,Y) of E is an exactk-separation, for k a positive integer, if
(3.39)min{|X|,|Y|}≥k,r(X)+r(Y)-r(M)=k-1.
Now, a matroid M can be written as a 1-sum of two of its proper minors if and only if M has an exact 1-separation, and M can be written as a 2-sum of two of its proper minors if and only if M has an exact 2-separation. The situation is more complicated in the case of 3-sum and we just want to point out that if a binary matroid M has an exact 3-separation (X,Y), with |X|, |Y|≥4, then there are binary matroids M1 and M2 such that M=M1⊕3M2.1-Sum
Forn matroids M1,M2,…,Mn, on disjoint sets E1,E2,…,En the direct sum M1⊕M2⊕⋯⊕Mn is the matroid (E,ℐ), where E is the union of the ground sets and ℐ={I1∪I2∪⋯∪In:Ii∈ℐ(Mi) for all i in {1,2,…,n}}.
Directly from (2.1) it follows that the Tutte polynomial of the 1-sum is given by
(3.40)TM1⊕M2⊕⋯⊕Mn(x,y)=∏i=1n‍TMi(x,y).2-Sum
LetM1=(E1,ℐ1) and M2=(E2,ℐ2) be matroids with E1∩E2={p}. If p is not a loop or an isthmus in M1 or M2, then the 2-sum   M1⊕2M2 of M1 and M2 is the matroid on E1∪E2∖{p} whose collection of independent sets is {I1∪I2:I1∈ℐ1, I2∈ℐ2, and either I1∪{p}∈ℐ1 or I2∪{p}∈ℐ2}.
In [9, 44, 45], we find recursive formulas for computing the Tutte polynomial of the matroid M1⊕2M2, in terms of the matroids M1 and M2. Here we present the formula given in [44] for TM1⊕2M2(x,y)(3.41)TM1⊕2M2=1xy-x-y[TM1/pTM1∖p][x-1-1-1y-1][TM2/pTM2∖p],
where here we omit the variables x,y of each Tutte polynomial to avoid a cumbersome notation.
As an example, let us takeR6 that is the 2-sum of U2,4 with itself. Observe that the election of the base point is irrelevant as its automorphism group is the symmetric group. The matroid U2,4/r≅U1,3 and U2,4∖r≅U2,3. Then, by using (3.41), we get that TU2,4⊕2U2,4 equals
(3.42)1xy-x-y[y2+y+xx2+x+y][x-1-1-1y-1][y2+y+xx2+x+y].
Thus, we obtain that TR6(x,y)=x3+3x2+4x+2xy+4y+3y2+y3.3-Sum
The best known variant of a 3-sum of two matroids is calledΔ-sum, see [43]. For 3-connected matroids M1 and M2, the Δ-sum is usually denoted M=M1⊕3ΔM2. When M1 and M2 are graphic matroids with corresponding graphs being G1 and G2, the graph G=G1⊕3ΔG2 is obtained by identifying a triangle of T1 of G1 with a triangle T2 of G2 into a triangle T, called the connector triangle. Finally, G is obtained by deleting the edges in the connector triangle.
Here, we present a formula to compute the Tutte polynomial of theΔ-sum of M1 and M2, in terms of the Tutte polynomial of certain minors of the original matroids. The expression for TM1⊕3ΔM2(x,y) was taken from [44] and its proof is based on the concept of bipointed matroids that is an extension of the pointed matroid introduced by Brylawski in [45]. Also, the work in [44] is more general as the author gives an expression for the Tutte polynomial of a certain type of general parallel connection of two matroids.
LetM1 and M2 be two matroids defined on E1 and E2, respectively. Let T be equal to E1∩E2={p,s,q}, a 3-circuit and N=M1|T=M2|T. We require that in M1 there exist circuits U1∪{s} with U1⊆E1∖T and U2∪{p} with U2⊆E1∖T; similarly, we need that in M2 there exist circuits V1∪{s} with V1⊆E2∖T and V2∪{p} with V2⊆E2∖T.
Fori=1,2,…,5, we consider the following 5 minors Qi of M1, on E1∖T, and 5 minors Pi of M2, on E2∖T(3.43)Q1=M1∖p∖s∖q,Q2=M1∖p/s∖q,Q3=M1/p∖s∖q,Q4=M1/p/s/q,Q5=M1∖p∖s/q,P1=M2∖p∖s∖q,P2=M2∖p/s∖q,P3=M2/p∖s∖q,P4=M2/p/s/q,P5=M2∖p∖s/q.
We take the vectors overℤ[x,y]:
(3.44)q→=[TQ1,TQ2,TQ3,TQ4,TQ5],p→=[TP1,TP2,TP3,TP4,TP5]t,
again here we omit the variables x,y of each Tutte polynomial to avoid a cumbersome notation.
Finally, the formula for the 3-sum (Δ-sum) of M1 and M2 is as follows:
(3.45)TM1⊕3ΔM2(x,y)=q→[1-1-x-y+xyC]p→,
where the matrix, C, is given by
(3.46)[(1-y)2-x-y+xy1-y-x-y+xy1-y-x-y+xy2-x-y+xy1-y-x-y+xy1-y-x-y+xy11-x-y+xy1-x-x-y+xy1-x-y+xy1-y-x-y+xy1-x-y+xy11-x-x-y+xy1-x-y+xy2-x-y+xy1-x-x-y+xy1-x-x-y+xy(1-x)2-x-y+xy1-x-x-y+xy1-y-x-y+xy1-x-y+xy1-x-y+xy1-x-x-y+xy1].
As an example, let us takeF8 that is the 3-sum of F7 and F7- along a 3-circuit. In this case we have for F7 the values of Table 1, where in the the first column we present the 5 minors we need, then the second column has the corresponding matroid, and the third column has the corresponding Tutte polynomial of that matroid.Table 1
Minor
Matroid
Polynomial
F
7
∖
p
∖
s
∖
q
U
3,4
x
3
+
x
2
+
x
+
y
F
7
∖
p
/
s
∖
q
U
1,2
⊕
U
1,2
(
x
+
y
)
2
F
7
/
p
∖
s
∖
q
U
1,2
⊕
U
1,2
(
x
+
y
)
2
F
7
/
p
/
s
/
q
U
1,4
y
3
+
y
2
+
y
+
x
F
7
∖
p
∖
s
/
q
U
1,2
⊕
U
1,2
(
x
+
y
)
2Similarly forF7- we have Table 2.Table 2
Minor
Matroid
Polynomial
F
7
-
∖
p
∖
s
∖
q
U
3,4
x
3
+
x
2
+
x
+
y
F
7
-
∖
p
/
s
∖
q
C
3 plus a parallel edge
x
2
+
x
+
x
y
+
y
+
y
2
F
7
-
/
p
∖
s
∖
q
U
1,2
⊕
U
1,2
(
x
+
y
)
2
F
7
-
/
p
/
s
/
q
U
1,4
y
3
+
y
2
+
y
+
x
F
7
-
∖
p
∖
s
/
q
U
1,2
⊕
U
1,2
(
x
+
y
)
2By using (3.45) with the values of Tables 1 and 2 we get TF8(x,y)=x4+4x3+10x2+8x+12xy+8y+10y2+4y3+y4.There is a general concept ofk-sum for matroids and a formula exists for this general notion; however, the formula is quit intricate, so we refer the reader to the original paper of Bonin and Mier in [46].Thickening, Stretch, and Tensor Product
Given a matroidM and a positive integer k, the matroid M(k) is the matroid obtained from M by replacing each nonloop element by k parallel elements and replacing each loop by k loops. The matroid M(k) is called the k-thickening of M. In [4], the following formula is given for the Tutte polynomial of M(k) in terms of that of M(3.47)TM(k)(x,y)=(yk-1+yk-2+⋯+y+1)r(M)TM(yk-1+yk-2+⋯+y+xyk-1+yk-2+⋯+y+1,yk).
