TheTutte and the Jones PolynomialsThe Tutte and the Jones Polynomials 4719 a graph, then V = V(G)is the vertex set of G, and E = E(G)is the edge set.An edge x,yis said to join the

Global Journal of Pure and Applied Mathematics.ISSN 0973-1768 Volume 12, Number 6 (2016), pp. 4717–4740© Research India Publicationshttp://www.ripublication.com/gjpam.htm

The Tutte and the Jones Polynomials

Young Chel Kwun

Department of Mathematics,Dong-A University,

Busan 49315, Korea.

Abdul Rauf Nizami

Division of Science and Technology,University of Education,Lahore 54000, Pakistan

Mobeen Munir

Division of Science and Technology,University of Education,Lahore 54000, Pakistan.

Waqas Nazeer

Division of Science and Technology,University of Education,Lahore 54000, Pakistan.

Shin Min Kang1

Department of Mathematics and RINS,Gyeongsang National University,

Jinju 52828, Korea.

1Corresponding author

AbstractThere is a remarkable connection between the Tutte and the Jones polynomials.Actually, up to a signed multiplication of a power of t the Jones polynomial VL(t)

of an alternating link L is equal to the Tutte polynomial TG(−t, −t−1).We give the general form of the Tutte polynomial of a family of positive-signed

connected planar graphs, and specialize it to the Jones polynomial of the alternatinglinks that correspond to these graphs. Moreover, we give combinatorial interpreta-tion of some of evaluations of the Tutte polynomial, and also recover the chromaticpolynomial from it as a special case.

AMS subject classification: Primary 05C31; Secondary 57M27.Keywords: Tutte polynomial, Jones polynomial, chromatic polynomial.

1. Introduction

The Tutte polynomial was introduced by Tutte [16] in 1954 as a generalization of chro-matic polynomials studied by Birkhoff [1] andWhitney [19]. This graph invariant becamepopular because of its universal property that any multiplicative graph invariant with adeletion/contraction reduction must be an evaluation of it, and because of its applicationsin computer science, engineering, optimization, physics, biology, and knot theory.

In 1985, Jones [8] revolutionized knot theory by defining the Jones polynomial as aknot invariant via Von Neumann algebras. However, in 1987 Kauffman [10] introduceda state-sum model construction of the Jones polynomial that was purely combinatorialand remarkably simple, we follow this construction.

Our primary motivation to study the Tutte polynomial came from the remarkableconnection between the Tutte and the Jones polynomials that up to a sign and multipli-cation by a power of t the Jones polynomial VL(t) of an alternating link L is equal to theTutte polynomial TG(−t, −t−1) [7, 12, 15]. T -equivalence of some families of laddertype graphs has been studied in [13].

This paper is organized as follows: In Section 2 we give some basic notions aboutgraphs and knots along with definitions of theTutte and the Jones polynomials. Moreover,in this section we give the relation between graphs and knots, and the relation between theTutte and the Jones polynomials. The main result is given in Section 3. In Section 4 wespecialize the Tutte polynomial to the Jones, chromatic, flow, and reliability polynomials.Moreover, the interpretations of evaluations of the Tutte polynomial are also given inthis section.

2. Preliminary Notions

2.1. Basic concepts of graphs

A graph G is an ordered pair of disjoint sets (V , E) such that E is a subset of the set V 2

of unordered pairs of V . The set V is the set of vertices and E is the set of edges. If G is

The Tutte and the Jones Polynomials 4719

a graph, then V = V (G) is the vertex set of G, and E = E(G) is the edge set. An edgex, y is said to join the vertices x and y, and is denoted by xy; the vertices x and y arethe end vertices of this edge. If xy ∈ E(G), then x and y are adjacent, or neighboring,vertices of G, and the vertices x and y are incident with the edge xy. Two edges areadjacent if they have exactly one common end vertex.

We say that G′ = (V ′, E′) is a subgraph of G = (V , E) if V ′ ⊂ V and E′ ⊂ E. Inthis case we write G′ ⊂ G. If G′ contains all edges of G that join two vertices in V ′then G′ is said to be the subgraph induced or spanned by V ′, and is denoted by G[V ′].Thus, a subgraph G′ of G is an induced subgraph if G′ = G[V (G′)]. If V = V ′ thenG′ is said to be a spanning subgraph of G.

Two graphs are isomorphic if there is a correspondence between their vertex setsthat preserves adjacency. Thus, G = (V , E) is isomorphic to G′ = (V ′, E′), denotedG � G′, if there is a bijection ϕ : V → V ′ such that xy ∈ E if and only if ϕ(xy) ∈ E′.

The dual notion of a cycle is that of cut or cocycle. If {V 1, V 2} is a partition of thevertex set, and the set C, consisting of those edges with one end in V1 and one end in V2,

is not empty, then C is called a cut. A cycle with one edge is called a loop and a cocyclewith one edge is called a bridge. We refer to an edge that is neither a loop nor a bridgeas ordinary.

