The Tutte Polynomial of Some Matroids

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.

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.

If 𝑀=(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 of 𝑀, 𝑇_𝑀(𝑥,𝑦), is defined as follows: 𝑇_𝑀𝑥,𝑦=_𝐴⊆𝐸(𝑥−1)^𝑧(𝐴)𝑦−1^𝑛(𝐴).(2.1)

Duality
Recall that if 𝑀=(𝐸,𝑟) is a matroid, then 𝑀^∗=(𝐸,𝑟^∗) is its dual matroid, where 𝑟^∗(𝐴)=|𝐴|−𝑟(𝐸)+𝑟(𝐸⧵𝐴). Because 𝑧_𝑀^∗(𝐴)=𝑛_𝑀(𝐸⧵𝐴) and 𝑛_𝑀^∗(𝐴)=𝑧_𝑀(𝐸⧵𝐴), you get the following equality 𝑇_𝑀𝑥,𝑦=𝑇_𝑀^∗𝑦,𝑥.(2.2) 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 matroids 𝑈_𝑟,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 𝑇_{𝑈𝑟,𝑛}𝑥,𝑦=^𝑟−1_𝑖=0𝑛𝑖(𝑥−1)^𝑟−𝑖+𝑛𝑟+^𝑛_𝑖=𝑟+1𝑛𝑖𝑦−1^𝑖−𝑟.(2.3)
Thus, for 𝑈_2,5 we get 𝑇_𝑈2,5𝑥,𝑦=(𝑥−1)²+5(𝑥−1)+10+10𝑦−1+5𝑦−1²+𝑦−1³=𝑥²+3𝑥+3𝑦+2𝑦²+𝑦³.(2.4)
As 𝑈_3,5=(𝑈_2,5)^∗, we get by using (2.2) 𝑇_𝑈3,5𝑥,𝑦=𝑥³+2𝑥²+3𝑥+3𝑦+𝑦².(2.5)

Matroid Relaxation
Given a matroid 𝑀=(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, 𝐹⁻₇ 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. 𝑇_𝑀′𝑥,𝑦=𝑇_𝑀𝑥,𝑦−𝑥𝑦+𝑥+𝑦.(2.6)
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. A paving matroid 𝑀=(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 𝑈_2,2⊕𝑈_0,1, 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. Let 𝑀 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 size 𝑟 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 if 𝑀 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 𝑇_𝑀𝑥,𝑦=^𝑟−1_𝑖=0𝑛𝑖(𝑥−1)^𝑟−𝑖+𝑛𝑟+𝜆𝑥𝑦−𝑥−𝑦+^𝑛_𝑖=𝑟+1𝑛𝑖𝑦−1^𝑖−𝑟.(2.7)

Clearly, given a 𝑟-rank paving matroid 𝑀 with 𝑛 elements, we can obtain the uniform matroid 𝑈_𝑟,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, 𝑟_𝑀+𝑒(𝑋)=𝑟_𝑀(𝑋),𝑟_𝑀+𝑒()=𝑟𝑋∪𝑒^𝑀(𝑋)+1,if𝑟_𝑀(𝑋)(𝑀),𝑟(𝑀)<𝑟,otherwise.(2.8) Again, the Tutte polynomial of 𝑀+𝑒 can be computed easily from the Tutte polynomial of 𝑀 by using (2.1). Consider 𝑇_𝑀+𝑒=𝑥𝑥,𝑦𝑇𝑥−1^𝑀+𝑥𝑥,𝑦𝑦−𝑇𝑥−1^𝑀1,𝑦.(2.9) Here the trick is to notice that 𝑇_𝑀(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 matroid 𝑀(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 𝑇_𝐾4⧵𝑒𝑥,𝑦=𝑥³+2𝑥²+𝑥+2𝑥𝑦+𝑦+𝑦².(2.10) Now, by using (2.9), we get the Tutte polynomial of the free extension: 𝑇_𝑄6=𝑥³+3𝑥²+4𝑥+2𝑥𝑦+4𝑦+3𝑦²+𝑦³.(2.11) 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. If 𝑀 is a matroid, and 𝑒 is an element that is neither a loop nor a coloop, then 𝑇_𝑀𝑥,𝑦=𝑇_𝑀⧵𝑒𝑥,𝑦+𝑇_𝑀/𝑒𝑥,𝑦.(2.12) If 𝑒 is a coloop, then 𝑇_𝑀𝑥,𝑦=𝑥𝑇_𝑀⧵𝑒𝑥,𝑦.(2.13) If 𝑒 is a loop, then 𝑇_𝑀𝑥,𝑦=𝑦𝑇_𝑀/𝑒𝑥,𝑦.(2.14)

The proof that Definitions 2.1 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 Cycle 𝐶_𝑛
As a first example of this subsection we consider the graphic matroid 𝑀(𝐶_𝑛). Here, by deleting an edge 𝑒 from 𝐶_𝑛 we obtained a path whose Tutte polynomial is 𝑥^𝑛−1, while if we contract 𝑒 we get a smaller cycle 𝐶_𝑛−1. Thus, as the Tutte polynomial of the 2-cycle is 𝑥+𝑦, we obtain 𝑇_𝐶𝑛𝑥,𝑦=^𝑛−1_𝑖=1𝑥^𝑖+𝑦.(2.15)
By duality, the Tutte polynomial of the graph 𝐶^∗_𝑛 with two vertices and 𝑛 parallel edges between them is 𝑇_𝐶^∗_𝑛𝑥,𝑦=^𝑛−1_𝑖=1𝑦^𝑖+𝑥.(2.16)

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 a parallel 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. Let 𝑋 be a parallel class of a matroid 𝑀 with |𝑋|=𝑝+1. If 𝑋 is not a cocircuit, then 𝑇_𝑀𝑥,𝑦=𝑇_𝑀⧵𝑋𝑥,𝑦+𝑦^𝑝+𝑦^𝑝−1+⋯+1𝑇_𝑀/𝑋𝑥,𝑦.(2.17)

The proof is by induction on 𝑝 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.

A series class in 𝑀 is just a parallel class in 𝑀^∗ and by duality we get the following.

Lemma 2.5. Let 𝑋 be a series class of a matroid 𝑀 with |𝑋|=𝑝+1. If 𝑋 is not a circuit, then 𝑇_𝑀𝑥,𝑦=𝑥^𝑝+𝑥^𝑝−1+⋯+1𝑇_𝑀⧵𝑋𝑥,𝑦+𝑇_𝑀/𝑋𝑥,𝑦.(2.18)

The Rectangular Lattice 𝐿_𝑚,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.
Let 𝑚 and 𝑛 be integers, 𝑚,𝑛>1. The grid or rectangular lattice 𝐿_𝑚,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 𝐿_𝑚,𝑛 can be represented as in Figure 1.

Figure 1 

The grid graph 𝐿𝑚,𝑛.

The vertices of 𝐿_𝑚,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 graphs 𝐿_𝑚,𝑛 when 𝑚=2. For 𝑚>3, the formulas are very complicated to use in practical calculations.

The graphs 𝐿_2,1, 𝐿_2,2, and 𝐿_2,3 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 integer 𝑛. The graph 𝑄_2,𝑛 is defined from the graph 𝐿_2,𝑛 by contraction of the edge {11,21} of 𝐿_2,𝑛. In particular, the graph 𝑄_2,1 is one isolated vertex, see Figure 2.

The initial conditions to construct the recurrence relations for the Tutte polynomial of the grid graphs 𝐿_2,𝑛 are.(i)𝑇_𝑄2,1(𝑥,𝑦)=1;(ii)𝑇_𝐿2,1(𝑥,𝑦)=𝑥; (iii)𝑇_𝐿2,2(𝑥,𝑦)=𝑥³+𝑥²+𝑥+𝑦.