A proof by using the recipe theorem can be read in the aforementioned reference. Here we hint a simple proof by noticing that any flat ofM(k) is the k-thickening of a flat of M. Thus, by (3.1)
(3.48)χ-M(k)(q,t)=χ-M(q,tk).
And thus, by using (3.4) and (3.3) we obtained the formula.
The dual operation tok-thickening is that of k-stretch that is defined similarly. The matroid M(k) is the matroid obtained by replacing each nonisthmus of M by k elements in series and replacing each isthmus by k isthmuses. The matroid M(k) is called the k-stretch. It is not difficult to prove that M(k)≅((M*)(k))* and so, we obtained the corresponding formula for TM(k)(3.49)TM(k)(x,y)=(xk-1+xk-2+⋯+x+1)r(M*)TM(xk,xk-1+xk-2+⋯+x+yxk-1+xk-2+⋯+x+1).
More generally we have the following operation called thetensor product. A pointed matroid Nd is a matroid on a ground set which includes a distinguished element, the point d, which will be assumed to be neither a loop nor coloop. For an arbitrary matroid M and a pointed matroid Nd, the tensor productM⊗Nd is the matroid obtained by taking a 2-sum of M with Nd at each point e of M and the distinguished point d of Nd. The Tutte polynomial of M⊗Nd, where M=(E,r) is then given by
(3.50)TM⊗Nd(x,y)=f|E|-r(E)gr(E)TM((x-1)f+gg,f+(y-1)gf),
where f=f(x,y) and g=g(x,y) are polynomials which are determined by the equations
(3.51)(x-1)f(x,y)+g(x,y)=TNd∖d(x,y),f(x,y)+(y-1)g(x,y)=TNd/d(x,y).
The proof of the formula uses a generalization of the recipe theorem to pointed matroids and can be found in [14], here we follow the exposition in [4, 47]. Observe that when a matroid N has a transitive automorphism group, the choice of the distinguished point d is immaterial. Thus, if N is Uk,k+1, k≥1, we get the k-stretch and if N is U1,k+1, k≥1, we get the k-thickening.


## 3.1. Equivalent Polynomials: The Coboundary Polynomial

For some families of matroids and graphs it is easier to compute polynomials that are equivalent to the Tutte polynomial. One such polynomial is Crapo’s coboundary polynomial, see also [26]. Consider
(3.1)χ-M(λ,t)=∑X∈ℱM‍t|X|χM/X(λ),
where ℱM is the set of flats of M and χN(λ) is the characteristic polynomial of the matroid N. The characteristic polynomial of a matroid M is defined by
(3.2)χM(λ)=∑A⊆E‍(-1)|X|λz(A).The Tutte and the coboundary polynomial are related as follows:(3.3)TM(x,y)=1(y-1)r(E)χ-M((x-1)(y-1),y),(3.4)χ-M(λ,t)=(t-1)r(E)TM(λ+t-1t-1,t).WhenM is a graphic matroid M(G) of a connected graph G, the characteristic polynomial is equivalent to the chromatic polynomial, that we denote χG:
(3.5)χG(λ)=λχM(G)(λ).And also, in this case, the coboundary is the bad-colouring polynomial. Thebad-colouring polynomial is the generating function
(3.6)BG(λ,t)=∑j‍bj(G;λ)tj,
where bj(G;λ) is the number of λ-colourings of G with exactly j bad edges. Note that when we set t=0 we obtain the chromatic polynomial. Then, the polynomials are related as follows:
(3.7)BG(λ,t)=λχ-M(G)(λ,t).Theq-Cone
We base this section on the work of Bonin and Qin, see [27].Definition 3.1.
LetM be a rank-r simple matroid representable over GF(q). A matroid N is a q-cone of M with base S and apex a if(1)
the restrictionPG(r,q)∣S of PG(r,q) to the subset S is isomorphic to M,(2)
the pointa∈PG(r,q) is not contained in the closure of S, CLPG(r,q)(S), in PG(r,q), and(3)
the matroidN is the restriction of PG(r,q) to the set ⋃s∈S‍ClPG(r,q)({a,s}).That is, one representsM as a set S of points in PG(r,q) and constructs N by restricting PG(r,q) to the set of points on the lines joining the points of S to the fixed point a outside the hyperplane of PG(r,q) spanned by S. The basic result is by Kung who proved it in [28].Theorem 3.2.
For everyq-cone N of a rank-r simple matroid M, one has
(3.8)χN(λ)=(λ-1)qrχM(λq).To extend this result toχ-, the authors in [27] identify the flats F of N and the contractions N/F of N by these flats. They prove the following formula
(3.9)χ-N(λ,t)=tχ-M(λ,tq)+qr(λ-1)χ-M(λq,t).
By using (3.3) we get a formula for the Tutte polynomial of N in terms of the Tutte polynomial of M.Theorem 3.3.
IfM is a rank-r matroid representable over GF(q) and N is a q-cone of M, then
(3.10)TN(x,y)=y(yq-1)r(y-1)r+1TM((x-1)(y-1)yq-1+1,yq)+qr(xy-x-y)y-1TM(x+1q+1,y).Thus, for example, the2-cone of the 3-point line is the matroid PG(2,2). The Tutte polynomial of the former matroid is just x2+x+y, then by using the above formula we get the Tutte polynomial of PG(2,2) to be
(3.11)x3+4x2+3x+7xy+3y+6y2+3y3+y4.
As the q-cone of PG(r-1,q) is PG(r,q), this method can be used to compute the Tutte polynomial of any PG(r,q), but we do this later using the coboundary polynomial directly.Complete Graphs
Given the apparent simplicity of many of the formulas for invariants in complete graphs, like the number of spanning trees or acyclic orientations, it is not so straightforward to compute the whole Tutte polynomial of complete graphs; many researches, however, have tried with different amounts of success. The amount of frustration after failing to compute this seemingly innocuous invariant of many of our colleagues was our original motivation for writing this paper.