A graph is connected if there is a path from one vertex to any other vertex of thegraph. A connected subgraph of a graph G is called the component of G. We denoteby k(G) the number of connected components of a graph G, and by c(G) the numberof non-trivial connected components, that is the number of connected components notcounting isolated vertices. A graph is k-connected if at least k vertices must be removedto disconnect the graph.

A tree is a connected graph without cycles. A forest is a graph whose connectedcomponents are all trees. Spanning trees in connected graphs play a fundamental rolein the theory of the Tutte polynomial. Observe that a loop in a connected graph can becharacterized as an edge that is in no spanning tree, while a bridge is an edge that is inevery spanning tree.

A graph is planar if it can be drawn in the plane without edges crossings. A drawingof a graph in the plane separates the plane into regions called faces. Every plane graph G

has a dual graph, G∗, formed by assigning a vertex of G∗ to each face of G and joiningtwo vertices of G∗ by k edges if and only if the corresponding faces of G share k edges intheir boundaries. Note that G∗ is always connected. If G is connected, then (G∗)∗ = G.If G is planar, it may have many dual graphs.

A graph invariant is a function f on the collection of all graphs such that f (G1) =f (G2), whenever G1

∼= G2. A graph polynomial is a graph invariant where the imagelies in a ring of polynomials.

2.2. The Tutte polynomial

The following two operations are essential to understand the Tutte polynomial definitionfor a graph G. These are edge deletion, denoted by G−e, and edge contraction, denoted

4720 Y. C. Kwun, A. R. Nizami, M. Munir, W. Nazeer and S. M. Kang

by G/e.

e1

e2

G − e3

e1

e

e

2

3

G

e1 2e

G/e3

Definition 2.1. ([16, 17, 18]) The Tutte polynomial of a graph G is a two-variablepolynomial TG(x, y) defined as follows:

TG(x, y) =

1 if E is empty,

xT (G/e) if e is a bridge,

yT (G − e) if e is a loop,

T (G − e) + T (G/e) if e is neither a bridge nor a loop.

Example 2.2. Here is the Tutte polynomial of the graph G=e

1e

e

2

3 .

T (e

1e

e

2

3 ) = T (e

1e

2

) + T (e

1 2e

)

= xT (e

1

) + T (e

1

) + T (e

1 )

= x2T ( ) + xT ( ) + y

= x2 + x + y.

Remark 2.3. The definition of the Tutte polynomial outlines a simple recursive proce-dure to compute it, but the order of the rules applied is not fixed.

Theorem 2.4. ([2]) T (G � G′) = T (G)T (G′) and T (G ∗ G′) = T (G)T (G′), whereG � G′ is the disjoint union of G and G′ and G ∗ G′ is the identification of G and G′into a single vertex.

2.3. Basic concepts of Knots

A knot is a circle embedded in R3, and a link is an embedding of a union of such circles.

Since knots are special cases of links, we shall often use the term link for both knots andlinks. Links are usually studied via projecting them on a plan; a projection with extrainformation of overcrossing and undercrossing is called the link diagram.

undercrossing

overcrossing

Trefoil knot Hopf link

Two links are called isotopic if one of them can be transformed to the other by adiffeomorphism of the ambient space onto itself. A fundamental result about the isotopic


link diagrams is: Two unoriented links L1 and L2 are equivalent if and only if a diagramof L1 can be transformed into a diagram of L2 by a finite sequence of ambient isotopiesof the plane and local (Reidemeister) moves of the following three types:

R1 R2

R3

The set of all links that are equivalent to a link L is called a class of L. By a link L

we shall always mean a class of the link L.

2.4. The Jones polynomial

The main question of knot theory is Which two links are equivalent and which are not?To address this question one needs a knot invariant, a function that gives one value onall links in a single class and gives different values (but not always) on links that belongto different classes. In 1985, Jones [8] revolutionized knot theory by defining the Jonespolynomial as a knot invariant via Von Neumann algebras. However, in 1987 Kauffman[10] introduced a state-sum model construction of the Jones polynomial that was purelycombinatorial and remarkably simple.

Definition 2.5. ([8, 9, 10]) The Jones polynomial VK(t) of an oriented link L is aLaurent polynomial in the variable

√t satisfying the skein relation

t−1VL+(t) − tVL−(t) = (t1/2 − t−1/2)VL0(t),

and that the value of the unknot is 1. Here L+, L−, and L0 are three oriented linkshaving diagrams that are isotopic everywhere except at one crossing where they differas in the figure below:

L+ L− L0

Example 2.6. It is easy to verify that the Jones polynomials of the Hopf link and thetrefoil knot are respectively

V ( ) = −t−5/2 − t−1/2 and V ( ) = −t−4 + t−3 + t.