The third condition is true because the graph 𝐿_2,2 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 𝐿_2,𝑛 is 𝑇_𝐿2,𝑛𝑥,𝑦=𝑥²+𝑥+1𝑇_{𝐿2,𝑛−1}𝑥,𝑦+𝑦𝑇_{𝑄2,𝑛−1},𝑇𝑥,𝑦_𝑄2,𝑛𝑥,𝑦=(𝑥+1)𝑇_{𝐿2,𝑛−1}𝑥,𝑦+𝑦𝑇_{𝑄2,𝑛−1}𝑥,𝑦.(2.19)

From this, we get a linear recurrence of order two, 𝑇_𝐿2,𝑛−𝑥²+𝑥+1+𝑦𝑇_{𝐿2,𝑛−1}+𝑥²𝑦𝑇_{𝐿2,𝑛−2}=0,(2.20) that is easy to solve. Thus, the general formula is 𝑇_𝐿2,𝑛=𝑎𝑥,𝑦¹𝜆₁+𝑎₂𝜆₁−𝜆₂𝜆₁^𝑛−2+𝑎₁𝜆₂+𝑎₂𝜆₂−𝜆₁𝜆₂^𝑛−2,(2.21) for 𝑛≥2, where 𝑎₁=𝑦+𝑥+𝑥²+𝑥³, 𝑎₂=−𝑦𝑥³, and 𝜆₁ and 𝜆₂ are 121+𝑦+𝑥+𝑥²±𝑦²+2𝑦1+𝑥−𝑥²+1+𝑥+𝑥^221/2.(2.22)

A recursive family of graphs 𝐺_𝑚 is a sequence of graphs with the property that the Tutte polynomials 𝑇_𝐺𝑚(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 {𝐿_2,𝑛} is such a family and in general, {𝐿_𝑘,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. If 𝑀 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 matroid 𝑀=(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,…,𝑒_𝑚}, where 𝑒_𝑖≺𝑒_𝑗 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 𝑡_𝑖𝑗 to be the number of bases with 𝑖 internally active elements and 𝑗 externally active elements. In [20–22] Tutte defined 𝑇_𝑀 using these concepts. A proof of the equivalence with Definition 2.1 can be found in [23].

Definition 2.6. If 𝑀=(𝐸,𝑟) is a matroid with a total order on its ground set, then 𝑇_𝑀𝑥,𝑦=_𝑖,𝑗𝑡_𝑖𝑗𝑥^𝑖𝑦^𝑗.(2.23) In particular, the coefficients 𝑡_𝑖𝑗 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 𝑇_{𝑈𝑟,𝑛}𝑥,𝑦=^𝑛−𝑟_𝑗=1𝑦𝑛−𝑗−1𝑟−1^𝑗+^𝑟_𝑖=1𝑥𝑛−𝑖−1𝑛−𝑟−1^𝑖,(2.24) when 0<𝑟<𝑛, while T_{𝑈𝑛,𝑛}(𝑥,𝑦)=𝑥^𝑛 and 𝑇_𝑈0,𝑛(𝑥,𝑦)=𝑦^𝑛. 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,…,𝑇_𝑘} 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. If 𝒯 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. Let 𝑀 be a rank-𝑟 matroid with 𝑛 elements and Tutte polynomial ∑_𝑖,𝑗𝑡_𝑖𝑗𝑥^𝑖𝑦^𝑗. Then, 𝑀 is paving if and only if 𝑡_𝑖𝑗=0 for all (𝑖,𝑗)≥(2,1). In addition if the (𝑟−1)-partition of 𝐸 has 𝑏_𝑘 blocks of cardinality 𝑘, for 𝑘=𝑟−1,…𝑛, then 𝑡_𝑖0=𝑡𝑛−𝑖−1𝑟−𝑖∀𝑖≥2,₁₀=^∞_𝑘=0𝑏𝑟−2+𝑘𝑟−2^{𝑘+𝑟−1}+𝑛,𝑛−2𝑟−1𝑟−1(2.25) and for all 𝑗>0, 𝑡_1𝑗=^∞_𝑘=0𝑏𝑟−2+𝑘𝑟−2^{𝑘+𝑗+𝑟−1},𝑡_0𝑗=−𝑛−𝑗−1𝑟−1^∞𝑘=0𝑏𝑟−1+𝑘𝑟−1^{𝑘+𝑗+𝑟−1}.(2.26)

Note that when 𝑀 is the uniform matroid 𝑈_𝑟,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 matroid 𝑅₉ 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 𝑇_𝑅9𝑥,𝑦=𝑥³+6𝑥²+8𝑥+11𝑥𝑦+2𝑥𝑦²+8𝑦+13𝑦²+10𝑦³+6𝑦⁴+3𝑦⁵+𝑦⁶.(2.27)

Figure 4 

The matroid 𝑅9.

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⋯𝑤_𝑛 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).
Let 𝐴 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,𝑦_𝑃) and (𝑥₀,𝑦_𝑄) of 𝑃 and 𝑄, respectively. If 𝑦_𝑃<𝑦_𝑄 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 paths 𝑃 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: 𝑐_𝑛=1𝑛+1𝑛2𝑛.(2.28)
Denote by [𝑛] the set {1,2,…,𝑛}. Consider the word 𝑤=𝑤₁𝑤₂⋯𝑤_𝑚+𝑛 that corresponds to a 𝑃𝑄-bounded lattice path. We can associate to 𝑤 a subset 𝑋_𝑤 of [𝑚+𝑛] by defining that 𝑖∈𝑋_𝑤 if 𝑤_𝑖=𝑁. 𝑋_𝑤 is the support set of the steps 𝑁 in 𝑤. The family of support sets 𝑋_𝑤 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 lines 𝑦=0 and 𝑦=𝑥, the matroid is denoted by 𝑀_𝑛 and is named Catalan matroid. Note that 𝑀_𝑛 has a loop that corresponds to the label 1 and a coloop that corresponds to the label 2𝑛, and that 𝑀_𝑛 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. Let 𝐵∈ℬ be a basis of 𝑀[𝑃,𝑄] and let 𝑤_𝐵 be the 𝑃𝑄-bounded lattice path associated with 𝐵. Then 𝑖(𝐵) is the number of times 𝑤_𝐵 meets the upper path 𝑄 in a north step and 𝑒(𝐵) is the number of times 𝑤_𝐵 meets the lower path 𝑃 in an east step.

Then the Tutte polynomial of the Catalan matroids 𝑀_𝑛 for 𝑛>1, is 𝑇_𝑀𝑛𝑥,𝑦=_𝑖,𝑗>0𝑖+𝑗−2𝑥𝑛−12𝑛−𝑖−𝑗−1𝑛−𝑖−𝑗+1^𝑖𝑦^𝑗.(2.29)

Note that the coefficient of 𝑥^𝑖𝑦^𝑗 in the Tutte polynomial of the matroid 𝑀_𝑛 depends only on 𝑛 and the sum 𝑖+𝑗.

