International Journal of CombinatoricsVolume 2012 (2012), Article ID 430859, 40 pagesdoi:10.1155/2012/430859

Review Article

The Tutte Polynomial of Some Matroids

Criel Merino,¹ Marcelino Ramírez-Ibáñez,² and Guadalupe Rodríguez-Sánchez³

¹Instituto de Matemáticas, Universidad Nacional Autónoma de México, Area de la Investigación Científica, Circuito Exterior, C.U., Coyoácan, 04510 México City, DF, Mexico²Escuela de Ciencias, Universidad Autónoma Benito Juárez de Oaxaca, 68120 Oaxaca, OAX, Mexico³Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana, Azcapozalco, Avenue San Pablo No. 180, Col. Reynosa Tamaulipas, Azcapotzalco, 02200 México City, DF, Mexico

Received 23 February 2012; Accepted 10 July 2012

Academic Editor: Cai Heng Li

Copyright © 2012 Criel Merino et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 as the graphic matroid .

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 is a matroid, where is the rank function of , and , we denote by and by , the latter is called the nullity of .

Definition 2.1. The Tutte polynomial of , , is defined as follows:

DualityRecall that if is a matroid, then is its dual matroid, where . Because and , you get the following equality This gives the first technique to compute a Tutte polynomial. If you know the Tutte polynomial of , then, you know the Tutte polynomial of the dual matroid .

Uniform MatroidsOur first example is the family of uniform matroids , where . Here the Tutte polynomial can be computed easily using (2.1) because all subsets of size are independent, so is zero; all subsets of size are spanning, so is zero and if a subset is independent and spanning, then it is a basis of the matroid. Consider the following Thus, for we get As , we get by using (2.2)

Matroid RelaxationGiven a matroid with a subset that is both a circuit and a hyperplane, we can define a new matroid as the matroid with basis . That is indeed a matroid is easy to check. For example, is the unique relaxation of . The Tutte polynomial of can be computed easily from the Tutte polynomial of by using (2.1) one has. Matroid relaxation will be used extensively in the last section.

Sparse Paving MatroidsOur second example can be considered a generalization of uniform matroids but it is a much larger and richer family. A paving matroid is a matroid whose circuits all have size at least . 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 , 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- matroid is sparse paving if is paving and for every pair of circuits and of size we have . 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 be a paving matroid of rank . Then is sparse paving if and only if all the hyperplanes of have size or .

Note that we can say a little more, any circuit of size is a hyperplane. Conversely, any proper subset of a hyperplane of size is independent and so such a hyperplane must be a circuit. Thus, the circuits of size are precisely the hyperplanes of size .

Many invariants that are usually difficult to compute for a general matroid are easy for sparse paving matroids. For example, observe that if is sparse paving, all subsets of size are independent, and all subsets of size are spanning. On the other hand, the subsets of size are either bases or circuit hyperplanes. Thus, the Tutte polynomial of a rank- sparse matroid with elements and circuit-hyperplanes is given by

Clearly, given a -rank paving matroid with elements, we can obtain the uniform matroid by a sequence of relaxations from . If has circuits-hyperplanes, (2.6) also gives (2.7).

Free ExtensionAnother easy formula that we can obtaine from the above definition involves the free extension, , of a matroid by an element , which consists of adding the element to as independently as possible without increasing the rank. Equivalently, the rank function of is given by the following equations: for a subset of , Again, the Tutte polynomial of can be computed easily from the Tutte polynomial of by using (2.1). Consider Here the trick is to notice that , where the sum is over all subsets of with , that is, spanning subsets of . The presentation given here is from [13], but the formula can also be found in [14].For example, the graphic matroid has as free extention the matroid . Because is sparse paving, by using (2.7), we can compute its Tutte polynomial and obtain Now, by using (2.9), we get the Tutte polynomial of the free extension: This result can be checked by using (2.7), as 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 is a matroid, and is an element that is neither a loop nor a coloop, then If is a coloop, then If is a loop, then

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 and , then you know the Tutte polynomial of the matroid . This way of computing the Tutte polynomial leads naturally to linear recursions. We present two examples where these linear recursions give formulas.