2.5. A Connection between Knots and Graphs

Corresponding to every connected link diagram we can find a connected signed planargraph and vice versa. The process is as follows: Suppose K is a knot and K ′ its projection.The projection K ′ divides the plane into several regions. Starting with the outermostregion, we can color the regions either white or black. By our convention, we color theoutermost region white. Now, we color the regions so that on either side of an edge thecolors never agree.

The graph G corresponding to the knot projection K ′

Next, choose a vertex in each black region. If two black regions R and R′ havecommon crossing points c1, c2, . . . , cn, then we connect the selected vertices of R andR′ by simple edges that pass through c1, c2, . . . , cn and lie in these two black regions.In this way, we obtain from K ′ a plane graph G [14].

In order to the plane graph to embody some of the characteristics of the knot, weneed to use the regular diagram rather than the projection. So, we need to consider theunder- and over-crossings. To this end, we assign to each edge of G either the sign +or − as you can see in the figure:

+ -

K G(K)

- --

A signed graph corresponding to a knot diagram

A signed plane graph that has been formed by means of the above process is said tobe the graph of the knot K [14].

Conversely, corresponding to a connected signed planar graph, we can find a con-nected planar link diagram. The construction is clear from the following figure.

+ -

+

+

-

G K(G)

A knot diagram corresponding to a signed graph


The fundamental combinatorial result connecting knots and graphs is:

Theorem 2.7. ([11, 12]) The collection of connected planar link diagrams is in one-to-one correspondence with the collection of connected signed planar graphs.

2.6. Connection between the Tutte and the Jones polynomials

The primary motivation to study the Tutte polynomial came from the following remark-able connection between the Tutte and the Jones polynomials.

Theorem 2.8. (Thistlethwaite) ([7, 12, 15]) Up to a sign and multiplication by a powerof t the Jones polynomial VL(t) of an alternating link L is equal to the Tutte polynomialTG(−t, −t−1).

For positive-signed connected graphs, we have the precise connection:

Theorem 2.9. ([2, 5, 6]) Let G be the positive-signed connected planar graph of analternating oriented link diagram L. Then the Jones polynomial of the link L is

VL(t) = (−1)wr(L)t(b(L)−a(L)+3wr(L))

4 TG(−t, −t−1),

where a(L) is the number of vertices in G, b(L) is the number of vertices in the dual ofG, and wr(L) is the writhe of L.

Remark 2.10. In this paper, we shall compute Jones polynomial of links that correspondonly to positive-signed graphs.

Example 2.11. Corresponding to the positive-signed graph G:++

+ , we receive the right-

handed trefoil knot L: . It is easy to check, by definitions, that V ( ; t) = −t4 + t3 + t

and T (++

+ ; x, y) = x2 + x + y. Further note that the number of vertices in G is 3,

number of vertices in the dual of G is 2, and withe of L is 3. Now notice that

V ( ; t) = (−1)3t2−3+3(3)

4 T (++

+ ; −t, −t−1) = −t2(t2 − t − t−1),

which agrees with the known value.

3. The Main Theorem

In this section we compute the Tutte polynomial of the following graph, which we denoteby G3,m,n, and present it in a closed form.

��

��

mn

G3,m,n


Here m, n are the number of additional edges to two adjacent edges in the cycle graph,C3.

Theorem 3.1. The Tutte polynomial of the graph G3,m,n is

TG3,m,n(x, y) = x2 + x

m∑i=1

yi +(

x +m+1∑i=1

yi

) n∑i=0

yi.

Proof. We prove it by using induction on m. First we fix n = 0 and vary m:For m = 1, we have

T (

��

�� ) = T (

��

��

�� ) + T ( �� )

= (x2 + x + y) + yT ( ��

��

)

= (x2 + x + y) + y[T ( ��

�� ) + T ( ��)]

= (x2 + x + y) + y(x + y)

= x2 + xy + (x + y + y2).

For m = 2, we have

T ( ) = T (

��

��

�� ) + T (

�� )

= x2 + xy + (x + y + y2) + y2(x + y)

= x2 + xy + (x + y + y2) + xy2 + y3

= x2 + x(y + y2) + (x + y + y2 + y3).

And, for m = 3, we have

T ( ) = T ( ) + T (�� )

= x2 + x(y + y2) + (x + y + y2 + y3) + y3(x + y)

= x2 + x(y + y2 + y3) + (x + y + y2 + y3 + y4)

= x2 + x

3∑i=1

yi +(

x +3+1∑i=1

yi

) n∑i=0

yi.