In the Figure 5 are shown the bases of 𝑀₃ with its internal and external activities. Thus the Tutte polynomial of 𝑀₃ is 𝑇_𝑀3𝑥,𝑦=𝑥³𝑦+𝑥²𝑦+𝑥²𝑦²+𝑥𝑦²+𝑥𝑦³.(2.30)

Figure 5 

Bases of 𝑀3 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 𝜒_𝑀(𝜆,𝑡)=_{𝑋∈ℱ𝑀}𝑡^|𝑋|𝜒_𝑀/𝑋(𝜆),(3.1) where ℱ_𝑀 is the set of flats of 𝑀 and 𝜒_𝑁(𝜆) is the characteristic polynomial of the matroid 𝑁. The characteristic polynomial of a matroid 𝑀 is defined by 𝜒_𝑀(𝜆)=_𝐴⊆𝐸(−1)^|𝑋|𝜆^𝑧(𝐴).(3.2)

The Tutte and the coboundary polynomial are related as follows: 𝑇_𝑀=1𝑥,𝑦𝑦−1^𝑟(𝐸)𝜒_𝑀(𝑥−1)𝑦−1,𝑦,(3.3)𝜒_𝑀(𝜆,𝑡)=(𝑡−1)^𝑟(𝐸)𝑇_𝑀𝜆+𝑡−1𝑡−1,𝑡.(3.4)

When 𝑀 is a graphic matroid 𝑀(𝐺) of a connected graph 𝐺, the characteristic polynomial is equivalent to the chromatic polynomial, that we denote 𝜒_𝐺: 𝜒_𝐺(𝜆)=𝜆𝜒_𝑀(𝐺)(𝜆).(3.5)

And also, in this case, the coboundary is the bad-colouring polynomial. The bad-colouring polynomial is the generating function 𝐵_𝐺(𝜆,𝑡)=_𝑗𝑏_𝑗(𝐺;𝜆)𝑡^𝑗,(3.6) where 𝑏_𝑗(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: 𝐵_𝐺(𝜆,𝑡)=𝜆𝜒_𝑀(𝐺)(𝜆,𝑡).(3.7)

The 𝑞-Cone
We base this section on the work of Bonin and Qin, see [27].

Definition 3.1. Let 𝑀 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 restriction PG(𝑟,𝑞)∣𝑆 of PG(𝑟,𝑞) to the subset 𝑆 is isomorphic to 𝑀,(2)the point 𝑎∈PG(𝑟,𝑞) is not contained in the closure of 𝑆, 𝐶𝐿_{PG(𝑟,𝑞)}(𝑆), in PG(𝑟,𝑞), and(3)the matroid 𝑁 is the restriction of PG(𝑟,𝑞) to the set ⋃_𝑠∈𝑆𝐶𝑙_{PG(𝑟,𝑞)}({𝑎,𝑠}).

That is, one represents 𝑀 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 every 𝑞-cone 𝑁 of a rank-𝑟 simple matroid 𝑀, one has 𝜒_𝑁(𝜆)=(𝜆−1)𝑞^𝑟𝜒_𝑀𝜆𝑞.(3.8)

To extend this result to 𝜒, 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 𝜒_𝑁(𝜆,𝑡)=𝑡𝜒_𝑀𝜆,𝑡^𝑞+𝑞^𝑟(𝜆−1)𝜒_𝑀𝜆.𝑞,𝑡(3.9) 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. If 𝑀 is a rank-𝑟 matroid representable over GF(𝑞) and 𝑁 is a 𝑞-cone of 𝑀, then 𝑇_𝑁=𝑦𝑦𝑥,𝑦^𝑞−1^𝑟𝑦−1^𝑟+1𝑇_𝑀(𝑥−1)𝑦−1𝑦^𝑞−1+1,𝑦^𝑞+𝑞^𝑟𝑥𝑦−𝑥−𝑦𝑇𝑦−1^𝑀𝑥+1𝑞+1,𝑦.(3.10)

Thus, for example, the 2-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 𝑥³+4𝑥²+3𝑥+7𝑥𝑦+3𝑦+6𝑦²+3𝑦³+𝑦⁴.(3.11) 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 𝐾_𝑛 by 𝑉 and by 𝐵_𝑛 its bad colouring. To compute 𝐵_𝑛, observe that any 𝜆-colouring partitions the vertices 𝑉 into 𝜆 color classes of subsets 𝑉_𝑖 of vertices each of cardinality 𝑛_𝑖, for 𝑖=1,…,𝜆. So, we have that 𝑛₁+⋯+𝑛_𝜆=𝑛. The number of bad edges with both ends in the set 𝑉_𝑖 is 𝑡^𝑛𝑖2. Thus, by the exponential formula we get the following formula _𝑛∈ℕ𝑡^𝑛2𝑢^𝑛𝑛!^𝜆=1+_𝑛≥1𝐵_𝑛()𝑢𝜆,𝑡^𝑛.𝑛!(3.12)
Now, we can use (3.7) to get _𝑛∈ℕ𝑡^𝑛2𝑢^𝑛𝑛!^𝜆=1+𝜆_𝑛≥1𝜒_𝑛()𝑢𝜆,𝑡^𝑛.𝑛!(3.13)
Let 𝑇_𝑛(𝑥,𝑦) be the Tutte polynomial of 𝐾_𝑛. Tutte in [22] and Welsh in [30] give the following exponential generating function for 𝑇_𝑛(𝑥,𝑦) that follows from the previous equation and (3.3) _𝑛≥0𝑦^𝑛2𝑢^𝑛𝑛!^{(𝑥−1)(𝑦−1)}=1+(𝑥−1)_𝑛≥1𝑦−1^𝑛𝑇_𝑛𝑢𝑥,𝑦^𝑛.𝑛!(3.14)

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(v̂‌(i*(i-1)/2)*x̂‌i/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 computing 𝑇₅(𝑥,𝑦) we obtain 𝑇₅𝑥,𝑦=𝑦⁶+4𝑦⁵+𝑥⁴+5𝑥𝑦³+10𝑦⁴+6𝑥³+10𝑥²𝑦+15𝑥𝑦²+15𝑦³+11𝑥²+20𝑥𝑦+15𝑦²+6𝑥+6𝑦.(3.15)

Complete Bipartite Graphs
It is just natural to use this technique for complete bipartite graphs 𝐾_𝑛,𝑚 with similar results as above. This time let the vertex set be 𝑉₁∪𝑉₂ and denote by 𝐵_𝑛,𝑚 its bad-colouring polynomial. To compute 𝐵_𝑛,𝑚, observe that any 𝜆-colouring partitioned the vertices in subsets 𝑉₁^(𝑖)∪𝑉₂^(𝑖) each of cardinality 𝑛_𝑖𝑚_𝑖. The number of the bad edges with both ends colour 𝑖 is 𝑛_𝑖𝑚_𝑖. Thus, by the exponential formula we get the following: _{()𝑛,𝑚∈ℕ2}𝑡^𝑛𝑚𝑢^𝑛𝑢𝑛!𝑚^𝑚!^𝜆=1+_{(𝑛,𝑚)∈ℕ2(𝑛,𝑚)≠(0,0)}𝐵_𝑛,𝑚()𝑢𝜆,𝑡^𝑛𝑣𝑛!𝑚^𝑚!=1+𝜆_{(𝑛,𝑚)∈ℕ2(𝑛,𝑚)≠(0,0)}𝜒_𝑛,𝑚()𝑢𝜆,𝑡^𝑛𝑣𝑛!^𝑚.𝑚!(3.16)
Thus, a formula for the Tutte polynomial of the bipartite complete graph can be found in a similar way as before. Let 𝑇_𝑛,𝑚(𝑥,𝑦) be the Tutte polynomial of 𝐾_𝑛,𝑚. The following formula is from Stanley’s book [29], see also [31] _{()𝑛,𝑚∈ℕ2}𝑦^𝑛𝑚𝑢^𝑛𝑣𝑛!𝑚^𝑚!^{(𝑥−1)(𝑦−1)}=1+(𝑥−1)_{()𝑛,𝑚∈ℕ2()≠()𝑛,𝑚0,0}𝑦−1^𝑛𝑚𝑇_𝑛,𝑚𝑢𝑥,𝑦^𝑛𝑣𝑛!^𝑚.𝑚!(3.17)
As before, (3.16) is quite easy to use in Maple to compute 𝑇_𝑛,𝑚(𝑥,𝑦) for small values of 𝑛 and 𝑚. In this way we get the following: 𝑇_3,3𝑥,𝑦=𝑥⁵+4𝑥⁴+10𝑥³+9𝑥²𝑦+11𝑥²+6𝑥𝑦²+15𝑥𝑦+5𝑥+𝑦⁴+5𝑦³+9𝑦²+5𝑦.(3.18)

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 𝑚𝑘_𝑞=𝑞^𝑚−1𝑞^𝑚−𝑞⋯𝑞^𝑚−𝑞^𝑘−1𝑞^𝑘−1𝑞^𝑘−𝑞⋯𝑞^𝑘−𝑞^𝑘−1.(3.19) Note that [^𝑚₀]_𝑞=1 since by convention an empty product is 1 and ⁰_𝑘𝑞=0.