The Cycle As a first example of this subsection we consider the graphic matroid . Here, by deleting an edge from we obtained a path whose Tutte polynomial is , while if we contract we get a smaller cycle . Thus, as the Tutte polynomial of the 2-cycle is , we obtain By duality, the Tutte polynomial of the graph with two vertices and parallel edges between them is

Parallel and Series ClassesAs 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 as maximal subset of such that any two distinct members of are parallel and no member of 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 be a parallel class of a matroid with . If is not a cocircuit, then

The proof is by induction on with the case being (2.12). The rest of the proof follows easily from the fact that each loop introduces a multiplicative factor of in the Tutte polynomial.

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

Lemma 2.5. Let be a series class of a matroid with . If is not a circuit, then

The Rectangular Lattice 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 and be integers, . The grid or rectangular lattice is a connected planar graph with vertices, such that its faces are squares, except one face that is a polygon with edges. The grid graph can be represented as in Figure 1.

Figure 1: The grid graph .

The vertices of are denoted as ordered pairs, such that the vertex in the intersection of the row and the column is denoted by .

As an example, we show a recursive formula for the Tutte polynomials of grid graphs when . For , the formulas are very complicated to use in practical calculations.

The graphs , , and are shown in Figure 2, with the labels that correspond to their vertices.

Figure 2: The first three terms are from the sequence and the last three are from .

For every positive integer . The graph is defined from the graph by contraction of the edge of . In particular, the graph is one isolated vertex, see Figure 2.

The initial conditions to construct the recurrence relations for the Tutte polynomial of the grid graphs are.(i);(ii); (iii).

The third condition is true because the graph is isomorphic to the cycle , so we can use (2.15). The formula for the recurrence relation to calculate the Tutte polynomial of the graphs is

From this, we get a linear recurrence of order two, that is easy to solve. Thus, the general formula is for , where , , and and are

A recursive family of graphs is a sequence of graphs with the property that the Tutte polynomials satisfy a linear homogeneous recursion relation in which the coefficients are polynomials in and with integral coefficients, independent of , see [17]. So, the sequence is such a family and in general, , for a fixed , is a recursive family.

Graphic and Representable MatroidsProbably the most common class of matroids, when evaluations of the Tutte polynomial are considered, is graphic matroids. If is the graphic matroid of a graph , then deleting an element from amounts to deleting the corresponding edge from , and contracting 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 , representable over a field , you can compute its Tutte polynomial using deletion and contraction. For any element , the matroids and are easily computed from the representation of , 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 , say , where if . Given a fixed basis , an element is called internally active if and it is the smallest edge with respect to in the only cocircuit disjoint from . Dually, an element is externally active if and it is the smallest element in the only circuit contained in . We define to be the number of bases with internally active elements and externally active elements. In [20–22] Tutte defined using these concepts. A proof of the equivalence with Definition 2.1 can be found in [23].

Definition 2.6. If is a matroid with a total order on its ground set, then In particular, the coefficients are independent of the total order used on the ground set.

Uniform Matroids AgainOur first use of (2.23) is to get a slightly different expression for the Tutte polynomial of uniform matroids. This time we get when , while and . This can also be established by expanding (2.3).

Paving MatroidsPaving 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 is large, the probability that you choose a paving matroid uniformly at random among all matroids with up to 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 and , a collection of subsets of a set , such that each member of has at least elements and each -element subset of is contained in a unique member of and is called an -partition of . The elements of the partition are called blocks. The following proposition is also from [9].

Proposition 2.7. If is an -partition of , then is the set of hyperplanes of a paving matroid of rank on . Moreover, for , the set of hyperplanes of every rank- paving matroid on is an -partition of .

Brylawski gives the following proposition in [25], but we mentioned that his proof does not use activities.

Proposition 2.8. Let be a rank- matroid with elements and Tutte polynomial . Then, is paving if and only if for all . In addition if the -partition of has blocks of cardinality , for , then and for all ,

Note that when is the uniform matroid , is a paving matroid with -partition all subsets of size . Thus the above formula gives us (2.24).

As an example, consider the matroid 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 is

Figure 4: The matroid .