Now we fix n = 1 and vary the value of m:


For m = 1,

T ( ) = T (

��

�� ) + T ( ��

��

)

= x2 + xy + (x + y + y2) + yT ( ��

)

= x2 + xy + (x + y + y2) + y[T ( ��

�� ) + T ( ��

)]= x2 + xy + (x + y + y2) + y(x + y + y2)

= x2 + xy + (x + y + y2)(1 + y)

= x2 + x

1∑i=1

yi +(

x +1+1∑i=1

yi

) 1∑i=0

yi.

For m = 2,

T ( ) = T ( ) + T (��

�� )

= x2 + xy + (x + y + y2)(1 + y) + y2T ( ��

��

)

= x2 + xy + (x + y + y2)(1 + y) + y2(x + y + y2)

= x2 + xy + xy2 + (x + y + y2)(1 + y) + y3(1 + y)

= x2 + x(y + y2) + (x + y + y2 + y3)(1 + y)

= x2 + x

2∑i=1

yi +(

x +2+1∑i=1

yi

) 1∑i=0

yi.

And, for m = 3 we have

T ( ) = T ( ) + T (��

��

)

= x2 + xy + (x + y + y2 + y3)(1 + y) + y3T ( �� )

= x2 + xy + (x + y + y2 + y3)(1 + y) + y3(x + y + y2)

= x2 + xy + xy2 + xy3 + (x + y + y2 + y3)(1 + y) + y4(1 + y)

= x2 + x(y + y2 + y3) + (x + y + y2 + y3 + y4)(1 + y)

= x2 + x

3∑i=1

yi +(

x +3+1∑i=1

yi

) 1∑i=0

yi.

Now n = 2. We change the values of m:


For m = 1,

T ( ) = T ( ) + T (�� )

= x2 + xy + (x + y + y2)(1 + y) + y2T ( �� )

= x2 + xy + (x + y + y2)(1 + y) + y2(x + y + y2)

= x2 + xy + (x + y + y2)(1 + y + y2)

= x2 + x

1∑i=1

yi +(

x +1+1∑i=1

yi

) 2∑i=0

yi.

For m = 2,

T ( ) = T ( ) + T ( �� )

= x2 + xy + (x + y + y2)(1 + y + y2) + y2[T ( ��

��

) + T (��

)]= x2 + xy + (x + y + y2)(1 + y + y2) + y2(x + y + y2 + y3)

= x2 + xy + xy2 + (x + y + y2)(1 + y + y2) + y3(1 + y + y2)

= x2 + x(y + y2) + (x + y + y2 + y3)(1 + y + y2)

= x2 + x

2∑i=1

yi +(

x +2+1∑i=1

yi

) 2∑i=0

yi.

Now let the result be true for m = k, that is

T ( ��

��

n k

) = x2 + x

k∑i=1

yi +(

x +k+1∑i=1

yi

) n∑i=0

yi.


Now for m = k + 1 we have

T (G3,k+1,n) = T ( ��

��

n k

) + T (��

��k+1

n

)

= x2 + x

k∑i=1

yi +(

x +k+1∑i=1

yi

) n∑i=0

yi + yk+1T ( ��

n

��

)

= x2 + x

k∑i=1

yi +(

x +k+1∑i=1

yi

) n∑i=0

yi

+ yk+1(x + y + y2 + · · · + yn+1)

= x2 + x

k∑i=1

yi +(

x +k+1∑i=1

yi

) n∑i=0

yi

+ xyk+1 + yk+1(y + y2 + · · · + yn+1)

= x2 + x

( k∑i=1

yi + yk+1)

+(

x +k+1∑i=1

yi

) n∑i=0

yi + yk+2n∑

i=0

yi

= x2 + x

k+1∑i=1

yi +(

x +k+1∑i=1

yi + yk+2) 2∑

i=0

yi

= x2 + x

k+1∑i=1

yi +(

x +k+2∑i=1

yi

) n∑i=0

yi.

This completes the proof. �

4. Specializations and Evaluations

4.1. Jones polynomial

The alternating links L that correspond to the graphs G3,m,n fall into three categories,the 1-component links (when both m, n are even), the 2-component links (when bothm, n are odd) 1-component links (when one of both m, n is even and other is odd).

Case I. (m and n are even)

The graphs along with the corresponding 1-component links (which are simply knots)


are given in the following table. For n = 2 and 4 we respectively receive the tables:

m G L a(L) b(L) wr(L)

2 3 6 7

4 3 8 9

6 3 10 11

8 3 12 13...

......

......

...

and


2 3 8 9

4 3 10 11

6 3 12 13

8 3 14 15...

......

......

...

Similarly, we can construct tables for n = 6, 8, 10, . . ..


Proposition 4.1. The Jones polynomial of the link L corresponding to the graph G3,m,n,

where m and n are even, is

VL(t) = 1

(1 + t)2(tm+4 + tm+3 + tm+2 + tn+4 + tn+3 + tn+2 + t − tm+n+6

− tm+n+5 − tm+n+4).