Let PG(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 [^𝑟_𝑘]_𝑞. The characteristic polynomial of PG(𝑟−1,𝑞) is known to be, see [4], 𝜒_{PG(𝑟−1,𝑞)}(𝜆)=^𝑟−1_𝑖=0𝜆−𝑞^𝑖.(3.20) Thus, using (3.1) we get 𝜒_{PG(𝑟−1,𝑞)}(𝜆,𝑡)=^𝑟_𝑘=0𝑡^𝑘1𝑞𝑟𝑘_𝑞^{𝑟−𝑘−1𝑖=0}𝜆−𝑞^𝑖.(3.21) 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): ||||.12435678910111213100122110011201011101110110010121212210(3.22) To compute its Tutte polynomial we could use the program in [19] or use the above formula to get 𝜒_PG(2,3)(𝜆,𝑡)=(𝜆−1)(𝜆−3)(𝜆−9)+13𝑡(𝜆−1)(𝜆−3)+13𝑡⁴(𝜆−1)+𝑡¹³.(3.23)

Figure 6 

The projective plane PG(2,3) with a marked hyperplane.

Finally the Tutte polynomial of PG(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: 𝑇_PG(2,3)𝑥,𝑦=𝑥³+10𝑥²+13𝑥𝑦²+26𝑥𝑦+16𝑥+16𝑦+32𝑦²+36𝑦³+28𝑦⁴+21𝑦⁵+15𝑦⁶+10𝑦⁷+6𝑦⁸+3𝑦⁹+𝑦¹⁰.(3.24)

Every time we have a projective geometry PG(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 plane AG(2,3).

The situation to compute the Tutte polynomial is very similar for the 𝑟-dimensional affine geometry over GF(𝑞), AG(𝑟,𝑞). It has rank 𝑟+1 and 𝑞^𝑟 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 𝑞^𝑟−𝑘[^𝑟_𝑘]_𝑞. The characteristic polynomial of AG(𝑟,𝑞) is known to be, see [4], 𝜒_{AG(𝑟,𝑞)}(𝜆)=(𝜆−1)^𝑟_𝑘=0(−1)^𝑘𝜆^{𝑟−𝑘𝑘−1}_𝑖=0𝑞^𝑟−𝑖.(3.25) Now, using the result of Mphako we can compute the coboundary polynomial of AG(𝑟,𝑞): 𝜒_{AG(𝑟,𝑞)}(𝜆,𝑡)=𝜒_{AG(𝑟,𝑞)}(𝜆)+^𝑟_𝑘=0𝑡^𝑞𝑘𝑞^𝑟−𝑘𝑟𝑘_𝑞^{𝑟−𝑘−1𝑖=0}𝜆−𝑞^𝑖.(3.26)

The affine plane AG(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. 𝜒_AG(2,3)(𝜆,𝑡)=(𝜆−1)𝜆²−8𝜆+16+9𝑡(𝜆−1)(𝜆−3)+12𝑡³(𝜆−1)+𝑡⁹.(3.27)

Again, the Tutte polynomial of AG(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)³𝑇_AG(2,3)𝑥,𝑦=𝑥³+6𝑥²+12𝑥𝑦+9𝑥+9𝑦+15y²+10𝑦³+6𝑦⁴+3𝑦⁵+𝑦⁶.(3.28)

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.

A directed graph or digraph 𝐺^→ is a triple (𝑉,𝐸,𝜙), where 𝑉={𝑣₁,…,𝑣_𝑝} 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 walk Γ in 𝐺^→ of length 𝑛 from 𝑢 to 𝑣 is a sequence 𝑒₁,…,𝑒_𝑛 of 𝑛 edges such that the final vertex of 𝑒_𝑖 is the initial vertex of 𝑒_𝑖+1, for 1≤𝑖≤𝑛−1.

Now let 𝑤∶𝐸→𝑅 be a weight function on 𝐸 with values in some commutative ring 𝑅. If Γ=𝑒₁,…,𝑒_𝑛 is a walk, then the weight of Γ is defined by 𝑤(Γ)=𝑤(𝑒₁)⋯𝑤(𝑒_𝑛). For 1≤𝑖,𝑗≤𝑝 and 𝑛∈ℕ, we define 𝐴_𝑖,𝑗(𝑛)=_Γ𝑤(Γ),(3.29) where the sum is over all walks Γ in 𝐺^→ of length 𝑛 from 𝑣_𝑖 to 𝑣_𝑗. In particular, 𝐴_𝑖,𝑗(0)=𝛿_𝑖𝑗. The fundamental problem treated by the transfer-matrix method is the evaluation of 𝐴_𝑖,𝑗(𝑛). The idea is to interpret 𝐴_𝑖,𝑗(𝑛) as an entry in a certain matrix. Define a 𝑝×𝑝 matrix 𝐷=(𝐷_𝑖,𝑗) by 𝐷_𝑖,𝑗=_𝑒𝑤(𝑒),(3.30) where the sum is over all edges 𝑒 satisfying that its initial vertex is 𝑣_𝑖 and its final vertex is 𝑣_𝑗. In other words, 𝐷_𝑖,𝑗=𝐴_𝑖,𝑗(1). The matrix 𝐷 is called the adjacency matrix of 𝐺^→, with respect to the weight function 𝑤.

Theorem 3.4. Let 𝑛∈ℕ. Then the (𝑖,𝑗)-entry of 𝐷^𝑛 is equal to 𝐴_𝑖,𝑗(𝑛). Here one defines 𝐷⁰=𝐼_𝑝 even if 𝐷 is not invertible, where 𝐼_𝑝 is the identity matrix.

Proof. See [34].

Rectangular Lattice Again
The transfer-matrix method gives us another way to compute 𝑇_{𝐿𝑚,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 indeterminates 𝑥 and 𝑦 and integers 𝑛,𝑚≥2, 𝑚 fixed, one has 𝑇_{𝐿𝑚,𝑛}𝑥+1,𝑦+1=𝑥^𝑛𝑚−1𝑋^𝑡_𝑚⋅Λ_𝑚^𝑛−1⋅^→1,(3.31) where 𝑋_𝑚, a vector of length 𝑐_𝑚, and Λ_𝑚, a 𝑐_𝑚×𝑐_𝑚 matrix, depend on 𝑥,𝑦, and 𝑚 but not 𝑛. And ^→1 is the vector of length 𝑐_𝑚 with all entries equal to 1.

The quantity 𝑐_𝑚 is the 𝑚th Catalan number, so the method is just practical for small values of 𝑚. Computing the vectors 𝑋_𝑚 and the matrix Λ_𝑚 can be easily done in a computer.