Catalan MatroidsOur 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 . The length of a word is , the number of letters in . Every word can be associated with a path on the plane , with an initial point and a final point , and . The letter is identified with a north step and the letter with an east step, such that the first step of to will be or .Let and be a couple of fixed points of . A lattice path is a path in from to using only steps or . The lattice paths from to have steps and steps . Let and be two lattice paths, with initial points and of and , respectively. If for every in ; then the set of lattice paths in this work is lattice paths from to bounded between and , that we called -bounded lattice paths.The lattice paths and can be substituted by lines. If is the line , the line and , then the number of -bounded lattice paths from to is the th Catalan number: Denote by the set . Consider the word that corresponds to a -bounded lattice path. We can associate to a subset of by defining that if . is the support set of the steps in . The family of support sets of -bounded lattice paths is the set of bases of a transversal matroid, denoted with ground set , see [13].If the bounds of the lattice paths are the lines and , the matroid is denoted by and is named Catalan matroid. Note that has a loop that corresponds to the label and a coloop that corresponds to the label , and that is autodual.The fundamental result to compute the Tutte polynomial of Catalan matroids is the following, see [13], where here, for a basis , we denote by the number of internally active elements and by the externally active elements.

Proposition 2.9. Let be a basis of and let be the -bounded lattice path associated with . Then is the number of times meets the upper path in a north step and is the number of times meets the lower path in an east step.

Then the Tutte polynomial of the Catalan matroids for , is

Note that the coefficient of in the Tutte polynomial of the matroid depends only on and the sum .

In the Figure 5 are shown the bases of with its internal and external activities. Thus the Tutte polynomial of is

Figure 5: Bases of 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 where is the set of flats of and is the characteristic polynomial of the matroid . The characteristic polynomial of a matroid is defined by

The Tutte and the coboundary polynomial are related as follows:

When is a graphic matroid of a connected graph , the characteristic polynomial is equivalent to the chromatic polynomial, that we denote :

And also, in this case, the coboundary is the bad-colouring polynomial. The bad-colouring polynomial is the generating function where is the number of -colourings of with exactly bad edges. Note that when we set we obtain the chromatic polynomial. Then, the polynomials are related as follows:

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

Definition 3.1. Let be a rank- simple matroid representable over . A matroid is a -cone of with base and apex if(1)the restriction of to the subset is isomorphic to ,(2)the point is not contained in the closure of , , in , and(3)the matroid is the restriction of to the set .

That is, one represents as a set of points in and constructs by restricting to the set of points on the lines joining the points of to the fixed point outside the hyperplane of spanned by . The basic result is by Kung who proved it in [28].

Theorem 3.2. For every -cone of a rank- simple matroid , one has

To extend this result to , the authors in [27] identify the flats of and the contractions of by these flats. They prove the following formula By using (3.3) we get a formula for the Tutte polynomial of in terms of the Tutte polynomial of .

Theorem 3.3. If is a rank- matroid representable over and is a -cone of , then

Thus, for example, the -cone of the 3-point line is the matroid . The Tutte polynomial of the former matroid is just , then by using the above formula we get the Tutte polynomial of to be As the -cone of is , this method can be used to compute the Tutte polynomial of any , but we do this later using the coboundary polynomial directly.

Complete GraphsGiven 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 by and by its bad colouring. To compute , observe that any -colouring partitions the vertices into color classes of subsets of vertices each of cardinality , for . So, we have that . The number of bad edges with both ends in the set is . Thus, by the exponential formula we get the following formula Now, we can use (3.7) to get Let be the Tutte polynomial of . Tutte in [22] and Welsh in [30] give the following exponential generating function for that follows from the previous equation and (3.3)

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 in no time:Coboundary:= proc(n,q,v) local i,x,g;                            g:=x->(add(v(i*(i-1)/2)*xi/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 we obtain

Complete Bipartite GraphsIt is just natural to use this technique for complete bipartite graphs with similar results as above. This time let the vertex set be and denote by its bad-colouring polynomial. To compute , observe that any -colouring partitioned the vertices in subsets each of cardinality . The number of the bad edges with both ends colour is . Thus, by the exponential formula we get the following: Thus, a formula for the Tutte polynomial of the bipartite complete graph can be found in a similar way as before. Let be the Tutte polynomial of . The following formula is from Stanley’s book [29], see also [31] As before, (3.16) is quite easy to use in Maple to compute for small values of and . In this way we get the following:

Projective Geometries and Affine GeometriesThe 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 and we define the Gaussian coefficients as Note that since by convention an empty product is 1 and .