Proof. It follows from the above tables that a(L) = 3, b(L) = m+n+ 2, and wr(L) =m + n + 3. Moreover, the factor (−1)wr(L)t

(b(L)−a(L)+3wr(L))4 reduces to −tm+n+2 and

TG3,m,n(−t, −t−1) = t2 − t (1 − tm)

tm(1 + t)− t (1 + tn+1)

tn(1 + t)− (1 + tm+1)(1 + tn+1)

tm+n+1(1 + t)2.

Now

VL(t) = −tm+n+2TG3,m,n(−t, −t−1)

= −tm+n+2[t2 − t (1 − tm)

tm(1 + t)− t (1 + tn+1)

tn(1 + t)− (1 + tm+1)(1 + tn+1)

tm+n+1(1 + t)2

]

= 1

(1 + t)2

[ − tm+n+2t2(1 + t)2 + tn+3(1 − tm)(1 + t)

+ tm+3(1 − tn+1)(1 + t) + t (1 + tm+1 + tn+1 + tm+n+2)]

= 1

(1 + t)2

( − tm+n+4 − 2tm+n+5 − tm+n+6 + tn+3 + tn+4 + tm

− tm+n+3 − tm+n+4 + tm+3 + tm+4 + tm+n+4 + tm+n+5

+ t + tm+2 + tn+2 + tm+n+3)= 1

(1 + t)2

(tm+4 + tm+3 + tm+2 + tn+4 + tn+3 + tn+2

+ t − tm+n+6 − tm+n+5 − tm+n+4).

�

Case II. (m and n are odd)

In this case we receive 2-component links, both are oriented clockwise or counter


clockwise. The tables for n = 1 and n = 3 are:


2 3 4 −1

4 3 6 1

6 3 8 3

8 3 10 5...

......

......

...

Here n = 3.


2 3 6 −3

4 3 8 −1

6 3 10 1

8 3 12 3...

......

......

...

Similarly, we can construct tables for n = 5, 7, 9, . . ..



where m and n are both odd, is

VL(t) = 1

(1 + t)2

(tm− 3n

2 −1 − tm− n2 +3 − tm− n

2 +2 − tm− n2 +1 − t−

n2 +1 − t−

n2

− tm− 3n2 +1 − tm− 3n

2 − t−n2 −1 − t−

3n2 −2).

Proof. In this case a(L) = 3, b(L) = m + n + 2, wr(L) = m − n − 1, and the factor

(−1)wr(L)t(b(L)−a(L)+3wr(L))

4 reduces to −tm− n2 −1 with

TG3,m,n(−t, −t−1) = 1

(1 + t)2

[t2(1 + t)2 + t1−m(1 + t)(1 + tm)

+ t1−n(1 + t)(1 − tn+1) − t−m−n−1(1 − tn+1)(1 − tm+1)].

So,

VL(t) = −tm− n2 −1TG3,m,n

(−t, −t−1)

= −tm− n2 −1

(1 + t)2

(t2 + 2t3 + t4 + t1−m + t2−m + t + t2 + t1−n

+ t2−n − t2 − t3 + t−n − t−m−n−1 + t−m − t)

= −tm− n2 −1

(1 + t)2

(t4 + t2 + t3 + t1−m + t2−m + t1−n + t2−n + t−n

+ t−m − t−m−n−1)= 1

(1 + t)2

(t−

3n2 −2 − tm− 3n

2 +1 − tm− 3n2 − tm− 3n

2 −1 − tm− n2 +3

− tm− n2 +2 − tm− n

2 +1 − t−n2 +1 − t−

n2 − t−

n2 −1).

�

If one link is oriented clockwise and other counter clockwise, then a(L) = 3, b(L) =m + n + 2, and wr(L) = n − m + 1. So the Jones polynomial becomes


where m and n are both odd, is

VL(t) = 1

(1 + t)2

(t

3m2 − 1

2 − tn− 3m2 + 5

2 − tn− 3m2 + 3

2 − tn− 3m2 + 1

2 − tn−m2 + 9

2

− tn−m2 + 7

2 − tn−m2 + 5

2 − t−3m2 + 5

2 − t−3m2 + 3

2 − t−3m2 + 1

2).

Case III. (m is even and n is odd)The graphs along with the corresponding 1-component links (which are simply knots)


are given in the table:


2 3 5 −2

4 7 6 −4

6 3 9 −6

8 3 11 −8...

......

......

...

In this case a(L) = 3, b(L) = m + n + 2, and wr(L) = n − m − 1.


where m is even and n is odd, is

VL(t) = 1

(1 + t)2

(tn−m

2 +3 + tn−m2 +2 + tn−m

2 −1 − tn− 3m2 − tn− 3n

2 +1

+ t−m2 + t−

m2 +1 + t−

3m2 −2 + t−

m2 −1 − tn− 3m

2 −1).