Using the exponential formula, see Stanley [29], we can give an exponential generating function for the Tutte polynomial of the complete graphs. Let us denote the vertices of Kn by V and by Bn its bad colouring. To compute Bn, observe that any λ-colouring partitions the vertices V into λ color classes of subsets Vi of vertices each of cardinality ni, for i=1,…,λ. So, we have that n1+⋯+nλ=n. The number of bad edges with both ends in the set Vi is t(ni2). Thus, by the exponential formula we get the following formula
(3.12)(∑n∈ℕ‍t(n2)unn!)λ=1+∑n≥1‍Bn(λ,t)unn!.
Now, we can use (3.7) to get
(3.13)(∑n∈ℕ‍t(n2)unn!)λ=1+λ∑n≥1‍χ-n(λ,t)unn!.
LetTn(x,y) be the Tutte polynomial of Kn. Tutte in [22] and Welsh in [30] give the following exponential generating function for Tn(x,y) that follows from the previous equation and (3.3)
(3.14)(∑n≥0‍y(n2)unn!)(x-1)(y-1)=1+(x-1)∑n≥1‍(y-1)nTn(x,y)unn!.Even that the previous formulas seem difficult to handle, (3.13) is quite easy to use in Maple or Mathematica. For example, the following program in Maple can compute T30(x,y) in no time:Coboundary:= proc(n,q,v) local i,x,g;g:=x->(add(v^(i*(i-1)/2)*x^i/i!,i=0..n))^q;simplify(eval(diff(g(x),x∖$n),x=0)/q); end proc;T:= proc(n,x,y)simplify((1/((y-1)^(n−1)))*subs({q=(x−1)*(y−1),v=y},Coboundary(n,q,v))) end proc;By computingT5(x,y) we obtain
(3.15)T5(x,y)=y6+4y5+x4+5xy3+10y4+6x3+10x2y+15xy2+15y3+11x2+20xy+15y2+6x+6y.Complete Bipartite Graphs
It is just natural to use this technique for complete bipartite graphsKn,m with similar results as above. This time let the vertex set be V1∪V2 and denote by Bn,m its bad-colouring polynomial. To compute Bn,m, observe that any λ-colouring partitioned the vertices in subsets V1(i)∪V2(i) each of cardinality nimi. The number of the bad edges with both ends colour i is nimi. Thus, by the exponential formula we get the following:
(3.16)(∑(n,m)∈ℕ2‍tnmunn!umm!)λ=1+∑(n,m)∈ℕ2(n,m)≠(0,0)Bn,m(λ,t)unn!vmm!=1+λ∑(n,m)∈ℕ2(n,m)≠(0,0)χ-n,m(λ,t)unn!vmm!.
Thus, a formula for the Tutte polynomial of the bipartite complete graph can be found in a similar way as before. LetTn,m(x,y) be the Tutte polynomial of Kn,m. The following formula is from Stanley’s book [29], see also [31]
(3.17)(∑(n,m)∈ℕ2‍ynmunn!vmm!)(x-1)(y-1)=1+(x-1)∑(n,m)∈ℕ2(n,m)≠(0,0)(y-1)nmTn,m(x,y)unn!vmm!.
As before, (3.16) is quite easy to use in Maple to compute Tn,m(x,y) for small values of n and m. In this way we get the following:
(3.18)T3,3(x,y)=x5+4x4+10x3+9x2y+11x2+6xy2+15xy+5x+y4+5y3+9y2+5y.Projective Geometries and Affine Geometries
The role played by complete graphs in graphic matroids is taken by projective geometries in representable matroids. Herein lies the importance of projective geometries. Even though a formula for their Tutte polynomial has been known for a while and has been discovered at least twice, not much work has been done in the actual combinatorial interpretations for the value of the different evaluations of the Tutte polynomial.
For this part we follow Mphako [32], see also [33]. For all nonnegative integers m and k we define the Gaussian coefficients as
(3.19)[mk]q=(qm-1)(qm-q)⋯(qm-qk-1)(qk-1)(qk-q)⋯(qk-qk-1).
Note that [m0]q=1 since by convention an empty product is 1 and [0k]q=0.LetPG(r-1,q) be the (r-1) dimensional projective geometry over GF(q). Then, as a matroid, it has rank r and [r1]q elements. Also, every rank-k flat X is isomorphic to PG(k-1,q) and the simplification of M/X is isomorphic to PG(r-k-1,q). The number of rank-k flats is [rk]q. The characteristic polynomial of PG(r-1,q) is known to be, see [4],
(3.20)χPG(r-1,q)(λ)=∏i=0r-1‍(λ-qi).
Thus, using (3.1) we get
(3.21)χ-PG(r-1,q)(λ,t)=∑k=0r‍t[k1]q[rk]q∏i=0r-k-1‍(λ-qi).
The matroid PG(2,3) has a geometric representation given in Figure 6, and it is isomorphic to the unique Steiner system S(2,4,13). The matroid is paving but not sparse paving and has the following representation over GF(3):
(3.22)124|35678910111213[100|1221100112010|1110111011001|0121212210].
To compute its Tutte polynomial we could use the program in [19] or use the above formula to get
(3.23)χ-PG(2,3)(λ,t)=(λ-1)(λ-3)(λ-9)+13t(λ-1)(λ-3)+13t4(λ-1)+t13.Figure 6
The projective planePG(2,3) with a marked hyperplane.Finally the Tutte polynomial ofPG(2,3) is obtained from the coboundary by a substitution λ=(x-1)(y-1) and t=y and by multiplying by the factor 1/(y-1)3:
(3.24)TPG(2,3)(x,y)=x3+10x2+13xy2+26xy+16x+16y+32y2+36y3+28y4+21y5+15y6+10y7+6y8+3y9+y10.Every time we have a projective geometryPG(r-1,q) we can get an affine geometry, AG(r-1,q), simply by deleting all the points in a hyperplane of PG(r-1,q). For example, by deleting all the points in the red line from PG(2,3) in Figure 6 we get AG(2,3) with geometric representation in Figure 7.Figure 7
Affine planeAG(2,3).The situation to compute the Tutte polynomial is very similar for ther-dimensional affine geometry over GF(q), AG(r,q). It has rank r+1 and qr elements. Any flat of X rank k+1 is isomorphic to AG(k,q) and the simplification of AG(r,q)/X is isomorphic to PG(r-k-1,q). The number of rank-k flats is qr-k[rk]q. The characteristic polynomial of AG(r,q) is known to be, see [4],
(3.25)χAG(r,q)(λ)=(λ-1)∑k=0r‍(-1)kλr-k∏i=0k-1‍(qr-i).
Now, using the result of Mphako we can compute the coboundary polynomial of AG(r,q):
(3.26)χ-AG(r,q)(λ,t)=χAG(r,q)(λ)+∑k=0r‍tqkqr-k[rk]q∏i=0r-k-1‍(λ-qi).The affine planeAG(2,3) is isomorphic to the unique Steiner triple system S(2,3,9), as every line contains exactly 3 points and every pair of points is in exactly one line. S(2,3,9) is a sparse paving matroid with 12 circuit hyperplanes so by using (2.7) we could compute its Tutte polynomial. However, we use the above formula to obtain the same result consider the following.