Let be the dimensional projective geometry over . Then, as a matroid, it has rank and elements. Also, every rank- flat is isomorphic to and the simplification of is isomorphic to . The number of rank- flats is . The characteristic polynomial of is known to be, see [4], Thus, using (3.1) we get The matroid has a geometric representation given in Figure 6, and it is isomorphic to the unique Steiner system . The matroid is paving but not sparse paving and has the following representation over : To compute its Tutte polynomial we could use the program in [19] or use the above formula to get

Figure 6: The projective plane with a marked hyperplane.

Finally the Tutte polynomial of is obtained from the coboundary by a substitution and and by multiplying by the factor :

Every time we have a projective geometry we can get an affine geometry, , simply by deleting all the points in a hyperplane of . For example, by deleting all the points in the red line from in Figure 6 we get with geometric representation in Figure 7.

Figure 7: Affine plane .

The situation to compute the Tutte polynomial is very similar for the -dimensional affine geometry over , . It has rank and elements. Any flat of rank is isomorphic to and the simplification of is isomorphic to . The number of rank- flats is . The characteristic polynomial of is known to be, see [4], Now, using the result of Mphako we can compute the coboundary polynomial of :

The affine plane is isomorphic to the unique Steiner triple system , as every line contains exactly 3 points and every pair of points is in exactly one line. 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.

Again, the Tutte polynomial of is obtained from the coboundary by a substitution and and by multiplying by the factor

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 is a triple , where is a set of vertices, is a finite set of directed edges or arcs, and is a map from to . If , then is called an edge from to , with initial vertex and final vertex . A directed walk in of length from to is a sequence of edges such that the final vertex of is the initial vertex of , for .

Now let be a weight function on with values in some commutative ring . If is a walk, then the weight of is defined by . For and , we define where the sum is over all walks in of length from to . In particular, . The fundamental problem treated by the transfer-matrix method is the evaluation of . The idea is to interpret as an entry in a certain matrix. Define a matrix by where the sum is over all edges satisfying that its initial vertex is and its final vertex is . In other words, . The matrix is called the adjacency matrix of , with respect to the weight function .

Theorem 3.4. Let . Then the -entry of is equal to . Here one defines even if is not invertible, where is the identity matrix.

Proof. See [34].

Rectangular Lattice AgainThe transfer-matrix method gives us another way to compute for a fixed width at point (,) which is described in Calkin et al. [35]. In this case we have the same restriction as before, a fixed width for small values of , but it has the advantage of being easily automatized.

Theorem 3.5 (see, Calkin et. al. [35]). For indeterminates and and integers , fixed, one has where , a vector of length , and , a matrix, depend on ,, and but not . And is the vector of length with all entries equal to 1.

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

For example, for we get that equals

Wheels and WhirlsThe 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, . The graph has vertices and edges, see Figure 8. The rim of the wheel graph is a circuit hyperplane of the corresponding graphic matroid, the relaxation of this circuit-hyperplane gives the matroid , the whirl matroid on elements. In [36], Chang and Shrock using results from [37] compute the Tutte polynomial of as follows:

Figure 8: The -wheel.

From this expression, it is easy to compute an expression for the Tutte polynomial of whirls using (2.6). Consider

Now, the way Chang and Shrock compute 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 when we have 3 colours. For this we define the matrix The idea is that the entry of the matrix will contain all the contributions of bad edges to the bad colouring polynomial for all the colourings , with , , and for the fan graph , obtained from by deleting one edge from the rim. Then, to get the bad-colouring polynomial of we just add all the entries in and multiply by . To get the bad-colouring polynomial of we take the trace of , and multiply by . 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 is , so . By computing the eigenvalues of , we obtain the bad colouring polynomial of with

For arbitrary we need to compute the eigenvalues of the matrix of given below

The eigenvalues of the matrix are each with multiplicity 1 and with multiplicity . Thus the bad colouring polynomial of is . 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 , a partition of is an exact -separation, for a positive integer, if Now, a matroid can be written as a 1-sum of two of its proper minors if and only if has an exact 1-separation, and can be written as a 2-sum of two of its proper minors if and only if 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 has an exact 3-separation , with , , then there are binary matroids and such that .