Proof. Similar to the proof of Proposition 4.3. �

Case IV. (n is even and m is odd)

The graphs along with the corresponding 1-component links (which are simply knots)


are given in the following table.


1 3 5 −2

3 7 6 0

5 3 9 2

7 3 11 4...

......

......

...


where n is even and m is odd, is

VL(t) = 1

(1 + t)2

(tm− n

2 +3 + tm− n2 +2 + tm− n

2 +1 − tm− 3n2 − tm− 3n

2 +1

+ t−n2 + t−

n2 +1 + t−

3n2 −2 + t−

n2 −1 − tm− 3n

2 −1).Proof. Here a(L) = 3, b(L) = m + n + 2, wr(L) = m − n − 1, and the factor

(−1)wr(L)t(b(L)−a(L)+3wr(L))

4 reduces to tm− n2 −1. Moreover

T (−t, −t−1) = 1

(1 + t)2

(t2(1 + t)2 + t1−m(1 + tm)(1 + t)

+ t1−n(1 + tn+1)(1 + t) + t−m−n−1(1 − tm+1)(1 + tn+1)).

�

4.2. Chromatic polynomial

A common problem in the study of graph theory is the coloring of the vertices of a graph.The coloring of a graph in such a way that no two adjacent vertices have the same color iscalled proper coloring of graphs.That is If we have a positive integer λ, then a λ-coloringof a graph G is a mapping of V (G) into the set {1, 2, 3, . . . , λ} of λ colors. Thus, thereare exactly λn colorings for a graph on n vertices.


The chromatic polynomial, because of its theoretical and applied importance, hasgenerated a large body of work. Chia [3] provides an extensive bibliography on thechromatic polynomial, and Dong et al. [4] give a comprehensive treatment.

Definition 4.6. The chromatic polynomial PG(λ) of a graph G is a one-variable graphinvariant and is defined by the following deletion-contraction relation:

PG(λ) = P(G − e) − P(G/e)

Since the chromatic polynomial counts the number of distinct ways to color a graphwith λ colors, we recover it from the Tutte polynomial. The following theorem gives arelation between these polynomials.

Theorem 4.7. ([2]) The chromatic polynomial of a graph G = (V , E) is

PG(λ) = (−1)|V |−k(G)λk(G)TG(1 − λ, 0),

where k(G) denote the number of connected components of G.

Proposition 4.8. The chromatic polynomial of the graph G3,m,n is

PG3,m,n(λ) = λ(1 − λ)2.

Proof. Although one can directly compute the chromatic polynomial of G3,m,n by defi-nition, we recover it from the Tutte polynomial of G3,m,n. Since |V | = 3 and k(G) = 1,

the factor (−1)|V |−k(G)λk(G) reduces to λ and TG(1 − λ, 0) reduces to (1 − λ)2. Thenwe get the final result as

PG3,m,n(λ) = λ(1 − λ)2.

�


4.3. Flow polynomial

A function which could count the number of flows in a connected graph is known as flowpolynomial.

Definition 4.9. Suppose that G be a graph with an arbitrary but fixed orientation, andlet K be an Abelian group of order |K| and with 0 as its identity element. A K-flow is amapping φ of the oriented edges

−→E (G) into the elements of the group K such that:

∑−→e =u→v

φ(−→e ) +

∑−→e =u←v

φ(−→e ) = 0 (4.1)

for every vertex v, and where the first sum is taken over all arcs towards v and the secondsum is over all arcs leaving v.

A K-flow is nowhere zero if φ never takes the value 0. The relation (4.1) is calledthe conservation law (that is, the Kirchhoff’s law is satisfied at each vertex of G).

It is well known [2] that the number of proper K-flows does not depend on thestructure of the group, it depends only on its cardinality, and this number is a polynomialfunction of |K| that we refer to as the flow polynomial.

The following, due to Tutte [16], relates the Tutte polynomial of G with the numberof nowhere zero flows of G over a finite Abelian group (which, in our case, is Zk).

Theorem 4.10. ([16]) Let G = (V , E) be a graph and K a finite Abelian group. IfFG(|K|) denotes the number of nowhere zero K-flows then

FG(|K|) = (−1)|E|−|V |+k(G)T (0, 1 − |K|).

Proposition 4.11. The Flow polynomial of the graph G3,m,n is

FG(|K|) = (−1)n+m+1 1

|K|2[(1 − |K|)m+n+3

− (1 − |K|)m+2 − (1 − |K|)n+2 + (1 − |K|)].

Proof. Since |V | = 3, |E| = m + n + 3 and k(G) = 1 the factor (−1)|E|−|V |+k(G)


reduces to (−1)n+m+1.

