the tutte polynomial graph polynomials 238900 winter 05/06

The Tutte Polynomial

Graph Polynomials 238900 winter 05/06

The Rank Generation Polynomial

Reminder

GEF

FnEkFk yxyxGS ,;

FkVFr

FkVFFn

Fk

Number of connected components in G(V,F)

The Rank Generation Polynomial

Theorem 2

else

loop a is

bridge a is

,;/,;

,;1

,;1

,; e

e

yxeGSyxeGS

yxeGSy

yxeGSx

yxGS

eG

eG

/

Is the graph obtained by omitting edge e

Is the graph obtained by contracting edge e

In addition, S(En;x,y) = 1 for G(V,E) where |V|=n and |E|=0

The Rank Generation Polynomial

Theorem 2 – ProofLet us define :

G’ = G – e G’’ = G / e

r’<G> = r<G’>n’<G> = n<G’>r’’<G> = r<G’’>n’’<G> = n<G’’>

The Rank Generation Polynomial

Theorem 2 – Proof

eEeE

eFeF

Let us denote :

The Rank Generation Polynomial

Observations

FrGr

FreEreFrEr

FnFn

FrFr

eEFEe

''''

''''

'

'

,

The Rank Generation Polynomial

Observations

FrGr

FreEreFrEr

FnFn

FrFr

eEFEe

''''

''''

'

'

,

Fe

The Rank Generation Polynomial

Observations

FrGr

FreEreFrEr

FnFn

FrFr

eEFEe

''''

''''

'

'

,

FkeFk

eEkEk

else

loop

VV

VV

1

The Rank Generation Polynomial

Observations

FrGr

FreEreFrEr

FnFn

FrFr

eEFEe

''''

''''

'

'

,

''''' GG

3

The Rank Generation Polynomial

Observations

else

bridge a is

'

1' e

eEr

eErEr

else

loop a is

''

1'' e

Fn

FneFn

kk

VV

EE

1

1

kk

VV

EE

1

1

2

The Rank Generation Polynomial

GEF

FnEkFk yxyxGS ,;

Reminder

GEF

FnFrEr yx

The Rank Generation Polynomial

yxGSyxGSyxGS ,;,;,; 10

FeGEF

FnFrEr yxyxGS,

0 ,;

FeGEF

FnFrEr yxyxGS,

1 ,;

Let us define

The Rank Generation Polynomial

otherwise

bridge a is

,;

'

'''

'

''1'

,0

e

yx

yx

yx

yxyxGS

GEF

FnFreEr

GEF

FnFreEr

eEF

FnFrEr

FeGEF

FnFrEr

1

The Rank Generation Polynomial

otherwise

bridge a is

,;

,;

otherwise

bridge a is

,;

'

'''

'

''1'

0

e

yxeGS

yxeGxS

e

yx

yx

yxGS

GEF

FnFreEr

GEF

FnFreEr

The Rank Generation Polynomial

otherwise

loop a is

,;

''

''''''

''

1''''''

,1

e

yx

yx

yx

yxyxGS

GEF

FnFrGr

GEF

FnFrGr

eEF

eFneFrEr

FeGEF

FnFrEr

2

3

The Rank Generation Polynomial

otherwise

loop a is

,;/

,;/

otherwise

loop a is

,;

''

''''''

''

1''''''

1

e

yxeGS

yxeGyS

e

yx

yx

yxGS

GEF

FnFrGr

GEF

FnFrGr

The Rank Generation Polynomial

otherwise

loop a is

,;/

,;/ e

yxeGS

yxeGyS

yxGSyxGSyxGS ,;,;,; 10

otherwise

bridge a is

,;

,; e

yxeGS

yxeGxS

+

The Rank Generation Polynomial

yxGSyxGSyxGS ,;,;,; 10

otherwise

loop a is

bridge a is

,;/,;

,;/,;

,;/,;

e

e

yxeGSyxeGS

yxeGySyxeGS

yxeGSyxeGxS

The Rank Generation Polynomial

yxGSyxGSyxGS ,;,;,; 10

otherwise

loop a is

bridge a is

,;/,;

,;,;

,;/,;

e

e

yxeGSyxeGS

yxeGySyxeGS

yxeGSyxeGxS

Obvious

The Rank Generation Polynomial

yxGSyxGSyxGS ,;,;,; 10

otherwise

loop a is

bridge a is

,;/,;

,;1

,;/,;

e

e

yxeGSyxeGS

yxeGSy

yxeGSyxeGxS

Obvious

The Rank Generation Polynomial

yxGSyxGSyxGS ,;,;,; 10

otherwise

loop a is

bridge a is

,;/,;

,;1

,;,;

e

e

yxeGSyxeGS

yxeGSy

yxeGSyxeGxS

FreErFreEr

FnFneEF

''''''

,'''

kkVVGG 11'''

The Rank Generation Polynomial

yxGSyxGSyxGS ,;,;,; 10

otherwise

loop a is

bridge a is

,;/,;