For example, for 𝐿_2,𝑛 we get that 𝑇_𝐿2,𝑛(𝑥+1,𝑦+1) equals 𝑥^2𝑛−1𝑥⁻¹𝑥,1⁻¹+3𝑥⁻²+𝑦𝑥⁻²1+2𝑥⁻¹𝑥⁻¹+2𝑥⁻²+𝑥⁻³1+2𝑥⁻¹+𝑥^−2𝑛−111.(3.32)

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, 𝑊_𝑛. The graph 𝑊_𝑛 has 𝑛+1 vertices and 2𝑛 edges, see Figure 8. The rim of the wheel graph 𝑊_𝑛 is a circuit hyperplane of the corresponding graphic matroid, the relaxation of this circuit-hyperplane gives the matroid 𝑊^𝑛, the whirl matroid on 𝑛 elements. In [36], Chang and Shrock using results from [37] compute the Tutte polynomial of 𝑊_𝑛 as follows: 𝑇_𝑊𝑛=1𝑥,𝑦2^𝑛(1+𝑥+𝑦)+1+𝑥+𝑦²−4𝑥𝑦^1/2𝑛+12^𝑛1+𝑥+𝑦−1+𝑥+𝑦²−4𝑥𝑦^1/2𝑛+𝑥𝑦−𝑥−𝑦−1.(3.33)

Figure 8 

The 𝑛-wheel.

From this expression, it is easy to compute an expression for the Tutte polynomial of whirls using (2.6). Consider 𝑇_𝑊^𝑛=1𝑥,𝑦2^𝑛1+𝑥+𝑦+1+𝑥+𝑦²−4𝑥𝑦^1/2𝑛+12^𝑛(1+𝑥+𝑦)+1+𝑥+𝑦²−4𝑥𝑦^1/2𝑛−1.(3.34)

Now, the way Chang and Shrock compute 𝑇_𝑊𝑛(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 for 𝑊_𝑛 when we have 3 colours. For this we define the 3×3 matrix 𝐷₃𝑡²11𝑡𝑡1𝑡1𝑡.(3.35) The idea is that the entry 𝑖𝑗 of the matrix (𝐷₃)^𝑛=𝐷^𝑛₃ 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 𝐹_𝑛, obtained from 𝑊_𝑛 by deleting one edge from the rim. Then, to get the bad-colouring polynomial of 𝐹_𝑛 we just add all the entries in 𝐷^𝑛₃ and multiply by 𝜆=3. To get the bad-colouring polynomial of 𝑊_𝑛 we take the trace of 𝐷^𝑛₃, 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 in 𝐷³₃ is 𝑡⁶+8𝑡³+6𝑡²+12𝑡, so 𝐵_𝑊3(3,𝑡)=3𝑡⁶+24𝑡³+18𝑡²+36𝑡. By computing the eigenvalues of 𝐷₃, we obtain the bad colouring polynomial of 𝑊_𝑛 with 𝜆=3𝐵_𝑊𝑛()=33,𝑡2^𝑛𝑡²+𝑡+1+𝑡⁴−2𝑡³−𝑡²+10𝑡+1^1/2𝑛+32^𝑛𝑡²+𝑡+1+𝑡⁴−2𝑡³−𝑡²+10𝑡+1^1/2𝑛+3(𝑡−1)^𝑛.(3.36)

For arbitrary 𝜆 we need to compute the eigenvalues of the matrix of 𝜆×𝜆 given below 𝑡²11⋯1𝑡𝑡1⋯1𝑡1𝑡⋯1⋮⋮⋮⋱⋮𝑡11⋯𝑡.(3.37)

The eigenvalues of the matrix are 𝑒_1,2=12𝑡²+𝑡+𝜆−2±𝑡⁴−2𝑡³−(2𝜆−5)𝑡²+(6𝜆−8)𝑡+(𝜆−2)^21/2,(3.38) each with multiplicity 1 and 𝑒₃=(𝑡−1) with multiplicity 𝜆−2. Thus the bad colouring polynomial of 𝑊_𝑛 is 𝜆(𝑒^𝑛₁+𝑒^𝑛₂+(𝜆−2)𝑒^𝑛₃). 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 matroid 𝑀=(𝐸,𝑟), a partition (𝑋,𝑌) of 𝐸 is an exact 𝑘-separation, for 𝑘 a positive integer, if 𝑋|,|𝑌𝑟(𝑋)(𝑌)(𝑀)min{||}≥𝑘,+𝑟−𝑟=𝑘−1.(3.39) 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
For 𝑛 matroids 𝑀₁,𝑀₂,…,𝑀_𝑛, on disjoint sets 𝐸₁,𝐸₂,…,𝐸_𝑛 the direct sum 𝑀₁⊕𝑀₂⊕⋯⊕𝑀_𝑛 is the matroid (𝐸,ℐ), where 𝐸 is the union of the ground sets and ℐ={𝐼₁∪𝐼₂∪⋯∪𝐼_𝑛∶𝐼_𝑖∈ℐ(𝑀_𝑖) for all 𝑖 in {1,2,…,𝑛}}.
Directly from (2.1) it follows that the Tutte polynomial of the 1-sum is given by 𝑇_{𝑀1⊕𝑀2⊕⋯⊕𝑀𝑛}𝑥,𝑦=^𝑛_𝑖=1𝑇_𝑀𝑖𝑥,𝑦.(3.40)

2-Sum
Let 𝑀₁=(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 𝑇_{𝑀1⊕2𝑀2}(𝑥,𝑦)𝑇_{𝑀1⊕2𝑀2}=1𝑇𝑥𝑦−𝑥−𝑦^𝑀1/𝑝𝑇_𝑀1⧵𝑝𝑥−1−1𝑇−1𝑦−1𝑇^𝑀2/𝑝_𝑀2⧵𝑝,(3.41) where here we omit the variables 𝑥,𝑦 of each Tutte polynomial to avoid a cumbersome notation.
As an example, let us take 𝑅₆ that is the 2-sum of 𝑈_2,4 with itself. Observe that the election of the base point is irrelevant as its automorphism group is the symmetric group. The matroid 𝑈_2,4/𝑟≅𝑈_1,3 and 𝑈_2,4⧵𝑟≅𝑈_2,3. Then, by using (3.41), we get that 𝑇_{𝑈2,4⊕2𝑈2,4} equals 1𝑦𝑥𝑦−𝑥−𝑦²+𝑦+𝑥𝑥²𝑦+𝑥+𝑦𝑥−1−1−1𝑦−1𝑥²+𝑦+𝑥₂+𝑥+𝑦.(3.42) Thus, we obtain that 𝑇_𝑅6(𝑥,𝑦)=𝑥³+3𝑥²+4𝑥+2𝑥𝑦+4𝑦+3𝑦²+𝑦³.