(3.27)χ-AG(2,3)(λ,t)=(λ-1)(λ2-8λ+16)+9t(λ-1)(λ-3)+12t3(λ-1)+t9.Again, the Tutte polynomial ofAG(2,3) is obtained from the coboundary by a substitution λ=(x-1)(y-1) and t=y and by multiplying by the factor 1/(y-1)3(3.28)TAG(2,3)(x,y)=x3+6x2+12xy+9x+9y+15y2+10y3+6y4+3y5+y6.


## 3.2. Transfer-Matrix Method

Using formula (2.1) quickly becomes prohibitive as the number of states grows exponentially with the size of the matroid. You can get around this problem when you have a family of graphs that are constructed using a simple graph that you repeat in a path-like fashion; the bookkeeping of the contribution of each state can be done with a matrix, the update can be done by matrix multiplication after the graph grows a little more. This is the essence of our second method.The theoretical background of the transfer-matrix method, taken from [34], is described below.Adirected graph or digraphG→ is a triple (V,E,ϕ), where V={v1,…,vp} is a set of vertices, E is a finite set of directed edges or arcs, and ϕ is a map from E to V×V. If ϕ(e)=(u,v), then e is called an edge from u to v, with initial vertex u and final vertex v. A directed walkΓ in G→ of lengthn from u to v is a sequence e1,…,en of n edges such that the final vertex of ei is the initial vertex of ei+1, for 1≤i≤n-1.Now letw:E→R be a weight function on E with values in some commutative ring R. If Γ=e1,…,en is a walk, then the weight of Γ is defined by w(Γ)=w(e1)⋯w(en). For 1≤i,j≤p and n∈ℕ, we define
(3.29)Ai,j(n)=∑Γ‍w(Γ),
where the sum is over all walks Γ in G→ of length n from vi to vj. In particular, Ai,j(0)=δij. The fundamental problem treated by the transfer-matrix method is the evaluation of Ai,j(n). The idea is to interpret Ai,j(n) as an entry in a certain matrix. Define a p×p matrix D=(Di,j) by
(3.30)Di,j=∑e‍w(e),
where the sum is over all edges e satisfying that its initial vertex is vi and its final vertex is vj. In other words, Di,j=Ai,j(1). The matrix D is called the adjacency matrix of G→, with respect to the weight function w.Theorem 3.4.
Letn∈ℕ. Then the (i,j)-entry of Dn is equal to Ai,j(n). Here one defines D0=Ip even if D is not invertible, where Ip is the identity matrix.Proof.
See [34].Rectangular Lattice Again
The transfer-matrix method gives us another way to computeTLm,n(x,y) for a fixed width m at point (x,y) which is described in Calkin et al. [35]. In this case we have the same restriction as before, a fixed width m for small values of m, but it has the advantage of being easily automatized.Theorem 3.5 (see, Calkin et. al. [35]).
For indeterminatesx and y and integers n,m≥2, m fixed, one has
(3.31)TLm,n(x+1,y+1)=xnm-1Xmt·(Λm)n-1·1→,
where Xm, a vector of length cm, and Λm, a cm×cm matrix, depend on x,y, and m but not n. And 1→ is the vector of length cm with all entries equal to 1.The quantitycm is the mth Catalan number, so the method is just practical for small values of m. Computing the vectors Xm and the matrix Λm can be easily done in a computer.For example, forL2,n we get that TL2,n(x+1,y+1) equals
(3.32)x2n-1(x-1,1)(x-1+3x-2+yx-21+2x-1x-1+2x-2+x-31+2x-1+x-2)n-1(11).Wheels and Whirls
The transfer-matrix method takes a nice turn when combined with the physics idea of boundary conditions. In this case more lineal algebra is required but the method is still suitable to use in a computer algebra package.A well-known family of self-dual graphs are wheel graphs,Wn. The graph Wn has n+1 vertices and 2n edges, see Figure 8. The rim of the wheel graph Wn is a circuit hyperplane of the corresponding graphic matroid, the relaxation of this circuit-hyperplane gives the matroid Wn, the whirl matroid on n elements. In [36], Chang and Shrock using results from [37] compute the Tutte polynomial of Wn as follows:
(3.33)TWn(x,y)=12n[(1+x+y)+((1+x+y)2-4xy)1/2]n+12n[(1+x+y)-((1+x+y)2-4xy)1/2]n+xy-x-y-1.Figure 8
Then-wheel.From this expression, it is easy to compute an expression for the Tutte polynomial of whirls using (2.6). Consider
(3.34)TWn(x,y)=12n[(1+x+y)+((1+x+y)2-4xy)1/2]n+12n[(1+x+y)+((1+x+y)2-4xy)1/2]n-1.Now, the way Chang and Shrock computeTWn(x,y) is by using the Potts model partition function together with the transfer matrix method. Remarkably, the Tutte polynomial and the Potts model partition function are equivalent. But rather than defining the Potts model we invite the reader to check the surveys in [38, 39]. Here we show how to do the computation using the coboundary polynomial and the transfer method.Let us compute the bad colouring polynomial forWn when we have 3 colours. For this we define the 3×3 matrix D3(3.35)(t211tt1t1t).
The idea is that the entry ij of the matrix (D3)n=D3n will contain all the contributions of bad edges to the bad colouring polynomial for all the colourings σ, with σ(h)=1, σ(1)=i, and σ(n)=j for the fan graph Fn, obtained from Wn by deleting one edge from the rim. Then, to get the bad-colouring polynomial of Fn we just add all the entries in D3n and multiply by λ=3. To get the bad-colouring polynomial of Wn we take the trace of D3n, and multiply by λ=3. This works because, by taking the trace you are considering just colourings with the same initial and final configuration, that is you are placing periodic boundary conditions.Thus, for example, the trace inD33 is t6+8t3+6t2+12t, so BW3(3,t)=3t6+24t3+18t2+36t. By computing the eigenvalues of D3, we obtain the bad colouring polynomial of Wn with λ=3(3.36)BWn(3,t)=32n[(t2+t+1)+(t4-2t3-t2+10t+1)1/2]n+32n[(t2+t+1)+(t4-2t3-t2+10t+1)1/2]n+3(t-1)n.For arbitraryλ we need to compute the eigenvalues of the matrix of λ×λ given below
(3.37)(t211⋯1tt1⋯1t1t⋯1⋮⋮⋮⋱⋮t11⋯t).The eigenvalues of the matrix are(3.38)e1,2=12[(t2+t+λ-2)±(t4-2t3-(2λ-5)t2+(6λ-8)t+(λ-2)2)1/2],
each with multiplicity 1 and e3=(t-1) with multiplicity λ-2. Thus the bad colouring polynomial of Wn is λ(e1n+e2n+(λ-2)e3n). To get formula (3.33) we just need to make the corresponding change of variables as seen at the beginning of the previous section.Other examples that were computed by using this technique include Möbius strips, cycle strips, and homogeneous clan graphs. For the corresponding formulas you can check [36, 37, 40–42].