1-SumFor matroids , on disjoint sets the direct sum is the matroid , where is the union of the ground sets and for all in .Directly from (2.1) it follows that the Tutte polynomial of the 1-sum is given by

2-SumLet and be matroids with . If is not a loop or an isthmus in or , then the 2-sum    of and is the matroid on whose collection of independent sets is , , and either or .In [9, 44, 45], we find recursive formulas for computing the Tutte polynomial of the matroid , in terms of the matroids and . Here we present the formula given in [44] for where here we omit the variables of each Tutte polynomial to avoid a cumbersome notation.As an example, let us take that is the 2-sum of with itself. Observe that the election of the base point is irrelevant as its automorphism group is the symmetric group. The matroid and . Then, by using (3.41), we get that equals Thus, we obtain that .

3-SumThe best known variant of a 3-sum of two matroids is called -, see [43]. For -connected matroids and , the -sum is usually denoted . When and are graphic matroids with corresponding graphs being and , the graph is obtained by identifying a triangle of of with a triangle of into a triangle , called the connector triangle. Finally, is obtained by deleting the edges in the connector triangle.Here, we present a formula to compute the Tutte polynomial of the -sum of and , in terms of the Tutte polynomial of certain minors of the original matroids. The expression for 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 and be two matroids defined on and , respectively. Let be equal to , a -circuit and . We require that in there exist circuits with and with ; similarly, we need that in there exist circuits with and with .For , we consider the following minors of , on , and minors of , on We take the vectors over : again here we omit the variables of each Tutte polynomial to avoid a cumbersome notation.Finally, the formula for the 3-sum (-sum) of and is as follows: where the matrix, , is given by As an example, let us take that is the 3-sum of and along a 3-circuit. In this case we have for 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

Similarly for we have Table 2.

Table 2

Minor
Matroid
Polynomial

plus a parallel edge

By using (3.45) with the values of Tables 1 and 2 we get .

There is a general concept of -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 ProductGiven a matroid and a positive integer , the matroid is the matroid obtained from by replacing each nonloop element by parallel elements and replacing each loop by loops. The matroid is called the -thickening of . In [4], the following formula is given for the Tutte polynomial of in terms of that of 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 is the -thickening of a flat of . Thus, by (3.1) And thus, by using (3.4) and (3.3) we obtained the formula.The dual operation to -thickening is that of -stretch that is defined similarly. The matroid is the matroid obtained by replacing each nonisthmus of by elements in series and replacing each isthmus by isthmuses. The matroid is called the -stretch. It is not difficult to prove that and so, we obtained the corresponding formula for More generally we have the following operation called the tensor product. A pointed matroid is a matroid on a ground set which includes a distinguished element, the point , which will be assumed to be neither a loop nor coloop. For an arbitrary matroid and a pointed matroid , the tensor product is the matroid obtained by taking a 2-sum of with at each point of and the distinguished point of . The Tutte polynomial of , where is then given by where and are polynomials which are determined by the equations 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 has a transitive automorphism group, the choice of the distinguished point is immaterial. Thus, if is , , we get the -stretch and if is , , we get the -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 The Tutte polynomial of the uniform matroid see Figure 9, can be computed using (2.3) Also this matroid is the 2-whirl so you can check this result using (3.34).

Figure 9: The uniform matroid .

Matroids , , , and The 3-wheel , which is isomorphic to the graphic matroid , is a sparse paving matroid with circuit-hyperplanes, so by using (2.7) we get Of course, the above polynomial can be checked using (3.33). The only relaxation of is the 3-whirl , so by using (2.6) we obtain its Tutte polynomial. This can be checked by using (3.34). Consider The 4-wheel, , and 4-whirl, , 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

Figure 10: The 4-wheel.

Figure 11: The 4-whirl.

Matroids , , , and We use (2.6) to compute the Tutte polynomial of , , and , see Figure 12 for a geometric representation of these matroids. The matroid is uniform, from (2.3) we get As is the only relaxation of , from (2.6) its Tutte polynomial is Similarly, is the only relaxation of . Thus, from the above equation and (2.6) we obtain It is worth mentioning that is a relaxation of , see Figure 13. Thus, and have the same Tutte polynomial. This is not at all uncommon, see [9, 48]. Also, is sparse paving so by (2.7) its Tutte polynomial is

Figure 12: , , , , and .

Figure 13: The matroid .