,;1

,;1

e

e

yxeGSyxeGS

yxeGSy

yxeGSx

Obvious

Q.E.D

The Tutte Polynomial

The Tutte Polynomial

Reminder

4

5

The Universal Tutte Polynomial

yx

GT

yxGU

GrGnGk ,;

,,,,;

The Universal Tutte Polynomial

otherwise

loop a is

bridge a is

,;/,;

,;

,;

,,,,;

e

e

yxeGUyxeGU

yxeGUy

yxeGUx

yxGU

Theorem 6

nn yxEU ,,,,;

The Universal Tutte Polynomial

Theorem 6 - Proof

nn

nVVVVV

n

yxET

yxEU

111

,;

,,,,;

0

5

The Universal Tutte Polynomial

1

1

bridge a is

GreGr

GneGn

GkeGk

e

If e is a bridge

yx

eGTx

yxGU

eGreGneGk ,;

,,,,;

11

4

The Universal Tutte Polynomial

If e is a bridge

,,,,;

,;

,;

,,,,;

11

yxeGxU

yxeGxT

yxeGT

x

yxGU

eGreGneGk

eGreGneGk

The Universal Tutte Polynomial

GreGr

GneGn

GkeGk

e 1loop a is

If e is a loop

yx

eGTy

yxGU

eGreGneGk ,;

,,,,;

1

4

The Universal Tutte Polynomial

If e is a bridge

,,,,;

,;

,;

,,,,;

1

yxeGyU

yxeGyT

yxeGT

y

yxGU

eGreGneGk

eGreGneGk

The Universal Tutte Polynomial

1/

/

1

/

loop a is

bridge a is

GreGr

GreGr

GneGn

GneGn

GkeGkeGk

e

e

If e is neither a loop nor a bridge

The Universal Tutte Polynomial

If e is neither a loop nor a bridge

4

yxeGT

yxeGT

yxGUGrGnGk

,;/,;

,,,,;

The Universal Tutte Polynomial

If e is neither a loop nor a bridge

yxeGT

yxeGT

yxGU

eGreGneGk

eGreGneGk

,;/

,;

,,,,;

1///

1

The Universal Tutte Polynomial

If e is neither a loop nor a bridge

,,,,;/

,,,,;

,,,,;

yxeGU

yxeGU

yxGU

The Universal Tutte Polynomial

otherwise

loop a is

bridge a is

,;/,;

,;

,;

,,,,;

e

e

yxeGUyxeGU

yxeGUy

yxeGUx

yxGU

Theorem 6 - Proof

Q.E.D

The Universal Tutte Polynomial

otherwise

loop a is

bridge a is

,;/,;

,;

,;

,,,,;..,,,,;

e

e

yxeGVyxeGV

yxeGVy

yxeGVx

yxGVtsyxGV

Theorem 7

nn yxEV ,,,,;

,,,,;,,,,; yxGUyxGV then

The Universal Tutte Polynomial

otherwise

loop a is

bridge a is

,;/,;

,;

,;

,,,,;

e

e

yxeGVyxeGV

yxeGVy

yxeGVx

yxGV

Theorem 7 – proof

V(G) is dependant entirely on V(G-e) and V(G/e).In addition – ,,,,;,,,,; yxEUyxEV nn

Q.E.D

Evaluations of the Tutte Polynomial

Proposition 8

1,1,GT

Let G be a connected graph.

is the number of spanning trees of G

Shown in the previous lecture

Evaluations of the Tutte Polynomial

Proposition 8

2,1,GT

Let G be a connected graph.

is the number of connected

spanning sub-graphs of G

GEF

FnEkFkGSGT 101,0;2,1;

EkFkGEFEkFkGEF

Fn 11

Evaluations of the Tutte Polynomial

Proposition 8

2,1,GT

Let G be a connected graph.

is the number of connected

spanning sub-graphs of G

EkFkGEF

GT 12,1;

sub-graphs of G connected

Evaluations of the Tutte Polynomial

Proposition 8

1,2,GT

Let G be a connected graph.

is the number of (edge sets forming)

spanning forests of G

GEF

FnEkFkGSGT 010,1;1,2;

0

1FnGEF Spanning forests

Evaluations of the Tutte Polynomial

Proposition 8

2,2,GT

Let G be a connected graph.

is the number of spanning

sub-graphs of G

GEF

FnEkFkGSGT 111,1;2,2;

GEF

1

The Universal Tutte Polynomial

Theorem 9

0,1;1; xGTxxG GkGr

The chromatic polynomial and the Tutte polynomial are related by the equation :

The Universal Tutte Polynomial

Theorem 9 – proof

1,1,,0,

1,; xx

xGUxG

Claim :

The Universal Tutte Polynomial

Theorem 9 – proof

We must show that :

1,1,,0,

1,; xx

xEUxE nn

The Universal Tutte Polynomial

Theorem 9 – proofand :

otherwise

loop a is

bridge a is

1;/

1;

0

1;