## 3.3. Splitting the Problem: 1-Sum, 2-Sum and 3-Sum

Our last technique is based on the recurrent idea of splitting a problem that otherwise may be too big. The notion of connectedness in matroid theory offers a natural setting for our purposes, but we do not explain this theory here and we direct the reader to the book of Oxley [9] for such a subject. Also, for 1-sum, 2-sum and 3-sum of binary matroids the book of Truemper [43] is the best reference.However, we would like to explain a little of the relation of 1-sum, 2-sum and 3-sum and the splitting of a matroid. For a matroidM=(E,r), a partition (X,Y) of E is an exactk-separation, for k a positive integer, if
(3.39)min{|X|,|Y|}≥k,r(X)+r(Y)-r(M)=k-1.
Now, a matroid M can be written as a 1-sum of two of its proper minors if and only if M has an exact 1-separation, and M can be written as a 2-sum of two of its proper minors if and only if M has an exact 2-separation. The situation is more complicated in the case of 3-sum and we just want to point out that if a binary matroid M has an exact 3-separation (X,Y), with |X|, |Y|≥4, then there are binary matroids M1 and M2 such that M=M1⊕3M2.1-Sum
Forn matroids M1,M2,…,Mn, on disjoint sets E1,E2,…,En the direct sum M1⊕M2⊕⋯⊕Mn is the matroid (E,ℐ), where E is the union of the ground sets and ℐ={I1∪I2∪⋯∪In:Ii∈ℐ(Mi) for all i in {1,2,…,n}}.
Directly from (2.1) it follows that the Tutte polynomial of the 1-sum is given by
(3.40)TM1⊕M2⊕⋯⊕Mn(x,y)=∏i=1n‍TMi(x,y).2-Sum
LetM1=(E1,ℐ1) and M2=(E2,ℐ2) be matroids with E1∩E2={p}. If p is not a loop or an isthmus in M1 or M2, then the 2-sum   M1⊕2M2 of M1 and M2 is the matroid on E1∪E2∖{p} whose collection of independent sets is {I1∪I2:I1∈ℐ1, I2∈ℐ2, and either I1∪{p}∈ℐ1 or I2∪{p}∈ℐ2}.
In [9, 44, 45], we find recursive formulas for computing the Tutte polynomial of the matroid M1⊕2M2, in terms of the matroids M1 and M2. Here we present the formula given in [44] for TM1⊕2M2(x,y)(3.41)TM1⊕2M2=1xy-x-y[TM1/pTM1∖p][x-1-1-1y-1][TM2/pTM2∖p],
where here we omit the variables x,y of each Tutte polynomial to avoid a cumbersome notation.
As an example, let us takeR6 that is the 2-sum of U2,4 with itself. Observe that the election of the base point is irrelevant as its automorphism group is the symmetric group. The matroid U2,4/r≅U1,3 and U2,4∖r≅U2,3. Then, by using (3.41), we get that TU2,4⊕2U2,4 equals
(3.42)1xy-x-y[y2+y+xx2+x+y][x-1-1-1y-1][y2+y+xx2+x+y].
Thus, we obtain that TR6(x,y)=x3+3x2+4x+2xy+4y+3y2+y3.3-Sum
The best known variant of a 3-sum of two matroids is calledΔ-sum, see [43]. For 3-connected matroids M1 and M2, the Δ-sum is usually denoted M=M1⊕3ΔM2. When M1 and M2 are graphic matroids with corresponding graphs being G1 and G2, the graph G=G1⊕3ΔG2 is obtained by identifying a triangle of T1 of G1 with a triangle T2 of G2 into a triangle T, called the connector triangle. Finally, G is obtained by deleting the edges in the connector triangle.
Here, we present a formula to compute the Tutte polynomial of theΔ-sum of M1 and M2, in terms of the Tutte polynomial of certain minors of the original matroids. The expression for TM1⊕3ΔM2(x,y) was taken from [44] and its proof is based on the concept of bipointed matroids that is an extension of the pointed matroid introduced by Brylawski in [45]. Also, the work in [44] is more general as the author gives an expression for the Tutte polynomial of a certain type of general parallel connection of two matroids.
LetM1 and M2 be two matroids defined on E1 and E2, respectively. Let T be equal to E1∩E2={p,s,q}, a 3-circuit and N=M1|T=M2|T. We require that in M1 there exist circuits U1∪{s} with U1⊆E1∖T and U2∪{p} with U2⊆E1∖T; similarly, we need that in M2 there exist circuits V1∪{s} with V1⊆E2∖T and V2∪{p} with V2⊆E2∖T.
Fori=1,2,…,5, we consider the following 5 minors Qi of M1, on E1∖T, and 5 minors Pi of M2, on E2∖T(3.43)Q1=M1∖p∖s∖q,Q2=M1∖p/s∖q,Q3=M1/p∖s∖q,Q4=M1/p/s/q,Q5=M1∖p∖s/q,P1=M2∖p∖s∖q,P2=M2∖p/s∖q,P3=M2/p∖s∖q,P4=M2/p/s/q,P5=M2∖p∖s/q.
We take the vectors overℤ[x,y]:
(3.44)q→=[TQ1,TQ2,TQ3,TQ4,TQ5],p→=[TP1,TP2,TP3,TP4,TP5]t,
again here we omit the variables x,y of each Tutte polynomial to avoid a cumbersome notation.
Finally, the formula for the 3-sum (Δ-sum) of M1 and M2 is as follows:
(3.45)TM1⊕3ΔM2(x,y)=q→[1-1-x-y+xyC]p→,
where the matrix, C, is given by
(3.46)[(1-y)2-x-y+xy1-y-x-y+xy1-y-x-y+xy2-x-y+xy1-y-x-y+xy1-y-x-y+xy11-x-y+xy1-x-x-y+xy1-x-y+xy1-y-x-y+xy1-x-y+xy11-x-x-y+xy1-x-y+xy2-x-y+xy1-x-x-y+xy1-x-x-y+xy(1-x)2-x-y+xy1-x-x-y+xy1-y-x-y+xy1-x-y+xy1-x-y+xy1-x-x-y+xy1].