Matroids , , and Their DualsOne of the most mentioned matroid in the literature is the Fano matroid , see Figure 14, also known as the projective plane or the unique Steiner system . 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 is The matroid is representable over any field of characteristic 2, so the above calculation can be checked using the program in [19] with the matrix By using (2.2), the Tutte polynomial of is

Figure 14: The Fano matroid and its dual.

The matroids and are the corresponding relaxations of , thus we get that

Matroids , and The matroid , 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, The matrix that represents over a field different from is with . Taking we have a representation of over , see [9]. Thus, the above calculation of the Tutte polynomial can be checked using the computer program in [19].

Figure 15: Geometric representation of and .

The matroid , shown on the right side of Figure 15, is also sparse paving and its representation over is as below, 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

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

Figure 16: Geometric representation of .

The matroid is representable over a field if and only if the characteristic of is different from 2. A matrix that represents over is Thus, we can compute its Tutte polynomial using [19] and obtain Note, however, that this matroid is paving, so we could have used Proposition 2.8 to get the same result.

Matroid The matroid is obtained from 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

Figure 17: Deletion and contraction reduction for .

Matroids , , , , , and The second affine plane that we consider here is , 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 ,, and , plus the two twisted planes and . Each of these planes is a circuit-hyperplane. It is not difficult to check that is isomorphic to the unique Steiner system , thus, as explained for , it is sparse paving and its Tutte polynomial is By using (2.6), we can compute the Tutte polynomial of , the unique relaxation of that here we obtained by relaxing the twisted plane . Thus, the resulting polynomial is Now, has two relaxations, and . The matroid is obtained by relaxing from the other twisted plane . On the other hand, is obtained from by relaxing a diagonal plane. The geometric representation of is shown in Figure 20.

Figure 18: The affine plane .

Figure 19: The cube with 8 points.

Figure 20: The matroid .

The matroid is representable over any field except for while is not representable; however, they have the same Tutte polynomial as both are relaxations of the same matroid. Consider

The unique relaxation of is and is obtained by relaxing one of the six diagonal planes of . From the previous equation and (2.6) we get To end this subsection we consider the matroid 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 and , see Figure 19. Thus, its Tutte polynomial is

Matroids , and The matroid has a geometric representation shown in Figure 21 together with its representation over . Its Tutte polynomial can be computed using the program in [19]. One has 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 , 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 with an edge contracted. If we call this graph , by using (2.12) we obtain . 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 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 and on the right-hand side the matrix that represents it over .

The matroid is representable over a field if and only if the characteristic is 3. We show its geometric representation in Figure 22. A representation of over is , where is the matrix of 1′s. This matroid is self-dual and sparse paving so its Tutte polynomial is

Figure 22: The matroid .

In Figure 23 we show the geometric representation of the matroid . It is a self-dual matroid that is not paving, as it has a 3-circuit, and its representation over is 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

Figure 23: The matroid .

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 minus an edge. Now, the matroid 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 MatroidThe 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 . 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 has the same ground set and is defined in the same way as but with added as a hyperplane. In fact, is obtained from by relaxing . Notice that both are self-dual. Thus the Tutte polynomial of these two matroids are

Figure 24: Geometric representation of .

Matroids , and The real affine cube, , is represented over all fields of characteristic other than two by the matrix The matroid is sparse paving and its Tutte polynomial is On the contrary, the ternary Reid geometry, , is not paving but it is representable if and only if the characteristic of the field is three. The matrix that represents over is and its Tutte polynomial can be computed using the program in [19]. Consider This was computed before in Section 2.3 with the same result.The unique 10-element regular matroid that is neither graphic nor cographic, , has the property that every single-element deletion is isomorphic to , and every single-element contraction is isomorphic to , then by using the polynomial in (3.18) and (2.2) we obtain Another important regular matroid that is neither graphic or cographic is , which has a matrix representation over given by As this matroid is not paving we use the program in [19] to compute its Tutte polynomial. One has

Pappus and Non-Pappus MatroidsThe 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 circuit-hyperplanes and its Tutte polynomial is 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

Figure 25: Geometric representation of the Pappus matroid.

Figure 26: Geometric representation of the non-Pappus matroid.

Matroid Non-DesarguesThe 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

Figure 27: Geometric representation of the non-Desargues matroid.