FG(|K|) = (−1)n+m+1 · TG(0, 1 − |K|)

= (−1)n+m+1(

m+1∑i=1

(1 − |K|)i)(n∑

i=0

(1 − |K|)i)

= (−1)n+m+1[(1 − |K|)[(1 − |K|)m+1 − 1]

(1 − |K| − 1)

][(1 − |K|)n − 1

(1 − |K| − 1)

]

= (−1)n+m+1 1

|K|2[(1 − |K|)[(1 − |K|)m+1 − 1][(1 − |K|)n+1 − 1]]

= (−1)n+m+1 1

|K|2[(1 − |K|)m+n+3 − (1 − |K|)m+2

− (1 − |K|)n+2 + (1 − |K|)].

�

4.4. Reliability polynomial

Definition 4.12. Let G be a connected graph or network with |V | vertices and |E| edges,and supposed that each edge is independently chosen to be active with probability p.Then the (all terminal) reliability polynomial of the network is

RG(p) =∑A

p|A|(1 − p)|E−A|

=|E|−|V |+1∑

i=o

gi(pi+|V |−1)(1 − p)|E|−i−|V |+1,

where A is the connected spanning subgraph of G and gi is the number of spanningconnected subgraphs with i + |V | − 1 edges.


Theorem 4.13. ([5]) If G is a connected graph with m edges and n vertices, then

RG(p) = p|V |−1(1 − p)|E|−|V |+1TG

(1,

1

1 − p

).

Proposition 4.14. The Reliability polynomial of the graph G3,m,n is

RG(p) = (1 − p)m+n+3 − p(1 − p)m+n+1 − (1 − p)n+2

− (1 − p)m+1 + (1 − p)m + 1.

Proof. Since |V | = 3 and |E| = m+n+ 3, the factor (p)|V |−1(1 −p)|E|−|V |+1 reducesto p2(1 − p)m+n+1.

RG(p) = p2(1 − p)m+n+1 · TG3,m,n

(1,

1

1 − p

)

= p2(1 − p)m+n+1[

1 +(

1

1 − p

) [( 11−p

)m − 1]1

1−p− 1

+ ( 11−p

)n+1 − 11

1−p− 1

+(

1

1 − p

) [( 11−p

)m+1 − 1]1

1−p− 1

[( 11−p

)n+1 − 1]1

1−p− 1

]

= p2(1 − p)m+n+1[

1 + (1 − (1 − p)m)(1 − p)

(1 − p)m+1(1 − 1 + p)+ (1 − (1 − p)n+1)(1 − p)

(1 − p)n+1(1 − 1 + p)

+ (1 − (1 − p)m+1)(1 − p)

(1 − p)m+2(1 − 1 + p)· (1 − (1 − p)n+1)

(1 − p)n+1(1 − 1 + p)

]

= p2(1 − p)m+n+1[

1 + (1 − (1 − p)m)

(1 − p)m(p)+ (1 − (1 − p)n+1)

(1 − p)n+1(p)

+ (1 − (1 − p)m+1)

(1 − p)m+1(p)· (1 − (1 − p)n+1)

(1 − p)n(p)

]

= (1 − p)m+n+1[p2 + p(1 − p)−m(1 − (1 − p)m)

+ p(1 − p)−n(1 − (1 − p)n+1)

+ (1 − p)−m−n−1(1 − (1 − p)m+1)(1 − (1 − p)n+1)]

= (1 − p)m+n+1(p2 + p(1 − p)−m − p + p(1 − p)−n − p(1 − p)

+ (1 − p)−m−n−1(1 − (1 − p)m+1 − (1 − p)n+1 + (1 − p)m+n+2)


= (1 − p)m+n+1(p2 + p(1 − p)−m − p + p(1 − p)−n − p(1 − p)

+ (1 − p)−m−n−1 + (1 − p) − (1 − p)−n − (1 − p)−m)

= p2(1 − p)m+n+1 + p(1 − p)n+1 − (1 − p)m+n+1 + p(1 − p)m+1

− p(1 − p)m+n+2 + 1 − (1 − p)m+1 − (1 − p)n+1 + (1 − p)m+n+2

= (1 − p)m+n+1(p2 + 1 − p − p) − (1 − p)n+1(1 − p)

− (1 − p)m(1 − p − p) − p(1 − p)m+n+1 + 1

= (1 − p)m+n+3 − p(1 − p)m+n+1 − (1 − p)n+2 − (1 − p)m+1

+ (1 − p)m + 1.

�

4.5. Evaluations

In this section we evaluate the Tutte polynomial at some special points to get usefulcombinatorial information about the graph G3,m,n.

Theorem 4.15. ([5]) If G = (V , E) is a connected graph, then

(1) TG(1, 1) is the number of spanning trees of G.

(2) TG(2, 1) equals the number of spanning forests of G.

(3) TG(1, 2) is the number of spanning connected subgraphs of G.