As an example, let us takeF8 that is the 3-sum of F7 and F7- along a 3-circuit. In this case we have for F7 the values of Table 1, where in the the first column we present the 5 minors we need, then the second column has the corresponding matroid, and the third column has the corresponding Tutte polynomial of that matroid.Table 1
Minor
Matroid
Polynomial
F
7
∖
p
∖
s
∖
q
U
3,4
x
3
+
x
2
+
x
+
y
F
7
∖
p
/
s
∖
q
U
1,2
⊕
U
1,2
(
x
+
y
)
2
F
7
/
p
∖
s
∖
q
U
1,2
⊕
U
1,2
(
x
+
y
)
2
F
7
/
p
/
s
/
q
U
1,4
y
3
+
y
2
+
y
+
x
F
7
∖
p
∖
s
/
q
U
1,2
⊕
U
1,2
(
x
+
y
)
2Similarly forF7- we have Table 2.Table 2
Minor
Matroid
Polynomial
F
7
-
∖
p
∖
s
∖
q
U
3,4
x
3
+
x
2
+
x
+
y
F
7
-
∖
p
/
s
∖
q
C
3 plus a parallel edge
x
2
+
x
+
x
y
+
y
+
y
2
F
7
-
/
p
∖
s
∖
q
U
1,2
⊕
U
1,2
(
x
+
y
)
2
F
7
-
/
p
/
s
/
q
U
1,4
y
3
+
y
2
+
y
+
x
F
7
-
∖
p
∖
s
/
q
U
1,2
⊕
U
1,2
(
x
+
y
)
2By using (3.45) with the values of Tables 1 and 2 we get TF8(x,y)=x4+4x3+10x2+8x+12xy+8y+10y2+4y3+y4.There is a general concept ofk-sum for matroids and a formula exists for this general notion; however, the formula is quit intricate, so we refer the reader to the original paper of Bonin and Mier in [46].Thickening, Stretch, and Tensor Product
Given a matroidM and a positive integer k, the matroid M(k) is the matroid obtained from M by replacing each nonloop element by k parallel elements and replacing each loop by k loops. The matroid M(k) is called the k-thickening of M. In [4], the following formula is given for the Tutte polynomial of M(k) in terms of that of M(3.47)TM(k)(x,y)=(yk-1+yk-2+⋯+y+1)r(M)TM(yk-1+yk-2+⋯+y+xyk-1+yk-2+⋯+y+1,yk).
A proof by using the recipe theorem can be read in the aforementioned reference. Here we hint a simple proof by noticing that any flat ofM(k) is the k-thickening of a flat of M. Thus, by (3.1)
(3.48)χ-M(k)(q,t)=χ-M(q,tk).
And thus, by using (3.4) and (3.3) we obtained the formula.
The dual operation tok-thickening is that of k-stretch that is defined similarly. The matroid M(k) is the matroid obtained by replacing each nonisthmus of M by k elements in series and replacing each isthmus by k isthmuses. The matroid M(k) is called the k-stretch. It is not difficult to prove that M(k)≅((M*)(k))* and so, we obtained the corresponding formula for TM(k)(3.49)TM(k)(x,y)=(xk-1+xk-2+⋯+x+1)r(M*)TM(xk,xk-1+xk-2+⋯+x+yxk-1+xk-2+⋯+x+1).
More generally we have the following operation called thetensor product. A pointed matroid Nd is a matroid on a ground set which includes a distinguished element, the point d, which will be assumed to be neither a loop nor coloop. For an arbitrary matroid M and a pointed matroid Nd, the tensor productM⊗Nd is the matroid obtained by taking a 2-sum of M with Nd at each point e of M and the distinguished point d of Nd. The Tutte polynomial of M⊗Nd, where M=(E,r) is then given by
(3.50)TM⊗Nd(x,y)=f|E|-r(E)gr(E)TM((x-1)f+gg,f+(y-1)gf),
where f=f(x,y) and g=g(x,y) are polynomials which are determined by the equations
(3.51)(x-1)f(x,y)+g(x,y)=TNd∖d(x,y),f(x,y)+(y-1)g(x,y)=TNd/d(x,y).
The proof of the formula uses a generalization of the recipe theorem to pointed matroids and can be found in [14], here we follow the exposition in [4, 47]. Observe that when a matroid N has a transitive automorphism group, the choice of the distinguished point d is immaterial. Thus, if N is Uk,k+1, k≥1, we get the k-stretch and if N is U1,k+1, k≥1, we get the k-thickening.


## 4. Aplication: Small Matroids

Let us put these techniques in practice and compute some Tutte polynomials for matroids with a small number of elements, these are matroids from the appendix in Oxley’s book [9]. We will try, whenever possible, to check the result by using two techniques. Soon, the reader will realize that a fair amount of the matroids considered are sparse paving and that computing the Tutte polynomial for them is quite easy.MatroidU2,4
The Tutte polynomial of the uniform matroidU2,4 see Figure 9, can be computed using (2.3)
(4.1)TU2,4(x,y)=x2+2x+2y+y2.
Also this matroid is the 2-whirlW2 so you can check this result using (3.34).Figure 9
The uniform matroidU2,4.MatroidsW3, W4, W3, and W4
The 3-wheelM(W3), which is isomorphic to the graphic matroid M(K4), is a sparse paving matroid with λ=4 circuit-hyperplanes, so by using (2.7) we get
(4.2)TW3(x,y)=x3+3x2+2x+4xy+2y+3y2+y3.
Of course, the above polynomial can be checked using (3.33). The only relaxation of M(W3) is the 3-whirl W3, so by using (2.6) we obtain its Tutte polynomial. This can be checked by using (3.34). Consider
(4.3)TW3(x,y)=x3+3x2+3x+3xy+3y+3y2+y3.
The 4-wheel,M(W4), and 4-whirl, W4, are matroids whose geometric representations are shown in Figures 10 and 11. Their Tutte polynomial can be computed using (3.33) and (3.34). One has
(4.4)TW4(x,y)=(1+x+y+(1+x+y)2-4xy)416+(1+x+y-(1+x+y)2-4xy)416+xy-x-y-1=x4+4x3+6x2+3x+4x2y+4xy2+9xy+3y+6y2+4y3+y4,TW4(x,y)=x4+4x3+6x2+4x+4x2y+4xy2+8xy+4y+6y2+4y3+y4.Figure 10
The 4-wheel.Figure 11
The 4-whirl.MatroidsQ6, P6, R6, and U3,6
We use (2.6) to compute the Tutte polynomial of Q6, P6, and U3,6, see Figure 12 for a geometric representation of these matroids. The matroid U3,6 is uniform, from (2.3) we get
(4.5)TU3,6(x,y)=x3+3x2+6x+6y+3y2+y3.
As U3,6 is the only relaxation of P6, from (2.6) its Tutte polynomial is
(4.6)TP6(x,y)=x3+3x2+5x+xy+5y+3y2+y3.
Similarly, P6 is the only relaxation of Q6. Thus, from the above equation and (2.6) we obtain
(4.7)TQ6(x,y)=x3+3x2+4x+2xy+4y+3y2+y3.
It is worth mentioning thatP6 is a relaxation of R6, see Figure 13. Thus, Q6 and R6 have the same Tutte polynomial. This is not at all uncommon, see [9, 48]. Also, R6 is sparse paving so by (2.7) its Tutte polynomial is
(4.8)TR6(x,y)=(x-1)3+6(x-1)2+13x-10+2xy+13y+6(y-1)2+(y-1)3=x3+3x2+4x+2xy+4y+3y2+y3.Figure 12
M
(
K
4
), W3, Q6, P6, and U3,6.Figure 13
The matroidR6.MatroidsF7, F7-, and Their Duals
One of the most mentioned matroid in the literature is the Fano matroidF7, see Figure 14, also known as the projective plane PG(2,2) or the unique Steiner system S(2,3,7). Being a Steiner triple system implies that the Fano matroid is sparse paving, see [49]. As it has 7 circuit-hyperplanes we obtain from (2.7) that the Tutte polynomial of F7 is
(4.9)TF7(x,y)=(x-1)3+7(x-1)2+14x-21+7xy+28y+21(y-1)2+7(y-1)3+(y-1)4=x3+4x2+3x+7xy+3y+6y2+3y3+y4.