The Steiner Systems and Finally, we consider two Steiner systems from the Appendix of Oxley’s book. In general any Steiner system is sparse paving, see [49]. Thus, the Steiner triple system is a rank-3 sparse paving matroid with 13 elements and circuit-hyperplanes, then its Tutte polynomial is The Steiner system has rank 6, 12 elements, and 132 circuit-hyperplanes. The Tutte polynomial is, by using (2.7),

Acknowledgment

This work was supported by CONACyT of Mexico project 83977.

References

J. A. Ellis-Monaghan and C. Merino, “Graph polynomials and their applications I: the Tutte polynomial,” in Structural Analysis of Complex Networks: Theory and Applications, M. Dehmer, Ed., Birkhäuser, Boston, Mass, USA, 2011.

D. J. A. Welsh, Complexity: Knots, Colourings and Counting, Cambridge University Press, Cambridge, UK, 1993.

S. D. Noble, “The complexity of graph polynomials,” in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh, G. R. Grimmett and C. J. H. McDiarmid, Eds., Oxford University Press, Oxford, UK, 2007.

T. Brylawski and J. Oxley, “The Tutte polynomial and its applications,” in Matroid Applications, Encyclopedia of Mathematics and its Applications, N. White, Ed., Cambridge University Press, Cambridge, UK, 1992.

J. Bonin and A. de Mier, “T-uniqueness of some families of k-chordal matroids,” Advances in Applied Mathematics, vol. 32, no. 1-2, pp. 10–30, 2004.

A. de Mier and M. Noy, “On graphs determined by their Tutte polynomials,” Graphs and Combinatorics, vol. 20, no. 1, pp. 105–119, 2004.

I. Sarmiento, “A characterisation of jointless Dowling geometries,” Discrete Mathematics, vol. 197-198, pp. 713–731, 1999.

R. Diestel, Graph Theory, Graduate Texts in Mathematics, Springer, New York, NY, USA, 2000.

J. G. Oxley, Matroid Theory, Oxford University Press, New York, NY, USA, 1992.

C. Merino, S. D. Noble, M. Ramírez-Ibáñez, and R. Villarroel, “On the structure of the h-vector of a paving matroid,” European Journal of Combinatorics, vol. 33, no. 8, pp. 1787–1799, 2012.

M. Jerrum, “Two remarks concerning balanced matroids,” Combinatorica, vol. 26, no. 6, pp. 733–742, 2006.

D. Mayhew and G. F. Royle, “Matroids with nine elements,” Journal of Combinatorial Theory, vol. 98, no. 2, pp. 415–431, 2008.

J. Bonin, A. de Mier, and M. Noy, “Lattice path matroids: enumerative aspects and Tutte polynomials,” Journal of Combinatorial Theory, vol. 104, no. 1, pp. 63–94, 2003.

T. H. Brylawski, “The tutte polynomial, part 1: general theory,” in Matroid Theory and Its Applications. Proceedings of the Third International Mathematical Summer Center (C.I.M.E. 1980), A. Barlotti, Ed., 1982.

M. K. Chari, “Matroid inequalities,” Discrete Mathematics, vol. 147, no. 1–3, pp. 283–286, 1995.

B. Jackson, “An inequality for Tutte polynomials,” Combinatorica, vol. 30, no. 1, pp. 69–81, 2010.

N. L. Biggs, R. M. Damerell, and D. A. Sands, “Recursive families of graphs,” Journal of Combinatorial Theory, vol. 12, pp. 123–131, 1972.

K. Sekine, H. Imai, and S. Tani, “Computing the Tutte polynomial of a graph of moderate size,” in Lecture Notes in Computer Science, Springer, Berlin, Germany, 1995.

http://www.math.umn.edu/~reiner/Tutte/TUTTE.html

W. T. Tutte, “A ring in graph theory,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 43, pp. 26–40, 1947.

W. T. Tutte, “A contribution to the theory of chromatic polynomials,” Canadian Journal of Mathematics, vol. 6, pp. 80–91, 1954.

W. T. Tutte, “On dichromatic polynomials,” Journal of Combinatorial Theory, vol. 2, no. 3, pp. 301–320, 1967.

A. Björner, “The homology and shellability of matroids and geometric lattices,” in Matroid Applications, Encyclopedia of Mathematics and Its Applications, N. White, Ed., Cambridge University Press, Cambridge, UK, 1992.