3-Sum
The best known variant of a 3-sum of two matroids is called Δ-sum, see [43]. For 3-connected matroids 𝑀₁ and 𝑀₂, the Δ-sum is usually denoted 𝑀=𝑀₁⊕^Δ₃𝑀₂. When 𝑀₁ and 𝑀₂ are graphic matroids with corresponding graphs being 𝐺₁ and 𝐺₂, the graph 𝐺=𝐺₁⊕^Δ₃𝐺₂ 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 the Δ-sum of 𝑀₁ and 𝑀₂, in terms of the Tutte polynomial of certain minors of the original matroids. The expression for 𝑇_𝑀1⊕^Δ_3𝑀2(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.
Let 𝑀₁ 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.
For 𝑖=1,2,…,5, we consider the following 5 minors 𝑄_𝑖 of 𝑀₁, on 𝐸₁⧵𝑇, and 5 minors 𝑃_𝑖 of 𝑀₂, on 𝐸₂⧵𝑇𝑄₁=𝑀₁⧵𝑝⧵𝑠⧵𝑞,𝑄₂=𝑀₁𝑄⧵𝑝/𝑠⧵𝑞,₃=𝑀₁/𝑝⧵𝑠⧵𝑞,𝑄₄=𝑀₁/𝑝/𝑠/𝑞,𝑄₅=𝑀₁𝑃⧵𝑝⧵𝑠/𝑞,₁=𝑀₂⧵𝑝⧵𝑠⧵𝑞,𝑃₂=𝑀₂𝑃⧵𝑝/𝑠⧵𝑞,₃=𝑀₂/𝑝⧵𝑠⧵𝑞,𝑃₄=𝑀₂/𝑝/𝑠/𝑞,𝑃₅=𝑀₂⧵𝑝⧵𝑠/𝑞.(3.43)
We take the vectors over ℤ[𝑥,𝑦]: 𝑞^→=𝑇_𝑄1,𝑇_𝑄2,𝑇_𝑄3,𝑇_𝑄4,𝑇_𝑄5,𝑝^→=𝑇_𝑃1,𝑇_𝑃2,𝑇_𝑃3,𝑇_𝑃4,𝑇_𝑃5^𝑡,(3.44) 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: 𝑇_𝑀1⊕^Δ_3𝑀2𝑥,𝑦=𝑞^→1𝐂𝑝−1−𝑥−𝑦+𝑥𝑦^→,(3.45) where the matrix, 𝐂, is given by 1−𝑦²−𝑥−𝑦+𝑥𝑦1−𝑦−𝑥−𝑦+𝑥𝑦1−𝑦2−𝑥−𝑦+𝑥𝑦−𝑥−𝑦+𝑥𝑦1−𝑦−𝑥−𝑦+𝑥𝑦1−𝑦11−𝑥−𝑦+𝑥𝑦−𝑥−𝑦+𝑥𝑦1−𝑥1−𝑥−𝑦+𝑥𝑦−𝑥−𝑦+𝑥𝑦1−𝑦1−𝑥−𝑦+𝑥𝑦1−𝑥−𝑦+𝑥𝑦1−𝑥1−𝑥−𝑦+𝑥𝑦2−𝑥−𝑦+𝑥𝑦−𝑥−𝑦+𝑥𝑦1−𝑥−𝑥−𝑦+𝑥𝑦1−𝑥()−𝑥−𝑦+𝑥𝑦1−𝑥²−𝑥−𝑦+𝑥𝑦1−𝑥−𝑥−𝑦+𝑥𝑦1−𝑦1−𝑥−𝑦+𝑥𝑦1−𝑥−𝑦+𝑥𝑦−𝑥−𝑦+𝑥𝑦1−𝑥1−𝑥−𝑦+𝑥𝑦.(3.46)
As an example, let us take 𝐹₈ that is the 3-sum of 𝐹₇ and 𝐹⁻₇ 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 

Similarly for 𝐹⁻₇ we have Table 2.

Table 2 

By using (3.45) with the values of Tables 1 and 2 we get 𝑇_𝐹8(𝑥,𝑦)=𝑥⁴+4𝑥³+10𝑥²+8𝑥+12𝑥𝑦+8𝑦+10𝑦²+4𝑦³+𝑦⁴.

There is a general concept of 𝑘-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 matroid 𝑀 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 𝑀𝑇_𝑀^(𝑘)𝑥,𝑦=𝑦^𝑘−1+𝑦^𝑘−2+⋯+𝑦+1^𝑟(𝑀)𝑇_𝑀𝑦^𝑘−1+𝑦^𝑘−2+⋯+𝑦+𝑥𝑦^𝑘−1+𝑦^𝑘−2+⋯+𝑦+1,𝑦^𝑘.(3.47)
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 of 𝑀^(𝑘) is the 𝑘-thickening of a flat of 𝑀. Thus, by (3.1) 𝜒_𝑀(𝑘)𝑞,𝑡=𝜒_𝑀𝑞,𝑡^𝑘.(3.48) And thus, by using (3.4) and (3.3) we obtained the formula.
The dual operation to 𝑘-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 𝑇_𝑀(𝑘)𝑇_𝑀(𝑘)𝑥,𝑦=𝑥^𝑘−1+𝑥^𝑘−2+⋯+𝑥+1^{𝑟(𝑀∗)}𝑇_𝑀𝑥^𝑘,𝑥^𝑘−1+𝑥^𝑘−2+⋯+𝑥+𝑦𝑥^𝑘−1+𝑥^𝑘−2.+⋯+𝑥+1(3.49)
More generally we have the following operation called the tensor product. A pointed matroid 𝑁_𝑑 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 𝑁_𝑑, the tensor product 𝑀⊗𝑁_𝑑 is the matroid obtained by taking a 2-sum of 𝑀 with 𝑁_𝑑 at each point 𝑒 of 𝑀 and the distinguished point 𝑑 of 𝑁_𝑑. The Tutte polynomial of 𝑀⊗𝑁_𝑑, where 𝑀=(𝐸,𝑟) is then given by 𝑇_{𝑀⊗𝑁𝑑}𝑥,𝑦=𝑓^{|𝐸|−𝑟(𝐸)}𝑔^𝑟(𝐸)𝑇_𝑀(𝑥−1)𝑓+𝑔𝑔,𝑓+𝑦−1𝑔𝑓,(3.50) where 𝑓=𝑓(𝑥,𝑦) and 𝑔=𝑔(𝑥,𝑦) are polynomials which are determined by the equations (𝑥−1)𝑓𝑥,𝑦+𝑔𝑥,𝑦=𝑇_{𝑁𝑑⧵𝑑},𝑓+𝑔𝑥,𝑦𝑥,𝑦𝑦−1𝑥,𝑦=𝑇_{𝑁𝑑/𝑑}𝑥,𝑦.(3.51)
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 𝑈_𝑘,𝑘+1, 𝑘≥1, we get the 𝑘-stretch and if 𝑁 is 𝑈_1,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.

Matroid 𝑈_2,4
The Tutte polynomial of the uniform matroid 𝑈_2,4 see Figure 9, can be computed using (2.3) 𝑇_𝑈2,4𝑥,𝑦=𝑥²+2𝑥+2𝑦+𝑦².(4.1)
Also this matroid is the 2-whirl 𝑊² so you can check this result using (3.34).

Figure 9 

The uniform matroid 𝑈2,4.

Matroids 𝑊₃, 𝑊₄, 𝑊³, and 𝑊⁴
The 3-wheel 𝑀(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 𝑇_𝑊3𝑥,𝑦=𝑥³+3𝑥²+2𝑥+4𝑥𝑦+2𝑦+3𝑦²+𝑦³.(4.2)
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 𝑇_𝑊³𝑥,𝑦=𝑥³+3𝑥²+3𝑥+3𝑥𝑦+3𝑦+3𝑦²+𝑦³.(4.3)
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 𝑇_𝑊4𝑥,𝑦=1+𝑥+𝑦+1+𝑥+𝑦²−4𝑥𝑦⁴+161+𝑥+𝑦−1+𝑥+𝑦²−4𝑥𝑦⁴16+𝑥𝑦−𝑥−𝑦−1=𝑥⁴+4𝑥³+6𝑥²+3𝑥+4𝑥²𝑦+4𝑥𝑦²+9𝑥𝑦+3𝑦+6𝑦²+4𝑦³+𝑦⁴,𝑇_𝑊⁴𝑥,𝑦=𝑥⁴+4𝑥³+6𝑥²+4𝑥+4𝑥²𝑦+4𝑥𝑦²+8𝑥𝑦+4𝑦+6𝑦²+4𝑦³+𝑦⁴.(4.4)

Figure 10 

The 4-wheel.