(4) TG(2, 2) equals 2|E|, and is the number of subgraphs of G.

Theorem 4.16. ([5]) If G = (V , E) is a connected graph, then

(1) TG(2, 0) equals the number of cyclic orientations of G, that is, orientation withoutoriented cycles.


(2) TG(1, 0) equals the number of acyclic orientations with exactly one predefinedsource.

(3) TG(0, 2) equals the number of totally cyclic orientations of G, that is, orientationin which every arc is a directed cyclic.

(4) TG(2, 1) equals the number of score vectors of orientations of G.

Theorem 4.17.

(1) The number of spanning trees of G3,m,n is 2m + 2n + mn + 3.

(2) The number of connected spanning subgraphs of 3,m,n is 6m + 6n + 4mn + 8.

(3) The number of spanning forests of G3,m,n is 4m + 3n + mn + 7.

(4) The number of subgraphs of G3,m,n is 8m + 8n + 4mn + 12.

(5) The number of orientations of G3,m,n without oriented cycles is 4.

(6) The number of acyclic orientations of G3,m,n with exactly one predefined source1.

(7) The number of orientation of G3,m,n in which every arc is a directed cyclic is4m + 4n + 4mn + 4.

(8) The number of score vectors of orientations of G3,m,n is 4m + 3n + mn + 7.

Proof. Following Theorems 3.1 and 4.15 we have

T (1, 1) = 12 + 1 +m∑

i=1

1i +(

1 +m+1∑i=1

1i

) n∑i=0

1i

= 1 + m + (1 + m + 1)(n + 1)

= 2m + 2n + mn + 3,

which is the required result.The proofs of the rest statements are similar. �

Acknowledgments

This work was supported by the Dong-A University research fund.


References

[1] G. D. Birkhoff, A determinant formula for the number of ways of coloring a map,Ann. Math., 14 (1912), pp. 42–46.

[2] B. Bollobás, Modern Graph Theory, Gratudate Texampleamplets in Mathematics,Springer, New York, 1998.

[3] G. L. Chia, A bibliography on chromatic polynomials, Discrete Math., 172 (1997),pp. 175–191.

[4] F. M. Dong, K. M. Koh and K. L. Teo, Chromatic polynomials and chromaticity ofgraphs, World Scientific, New Jersey, 2005.

[5] J. A. Ellis-Monaghan and C. Marino, Graph polynomials and their applications I,The Tutte polynomial, arXiv:0803.3079[math.CO], arXiv:0803.3079v2[math.CO],2008.

[6] S. Jablan, L. Radovic and R. Sazdanovic, Tutte and Jones polynomials of linkfamilies, arXiv:0902.1162[math.GT], 2009, arXiv:0902.1162v2[math.GT], 2010.

[7] F. Jaeger, Tutte polynomials and link polynomials, Proc. Amer. Math. Soc., 103(1988), pp. 647–654.

[8] V. F. R. Jones, A polynomial invariant for Knots via Von Neumann algebras, Bull.Amer. Math. Soc., 12 (1985), pp. 103–111.

[9] V. F. R. Jones, The Jones polynomial, Discrete Math., 294 (2005), pp. 275–277.[10] L. H. Kauffman, State models and the Jones Polynomial, Topology, 26 (1987), pp.

395–407.[11] L. H. Kauffman, New invariants in knot theory, Amer. Math. Monthly, 95 (1988),

pp. 195–242.[12] L. H. Kauffman, A Tutte polynomial for signed graphs, Discrete Appl. Math., 25

(1989), pp. 105–127.[13] Y. C. Kwun, M. Munir, A. R. Nizami, W. Nazeer and S. M. Kang, On T -equivalence

of some families of Ladder-type Graphs, Global J. Pure Appl. Math., 12 (2016),pp. 4273–4284.

[14] K. Murasugi, Knot Theory and Its Applications, Birkhauser, Boston, 1996.[15] M. Thistlethwaite, A spanning tree exampleamplepansion for the Jones polynomial,

Topology, 26 (1987), pp. 297–309.[16] W. T. Tutte, A contribution to the theory of chromatic polynomials, Canadian J.

Math., 6 (1954), pp. 80–91.[17] W. T. Tutte, On dichromatic polynomials, J. Combinatorial Theory, 2 (1967), pp.

301–320.[18] W. T. Tutte, Graph-polynomials, Special issue on the Tutte polynomial, Adv. in

Appl. Math., 32 (2004), pp. 5–9.[19] H. Whitney,A logical expansion in mathematics, Bull.Amer. Math. Soc., 38 (1932),

pp. 572–579.

Documents

TheTutte and the Jones PolynomialsThe Tutte and the Jones Polynomials 4719 a graph, then V = V(G)is the vertex set of G, and E = E(G)is the edge set.An edge x,yis said to join the