The matroid F7 is representable over any field of characteristic 2, so the above calculation can be checked using the program in [19] with the matrix
(4.10)[I3|110110110111].
By using (2.2), the Tutte polynomial of F7* is
(4.11)TF7*(x,y)=x4+3x3+6x2+3x+7xy+3y+4y2+y3.Figure 14
The Fano matroid and its dual.The matroidsF7- and (F7-)* are the corresponding relaxations of F7yF7*, thus we get that
(4.12)TF7-(x,y)=x3+4x2+4x+6xy+4y+6y2+3y3+y4,T(F7-)*(x,y)=x4+3x3+6x2+4x+6xy+4y+4y2+y3.MatroidsP7,P8, and Q3
The matroidP7, shown on the left side of Figure 15, is a rank-3 sparse paving matroid, thus by using (2.7) we get its Tutte polynomial,
(4.13)TP7(x,y)=x3+4x2+5x+5xy+5y+6y2+3y3+y4.
The matrix that represents P7 over a field different from GF(2) is
(4.14)[I3|10111101a1a-10]
with a∉{0,1}. Taking a=2 we have a representation of P7 over GF(3), see [9]. Thus, the above calculation of the Tutte polynomial can be checked using the computer program in [19].Figure 15
Geometric representation ofP7 and P8.The matroidP8, shown on the right side of Figure 15, is also sparse paving and its representation over GF(3) is as below,
(4.15)[I4|011-110111101-1110].
Thus, its Tutte polynomial can be computed using either (2.7) or the program in [19], and you get as a result the following polynomial
(4.16)TP8(x,y)=x4+4x3+10x2+10x+10xy+10y+10y2+4y3+y4.The rank-3 ternary Dowling geometryQ3 has geometric representation shown in Figure 16.Figure 16
Geometric representation ofQ3.The matroidQ3 is representable over a field F if and only if the characteristic of F is different from 2. A matrix that represents Q3 over GF(3) is
(4.17)123|456789[100|111100010|1-10011001|001-1-11].
Thus, we can compute its Tutte polynomial using [19] and obtain
(4.18)TQ3(x,y)=x3+6x2+8x+3xy2+10xy+8y+12y2+10y3+6y4+3y5+y6.
Note, however, that this matroid is paving, so we could have used Proposition 2.8 to get the same result.MatroidW+3
The matroidW+3 is obtained from W3 by adding an element in parallel and so, the matroid is not paving. In this case, we can use Definition 2.3 to compute the Tutte polynomial. In Figure 17 we show how we are using deletion and contraction to find matroids where the Tutte polynomial is either known or easy to compute. Consider
(4.19)TW+3(x,y)=TW3(x,y)+y·(TU2,4(x,y))+TU1,3⊕U0,2(x,y)=(x3+3x2+3xy+3x+3y+3y2+y3)+y·(x2+2x+2y+y2)+(y4+y3+xy2)=x3+3x2+3x+x2y+5xy+xy2+3y+5y2+3y3+y4.Figure 17
Deletion and contraction reduction forW+3.MatroidsAG(3,2), AG(3,2)′, R8, Q8, F8, and L8
The second affine plane that we consider here isAG(3,2), Figures 18 and 19 show two ways of representing the matroid. For example, in Figure 19, the planes with 4 points are the six faces of the cube, the six diagonal planes {1,2,7,8},{2,3,5,8},{3,4,5,6},{1,4,6,7},{1,3,5,7}, and {2,4,6,8}, plus the two twisted planes {1,3,6,8} and {2,4,5,7}. Each of these planes is a circuit-hyperplane. It is not difficult to check that AG(3,2) is isomorphic to the unique Steiner system S(3,4,8), thus, as explained for AG(2,3), it is sparse paving and its Tutte polynomial is
(4.20)TAG(3,2)(x,y)=x4+4x3+10x2+6x+14xy+6y+10y2+4y3+y4.
By using (2.6), we can compute the Tutte polynomial of AG(3,2)′, the unique relaxation of AG(3,2) that here we obtained by relaxing the twisted plane {2,4,5,7}. Thus, the resulting polynomial is
(4.21)TAG(3,2)′(x,y)=x4+4x3+10x2+7x+13xy+7y+10y2+4y3+y4.
Now,AG(3,2)′ has two relaxations, R8 and F8. The matroid R8 is obtained by relaxing from AG(3,2)′ the other twisted plane {1,3,6,8}. On the other hand, F8 is obtained from AG(3,2)′ by relaxing a diagonal plane. The geometric representation of F8 is shown in Figure 20.Figure 18
The affine planeAG(3,2).Figure 19
The cube with 8 points.Figure 20
The matroidF8.The matroidR8 is representable over any field except for GF(2) while F8 is not representable; however, they have the same Tutte polynomial as both are relaxations of the same matroid. Consider
(4.22)TR8(x,y)=TF8(x,y)=x4+4x3+10x2+8x+12xy+8y+10y2+4y3+y4.The unique relaxation ofR8 is Q8 and is obtained by relaxing one of the six diagonal planes of R8. From the previous equation and (2.6) we get
(4.23)TQ8(x,y)=x4+4x3+10x2+7x+11xy+7y+10y2+4y3+y4.
To end this subsection we consider the matroid L8 which is a rank-4 sparse paving matroid with 8 elements and its circuit-hyperplanes are the six faces of the cube plus the two twisted planes {1,8,3,6} and {2,7,4,5}, see Figure 19. Thus, its Tutte polynomial is
(4.24)TL8(x,y)=(x-1)4+8(x-1)3+28(x-1)2+48x-42+8xy+48y+28(y-1)2+8(y-1)3+(y-1)4=x4+4x3+10x2+12x+8xy+12y+10y2+4y3+y4.MatroidsS8,T8, and J
The matroidS8 has a geometric representation shown in Figure 21 together with its representation over GF(2). Its Tutte polynomial can be computed using the program in [19]. One has
(4.25)TS8(x,y)=x4+4x3+7x2+4x+10xy+3xy2+3x2y+4y+7y2+4y3+y4.
Note that the matroid is self-dual but is not paving as it has a 3-circuit. However, we can check the above computation using deletion and contraction. When we contract the forth column in the representation we obtain the representation ofF7, and we have already computed the Tutte polynomial of this matroid. Now, when we delete the same element, we obtain a rank-4 graphic matroid. The graph is K2,4 with an edge contracted. If we call this graph H, by using (2.12) we obtain TH=TK2,4-yTK2,3. These polynomials can be computed either using the general method for complete bipartite graphs given in (3.17) or by using the formula for computing the Tutte polynomial of the 2-stretching of the graphs with two and three parallel edges, respectively, given in (3.49). In both cases, we get
(4.26)TH=x4+3x3+3x2+x+3xy+3x2y+3xy2+y+y2+y3.