Figure 11 

The 4-whirl.

Matroids 𝑄₆, 𝑃₆, 𝑅₆, and 𝑈_3,6
We use (2.6) to compute the Tutte polynomial of 𝑄₆, 𝑃₆, and 𝑈_3,6, see Figure 12 for a geometric representation of these matroids. The matroid 𝑈_3,6 is uniform, from (2.3) we get 𝑇_𝑈3,6𝑥,𝑦=𝑥³+3𝑥²+6𝑥+6𝑦+3𝑦²+𝑦³.(4.5) As 𝑈_3,6 is the only relaxation of 𝑃₆, from (2.6) its Tutte polynomial is 𝑇_𝑃6𝑥,𝑦=𝑥³+3𝑥²+5𝑥+𝑥𝑦+5𝑦+3𝑦²+𝑦³.(4.6) Similarly, 𝑃₆ is the only relaxation of 𝑄₆. Thus, from the above equation and (2.6) we obtain 𝑇_𝑄6𝑥,𝑦=𝑥³+3𝑥²+4𝑥+2𝑥𝑦+4𝑦+3𝑦²+𝑦³.(4.7)
It is worth mentioning that 𝑃₆ 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 𝑇_𝑅6𝑥,𝑦=(𝑥−1)³+6(𝑥−1)²+13𝑥−10+2𝑥𝑦+13𝑦+6𝑦−1²+𝑦−1³=𝑥³+3𝑥²+4𝑥+2𝑥𝑦+4𝑦+3𝑦²+𝑦³.(4.8)

Figure 12 

𝑀(𝐾4), 𝑊3, 𝑄6, 𝑃6, and 𝑈3,6.

Figure 13 

The matroid 𝑅6.

Matroids 𝐹₇, 𝐹⁻₇, and Their Duals
One of the most mentioned matroid in the literature is the Fano matroid 𝐹₇, 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 𝑇_𝐹7𝑥,𝑦=(𝑥−1)³+7(𝑥−1)²+14𝑥−21+7𝑥𝑦+28𝑦+21𝑦−1²+7𝑦−1³+𝑦−1⁴=𝑥³+4𝑥²+3𝑥+7𝑥𝑦+3𝑦+6𝑦²+3𝑦³+𝑦⁴.(4.9) 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 𝐼₃110110110111.(4.10) By using (2.2), the Tutte polynomial of 𝐹^∗₇ is 𝑇_𝐹^∗₇𝑥,𝑦=𝑥⁴+3𝑥³+6𝑥²+3𝑥+7𝑥𝑦+3𝑦+4𝑦²+𝑦³.(4.11)

Figure 14 

The Fano matroid and its dual.

The matroids 𝐹⁻₇ and (𝐹⁻₇)^∗ are the corresponding relaxations of 𝐹₇ 𝑦 𝐹^∗₇, thus we get that 𝑇_𝐹⁻₇𝑥,𝑦=𝑥³+4𝑥²+4𝑥+6𝑥𝑦+4𝑦+6𝑦²+3𝑦³+𝑦⁴,𝑇_(𝐹⁻₇₎^∗𝑥,𝑦=𝑥⁴+3𝑥³+6𝑥²+4𝑥+6𝑥𝑦+4𝑦+4𝑦²+𝑦³.(4.12)

Matroids 𝑃₇,𝑃₈, and 𝑄₃
The matroid 𝑃₇, 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, 𝑇_𝑃7𝑥,𝑦=𝑥³+4𝑥²+5𝑥+5𝑥𝑦+5𝑦+6𝑦²+3𝑦³+𝑦⁴.(4.13) The matrix that represents 𝑃₇ over a field different from GF(2) is 𝐼₃10111101𝑎1𝑎−10(4.14) 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 of 𝑃7 and 𝑃8.

The matroid 𝑃₈, shown on the right side of Figure 15, is also sparse paving and its representation over GF(3) is as below, 𝐼₄011−110111101−1110.(4.15) 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 𝑇_𝑃8𝑥,𝑦=𝑥⁴+4𝑥³+10𝑥²+10𝑥+10𝑥𝑦+10𝑦+10𝑦²+4𝑦³+𝑦⁴.(4.16)

The rank-3 ternary Dowling geometry 𝑄₃ has geometric representation shown in Figure 16.

Figure 16 

Geometric representation of 𝑄3.

The matroid 𝑄₃ 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 ||||.1234567891001111000101−10011001001−1−11(4.17) Thus, we can compute its Tutte polynomial using [19] and obtain 𝑇_𝑄3𝑥,𝑦=𝑥³+6𝑥²+8𝑥+3𝑥𝑦²+10𝑥𝑦+8𝑦+12𝑦²+10𝑦³+6𝑦⁴+3𝑦⁵+𝑦⁶.(4.18) Note, however, that this matroid is paving, so we could have used Proposition 2.8 to get the same result.

Matroid 𝑊³₊
The matroid 𝑊³₊ 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 𝑇_𝑊³₊𝑥,𝑦=𝑇_𝑊³𝑥,𝑦+𝑦⋅𝑇_𝑈2,4𝑥,𝑦+𝑇_{𝑈1,3⊕𝑈0,2}=𝑥𝑥,𝑦₃+3𝑥²+3𝑥𝑦+3𝑥+3𝑦+3𝑦²+𝑦³+𝑦⋅𝑥²+2𝑥+2𝑦+𝑦²+𝑦⁴+𝑦³+𝑥𝑦²=𝑥³+3𝑥²+3𝑥+𝑥²𝑦+5𝑥𝑦+𝑥𝑦²+3𝑦+5𝑦²+3𝑦³+𝑦⁴.(4.19)

Figure 17 

Deletion and contraction reduction for 𝑊3+.

Matroids 𝐴𝐺(3,2), 𝐴𝐺(3,2)^′, 𝑅₈, 𝑄₈, 𝐹₈, and 𝐿₈
The second affine plane that we consider here is AG(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 𝑇_AG(3,2)𝑥,𝑦=𝑥⁴+4𝑥³+10𝑥²+6𝑥+14𝑥𝑦+6𝑦+10𝑦²+4𝑦³+𝑦⁴.(4.20)
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 𝑇_AG(3,2)′𝑥,𝑦=𝑥⁴+4𝑥³+10𝑥²+7𝑥+13𝑥𝑦+7𝑦+10𝑦²+4𝑦³+𝑦⁴.(4.21)
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 plane AG(3,2).

Figure 19 

The cube with 8 points.

Figure 20 

The matroid 𝐹8.

The matroid 𝑅₈ 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 𝑇_𝑅8𝑥,𝑦=𝑇_𝐹8𝑥,𝑦=𝑥⁴+4𝑥³+10𝑥²+8𝑥+12𝑥𝑦+8𝑦+10𝑦²+4𝑦³+𝑦⁴.(4.22)

The unique relaxation of 𝑅₈ is 𝑄₈ and is obtained by relaxing one of the six diagonal planes of 𝑅₈. From the previous equation and (2.6) we get 𝑇_𝑄8𝑥,𝑦=𝑥⁴+4𝑥³+10𝑥²+7𝑥+11𝑥𝑦+7𝑦+10𝑦²+4𝑦³+𝑦⁴.(4.23) 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 𝑇_𝐿8𝑥,𝑦=(𝑥−1)⁴+8(𝑥−1)³+28(𝑥−1)²+48𝑥−42+8𝑥𝑦+48𝑦+28𝑦−1²+8𝑦−1³+𝑦−1⁴=𝑥⁴+4𝑥³+10𝑥²+12𝑥+8𝑥𝑦+12𝑦+10𝑦²+4𝑦³+𝑦⁴.(4.24)

Matroids 𝑆₈,𝑇₈, and 𝐽
The matroid 𝑆₈ 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 𝑇_𝑆8𝑥,𝑦=𝑥⁴+4𝑥³+7𝑥²+4𝑥+10𝑥𝑦+3𝑥𝑦²+3𝑥²𝑦+4𝑦+7𝑦²+4𝑦³+𝑦⁴.(4.25)
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 of 𝐹₇, 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 𝐾_2,4 with an edge contracted. If we call this graph 𝐻, by using (2.12) we obtain 𝑇_𝐻=𝑇_𝐾2,4−𝑦𝑇_𝐾2,3. 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 𝑇_𝐻=𝑥⁴+3𝑥³+3𝑥²+𝑥+3𝑥𝑦+3𝑥²𝑦+3𝑥𝑦²+𝑦+𝑦²+𝑦³.(4.26) By adding (4.26) and (4.9) we get (4.25).