D. Mayhew, M. Newman, D. Welsh, and G. Whittle, “On the asymptotic proportion of connected matroids,” European Journal of Combinatorics, vol. 32, no. 6, pp. 882–890, 2011.

T. H. Brylawski, “A decomposition for combinatorial geometries,” Transactions of the American Mathematical Society, vol. 171, pp. 235–282, 1972.

T. H. Brylawski, “Intersection theory for embeddings of matroids into uniform geometries,” Studies in Applied Mathematics, vol. 61, no. 3, pp. 211–244, 1979.

J. E. Bonin and H. Qin, “Tutte polynomials of q-cones,” Discrete Mathematics, vol. 232, no. 1–3, pp. 95–103, 2001.

J. P. S. Kung, “Critical problems,” in Matroid Theory, J. Bonin, J. G. Oxley, and B. Servatius, Eds., American Mathematical Society, Providence, RI, USA, 1996.

R. P. Stanley, Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge, UK, 1999.

D. J. A. Welsh, “Counting colourings and flows in random graphs,” in Combinatorics, Paul Erdös is Eighty, D. Miklós, V. T. Sós, and T. Szönyi, Eds., pp. 491–505, János Bolyai Mathematical Society, Budapest, Hungary, 1996.

J. L. Martin and V. Reiner, “Cyclotomic and simplicial matroids,” Israel Journal of Mathematics, vol. 150, pp. 229–240, 2005.

E. G. Mphako, “Tutte polynomials of perfect matroid designs,” Combinatorics, Probability and Computing, vol. 9, no. 4, pp. 363–367, 2000.

M. Barany and V. Reiner, “The Tutte polynomial of a finite projective space,” preprint

R. P. Stanley, Enumerative Combinatorics, vol. 1, Cambridge University Press, Cambridge, UK, 1997.

N. Calkin, C. Merino, S. Noble, and M. Noy, “Improved bounds for the number of forests and acyclic orientations in the square lattice,” The Electronic Journal of Combinatorics, vol. 10, no. 1, 2003.

S.-C. Chang and R. Shrock, “Zeros of Jones polynomials for families of knots and links,” Physica A, vol. 301, no. 1–4, pp. 196–218, 2001.

R. Shrock, “Exact Potts model partition functions on ladder graphs,” Physica A, vol. 283, no. 3-4, pp. 388–446, 2000.

D. J. A. Welsh and C. Merino, “The Potts model and the Tutte polynomial,” Journal of Mathematical Physics, vol. 41, no. 3, pp. 1127–1152, 2000.

L. Beaudin, J. Ellis-Monaghan, G. Pangborn, and R. Shrock, “A little statistical mechanics for the graph theorist,” Discrete Mathematics, vol. 310, no. 13-14, pp. 2037–2053, 2010.

S.-C. Chang and R. Shrock, “Exact Potts model partition function on strips of the triangular lattice,” Physica A, vol. 286, no. 1-2, pp. 189–238, 2000.

S.-C. Chang and R. Shrock, “Exact Potts model partition functions on wider arbitrary-length strips of the square lattice,” Physica A, vol. 296, no. 1-2, pp. 234–288, 2001.

S.-C. Chang and R. Shrock, “Tutte polynomials and related asymptotic limiting functions for recursive families of graphs,” Advances in Applied Mathematics, vol. 32, no. 1-2, pp. 44–87, 2004.

K. Truemper, Matroid Decomposition, Academic Press, 1992.

A. Andrzejak, “Splitting formulas for Tutte polynomials,” Journal of Combinatorial Theory, vol. 70, no. 2, pp. 346–366, 1997.

T. H. Brylawski, “A combinatorial model for series-parallel networks,” Transactions of the American Mathematical Society, vol. 154, pp. 1–22, 1971.

J. Bonin and A. de Mier, “Tutte polynomials of generalized parallel connections,” Advances in Applied Mathematics, vol. 32, no. 1-2, pp. 31–43, 2004.

F. Jaeger, D. L. Vertigan, and D. J. A. Welsh, “On the computational complexity of the Jones and Tutte polynomials,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 108, no. 1, pp. 35–53, 1990.

J. E. Bonin, “Strongly inequivalent representations and Tutte polynomials of matroids,” Algebra Universalis, vol. 49, no. 3, pp. 289–303, 2003.

D. J. A. Welsh, Matroid Theory, Dover publications, New York, NY, USA, 2010.