By adding (4.26) and (4.9) we get (4.25).On the left-hand side we give the geometric representation of the matroidS8 and on the right-hand side the matrix that represents it over GF(2).
(a)
(b)The matroidT8 is representable over a field 𝔽 if and only if the characteristic is 3. We show its geometric representation in Figure 22. A representation of T8 over GF(3) is [I4|J4-I4], where J4 is the matrix of 1′s. This matroid is self-dual and sparse paving so its Tutte polynomial is
(4.27)TT8(x,y)=x4+4x3+10x2+9x+11xy+9y+10y2+4y3+y4.Figure 22
The matroidT8.In Figure23 we show the geometric representation of the matroid J. It is a self-dual matroid that is not paving, as it has a 3-circuit, and its representation over GF(3) is
(4.28)[I4|1001111001010011],
where the labelling of the columns corresponds to the labelling of the elements in the geometric representation of Figure 23. The Tutte polynomial can be computed using [19]. Consider
(4.29)TJ(x,y)=x4+4x3+7x2+6x+3x2y+3xy2+8xy+6y+7y2+4y3+y4.Figure 23
The matroidJ.This computation can be checked by, for example, contracting the element labelled 2 to obtain the graphic matroid of the graph, that is, the 2-stretching ofC4 minus an edge. Now, the matroid J∖2 is sparse paving and contains 5 circuit-hyperplanes. By adding the Tutte polynomial of these two matroids you get the same result as above.The Vámos Matroid
The 8-element rank-4 matroid whose geometric representation is shown in Figure24 is known as the Vámos matroid and it is usually denoted by V8. It occurs quite frequently in Oxley’s book and has many interesting properties, for example, is not representable over any field and it is sparse paving, see [9]. A related matroid V8+ has the same ground set and is defined in the same way as V8 but with {5,6,7,8} added as a hyperplane. In fact, V8 is obtained from V8+ by relaxing {5,6,7,8}. Notice that both are self-dual. Thus the Tutte polynomial of these two matroids are
(4.30)TV8(x,y)=(x-1)4+8(x-1)3+(82)(x-1)2+(83)(x-1)+(84)+5(xy-x-y)+(85)(y-1)+(86)(y-1)2+8(y-1)3+(y-1)4=x4+4x3+10x2+15x+5xy+15y+10y2+4y3+y4,TV8+(x,y)=x4+4x3+10x2+14x+6xy+14y+10y2+4y3+y4.Figure 24
Geometric representation ofV8.MatroidsR8,R9,R10, and R12
The real affine cube,R8, is represented over all fields of characteristic other than two by the matrix
(4.31)[I4|-11111-11111-11111-1].
The matroid is sparse paving and its Tutte polynomial is
(4.32)TR8(x,y)=x4+4x3+10x2+8x+12xy+8y+10y2+4y3+y4.
On the contrary, the ternary Reid geometry,R9, is not paving but it is representable if and only if the characteristic of the field is three. The matrix that represents R9 over GF(3) is
(4.33)[I3|1111111-1-1-110001-11-1]
and its Tutte polynomial can be computed using the program in [19]. Consider
(4.34)TR9(x,y)=x3+6x2+8x+11xy+2xy2+8y+13y2+10y3+6y4+3y5+y6.
This was computed before in Section 2.3 with the same result.
The unique 10-element regular matroid that is neither graphic nor cographic,R10, has the property that every single-element deletion is isomorphic to M(K3,3), and every single-element contraction is isomorphic to M*(K3,3), then by using the polynomial in (3.18) and (2.2) we obtain
(4.35)TR10(x,y)=x5+5x4+15x3+20x2+10x+15x2y+30xy+15xy2+10y+20y2+15y3+5y4+y5.
Another important regular matroid that is neither graphic or cographic isR12, which has a matrix representation over GF(2) given by
(4.36)[I6|111000110100100010010001001011000111].
As this matroid is not paving we use the program in [19] to compute its Tutte polynomial. One has
(4.37)TR12(x,y)=x6+6x5+19x4+35x3+35x2+14x+2x4y+19x3y+53x2y+17x2y2+56xy+53xy2+19xy3+2xy4+14y+35y2+35y3+19y4+6y5+y6.Pappus and Non-Pappus Matroids
The geometric representation of thePappus matroid is shown in Figure 25. From the picture it is clear that any two points are in a unique line and that each line is a circuit-hyperplane in a rank-3 matroid. We conclude that the matroid is sparse paving with λ=9 circuit-hyperplanes and its Tutte polynomial is
(4.38)TM(x,y)=(x-1)3+9(x-1)2+27x-78+9xy+117y+126(y-1)2+84(y-1)3+36(y-1)4+9(y-1)5+(y-1)6=x3+6x2+12x+9xy+12y+15y2+10y3+6y4+3y5+y6.
Thenon-Pappus matroid is a relaxation of the Pappus matroid and its geometric representation is shown in Figure 26. From the previous equation and (2.6) we obtained
(4.39)TM(x,y)=x3+6x2+12x+9xy+12y+15y2+10y3+6y4+3y5+y6-xy+x+y=x3+6x2+13x+8xy+13y+15y2+10y3+6y4+3y5+y6.Figure 25
Geometric representation of the Pappus matroid.Figure 26
Geometric representation of the non-Pappus matroid.Matroid Non-Desargues
Thenon-Desargues has the geometric representation shown in Figure 27. It is a rank-3 matroid with 10 elements that is sparse paving with the 9 circuit-hyperplanes that are the 9 lines in the picture. Its Tutte polynomial is
(4.40)TM(x,y)=x3+7x2+19x+9xy+19y+21y2+15y3+10y4+6y5+3y6+y7.Figure 27
Geometric representation of the non-Desargues matroid.The Steiner SystemsS(2,3,13) and S(5,6,12)
Finally, we consider two Steiner systems from the Appendix of Oxley’s book. In general any Steiner systemS(v-1,v,n) is sparse paving, see [49]. Thus, the Steiner triple system S(2,3,13) is a rank-3 sparse paving matroid with 13 elements and λ=26 circuit-hyperplanes, then its Tutte polynomial is
(4.41)TS(2,3,13)(x,y)=x3+10x2+29x+26xy+29y+45y2+36y3+28y4+21y5+15y6+10y7+6y8+3y9+y10.
The Steiner systemS(5,6,12) has rank 6, 12 elements, and 132 circuit-hyperplanes. The Tutte polynomial is, by using (2.7),
(4.42)TS(5,6,12)(x,y)=x6+6x5+21x4+56x3+126x2+120x+132xy+120y+126y2+56y3+21y4+6y5+y6.



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*Source: 430859-2012-10-04.xml*