(a)

(b)

Figure 21 

On the left-hand side we give the geometric representation of the matroid 𝑆8 and on the right-hand side the matrix that represents it over GF(2).

The matroid 𝑇₈ 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 𝑇_𝑇8𝑥,𝑦=𝑥⁴+4𝑥³+10𝑥²+9𝑥+11𝑥𝑦+9𝑦+10𝑦²+4𝑦³+𝑦⁴.(4.27)

Figure 22 

The matroid 𝑇8.

In Figure 23 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 𝐼₄1001111001010011,(4.28) 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 𝑇_𝐽𝑥,𝑦=𝑥⁴+4𝑥³+7𝑥²+6𝑥+3𝑥²𝑦+3𝑥𝑦²+8𝑥𝑦+6𝑦+7𝑦²+4𝑦³+𝑦⁴.(4.29)

Figure 23 

The matroid 𝐽.

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 of 𝐶₄ 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 Figure 24 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 𝑉⁺₈ 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 𝑉⁺₈ by relaxing {5,6,7,8}. Notice that both are self-dual. Thus the Tutte polynomial of these two matroids are 𝑇_𝑉8𝑥,𝑦=(𝑥−1)⁴+8(𝑥−1)³+82(𝑥−1)²+83()+84+85+86𝑥−1+5𝑥𝑦−𝑥−𝑦𝑦−1𝑦−1²+8𝑦−1³+𝑦−1⁴=𝑥⁴+4𝑥³+10𝑥²+15𝑥+5𝑥𝑦+15𝑦+10𝑦²+4𝑦³+𝑦⁴,𝑇_𝑉⁺₈𝑥,𝑦=𝑥⁴+4𝑥³+10𝑥²+14𝑥+6𝑥𝑦+14𝑦+10𝑦²+4𝑦³+𝑦⁴.(4.30)

Figure 24 

Geometric representation of 𝑉8.

Matroids 𝑅₈,𝑅₉,𝑅₁₀, and 𝑅₁₂
The real affine cube, 𝑅₈, is represented over all fields of characteristic other than two by the matrix 𝐼₄−11111−11111−11111−1.(4.31) The matroid is sparse paving and its Tutte polynomial is 𝑇_𝑅8𝑥,𝑦=𝑥⁴+4𝑥³+10𝑥²+8𝑥+12𝑥𝑦+8𝑦+10𝑦²+4𝑦³+𝑦⁴.(4.32)
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 𝐼₃1111111−1−1−110001−11−1(4.33) and its Tutte polynomial can be computed using the program in [19]. Consider 𝑇_𝑅9𝑥,𝑦=𝑥³+6𝑥²+8𝑥+11𝑥𝑦+2𝑥𝑦²+8𝑦+13𝑦²+10𝑦³+6𝑦⁴+3𝑦⁵+𝑦⁶.(4.34) 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(𝐾_3,3), and every single-element contraction is isomorphic to 𝑀^∗(𝐾_3,3), then by using the polynomial in (3.18) and (2.2) we obtain 𝑇_𝑅10𝑥,𝑦=𝑥⁵+5𝑥⁴+15𝑥³+20𝑥²+10𝑥+15𝑥²𝑦+30𝑥𝑦+15𝑥𝑦²+10𝑦+20𝑦²+15𝑦³+5𝑦⁴+𝑦⁵.(4.35)
Another important regular matroid that is neither graphic or cographic is 𝑅₁₂, which has a matrix representation over GF(2) given by 𝐼₆111000110100100010010001001011000111.(4.36) As this matroid is not paving we use the program in [19] to compute its Tutte polynomial. One has 𝑇_𝑅12𝑥,𝑦=𝑥⁶+6𝑥⁵+19𝑥⁴+35𝑥³+35𝑥²+14𝑥+2𝑥⁴𝑦+19𝑥³𝑦+53𝑥²𝑦+17𝑥²𝑦²+56𝑥𝑦+53𝑥𝑦²+19𝑥𝑦³+2𝑥𝑦⁴+14𝑦+35𝑦²+35𝑦³+19𝑦⁴+6𝑦⁵+𝑦⁶.(4.37)

Pappus and Non-Pappus Matroids
The geometric representation of the Pappus 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 𝑇_𝑀𝑥,𝑦=(𝑥−1)³+9(𝑥−1)²+27𝑥−78+9𝑥𝑦+117𝑦+126𝑦−1²+84𝑦−1³+36𝑦−1⁴+9𝑦−1⁵+𝑦−1⁶=𝑥³+6𝑥²+12𝑥+9𝑥𝑦+12𝑦+15𝑦²+10𝑦³+6𝑦⁴+3𝑦⁵+𝑦⁶.(4.38)
The non-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 𝑇_𝑀𝑥,𝑦=𝑥³+6𝑥²+12𝑥+9𝑥𝑦+12𝑦+15𝑦²+10𝑦³+6𝑦⁴+3𝑦⁵+𝑦⁶−𝑥𝑦+𝑥+𝑦=𝑥³+6𝑥²+13𝑥+8𝑥𝑦+13𝑦+15𝑦²+10𝑦³+6𝑦⁴+3𝑦⁵+𝑦⁶.(4.39)

Figure 25 

Geometric representation of the Pappus matroid.

Figure 26 

Geometric representation of the non-Pappus matroid.

Matroid Non-Desargues
The non-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 𝑇_𝑀𝑥,𝑦=𝑥³+7𝑥²+19𝑥+9𝑥𝑦+19𝑦+21𝑦²+15𝑦³+10𝑦⁴+6𝑦⁵+3𝑦⁶+𝑦⁷.(4.40)

Figure 27 

Geometric representation of the non-Desargues matroid.

The Steiner Systems 𝑆(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 system 𝑆(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 𝑇_𝑆(2,3,13)𝑥,𝑦=𝑥³+10𝑥²+29𝑥+26𝑥𝑦+29𝑦+45𝑦²+36𝑦³+28𝑦⁴+21𝑦⁵+15𝑦⁶+10𝑦⁷+6𝑦⁸+3𝑦⁹+𝑦¹⁰.(4.41)
The Steiner system 𝑆(5,6,12) has rank 6, 12 elements, and 132 circuit-hyperplanes. The Tutte polynomial is, by using (2.7), 𝑇_𝑆(5,6,12)𝑥,𝑦=𝑥⁶+6𝑥⁵+21𝑥⁴+56𝑥³+126𝑥²+120𝑥+132𝑥𝑦+120𝑦+126𝑦²+56𝑦³+21𝑦⁴+6𝑦⁵+𝑦⁶.(4.42)

Acknowledgment

This work was supported by CONACyT of Mexico project 83977.