1

1;

e

e

x

xeG

x

xeG

x

xeG

x

x

x

xG

The Universal Tutte Polynomial

Theorem 9 – proof

1,1,,0,

1,; xx

xEUxE nn

nn xxE ; Obvious. (empty graphs)

nnn xxx

xEU

1,1,,0,1

,

The Universal Tutte Polynomial

Theorem 9 – proof

0;loop xGe Obvious

xeGxeGxGe ;/;;loopbridge

Chromatic polynomial property

xeGx

xxGe ;

1;bridge

Chromatic polynomial property

The Universal Tutte Polynomial

Theorem 9 – proofThus :

otherwise

loop a is

bridge a is

1;/

1;

0

1;

1

1;

e

e

x

xeG

x

xeG

x

xeG

x

x

x

xG

The Universal Tutte Polynomial

Theorem 9 – proof

And so, due to the uniqueness we shownin Theorem 7 :

1,1,,0,

1,; xx

xGUxG

The Universal Tutte Polynomial

Theorem 9 – proof

Remembering that :

yx

GT

yxGU

GrGnGk ,;

,,,,;

The Universal Tutte Polynomial

Theorem 9 – proof

Remembering that :

Q.E.D

0,1,1

0,1,11

1,1,,0,1

,;

xGTx

xGTx

xx

xGUxG

GrGk

GrGnGk

Evaluations of the Tutte Polynomial

0,2,GT

Let G be a connected graph.

is the number of acyclic orientations

of G.

We know that (Theorem 9) –

0,1;1; xGTxxG GkGr

Evaluations of the Tutte Polynomial

0,11,0,2, GTGTSo –

And thus –

1;11;1

1

1;

1

;0,2,

GG

G

x

xGGT

VV

VGkGrGkGr

Acyclic Orientations of Graphs

Let G be a connected graph without loops or multiple edges.

An orientation of a graph is received afterassigning a direction to each edge.

An orientation of a graph is acyclic if it doesnot contain any directed cycles.

Acyclic Orientations of Graphs

xG, is the number of pairs where

is a map

Proposition 1.1

,

xV ,...,3,2,1: and isan orientation of G, subject to the following :

• The orientation is acyclic• vuvu

Acyclic Orientations of Graphs

Proof

The second condition forces the map to be aproper coloring.

The second condition is immediately impliedfrom the first one.

Acyclic Orientations of Graphs

Proof

Conversely, if the map is proper, than thesecond condition defines a unique acyclicorientation of G.

Hence, the number of allowed mappings issimply the number of proper coloring with xcolors, which is by definition xG,

Acyclic Orientations of Graphs

xG,~ be the number of pairs where

is a map

,

xV ,...,3,2,1: and isan orientation of G, subject to the following :

• The orientation is acyclic• vuvu

Acyclic Orientations of Graphs

xGxGNx V ,1,~

Theorem 1.2

0,2,GT

Acyclic Orientations of Graphs

Proof

The chromatic polynomial is uniquelydetermined by the following :

xxG ,0 G0 is the one vertex graph

xHxGxHG ,,, Disjoint union

xeGxeGxG ,/,,

Acyclic Orientations of Graphs

Proof

We now have to show for the new polynomial :

xxG ,~0 G0 is the one vertex graph

xHxGxHG ,~,~,~ Disjoint union

xeGxeGxG ,/~,~,~

Obvious

Obvious

Acyclic Orientations of Graphs

Proof

We need to show that :

xeGxeGxG ,/~,~,~

xeGV ,...,3,2,1: Let :

Acyclic Orientations of Graphs

Proof

xeGV ,...,3,2,1: Let :

vue ,

Let be an acyclic orientation of G-ecompatible with

Let :

Acyclic Orientations of Graphs

Proof

Let be an orientation of G after adding{u v} to

1

Let be an orientation of G after adding{v u} to

2

Acyclic Orientations of Graphs

Proof

We will show that for each pair exactly one of the orientations is acyclic and compatible with ,expect forof them, in which case both are acyclic orientations compatible with

, 21,

21,

xeG ,/~

Acyclic Orientations of Graphs

Proof

Once this is done, we will know that –

due to the definition of

xeGxeGxG ,/~,~,~ xG,~

Acyclic Orientations of Graphs

Proof

For each pair where -

and is an acyclic orientation compatiblewith one of these three scenarios musthold :

,

xeGV ,...,3,2,1:

Acyclic Orientations of Graphs

Proof

vu Case 1 –

Clearly is not compatible with whileis compatible. Moreover, is acyclic :

2 1

1uwwvu 21

uwwvu 21

Impossible cicle

Acyclic Orientations of Graphs

Proof

vu Case 2 –

Clearly is not compatible with whileis compatible. Moreover, is acyclic :

1 2

2vwwuv 21

vwwuv 21

Impossible cicle

Acyclic Orientations of Graphs

Proof

vu Case 3 –

Both are compatible with At least one is also acyclic. Suppose not, then:

vwwuv '2'1

uwwvu 21

21 contains

contains

Acyclic Orientations of Graphs

Proof

vwwuv '2'1

uwwvu 21

21 contains

contains

uwvwu 1'1

contains

Impossible cicle

Acyclic Orientations of Graphs

ProofWe now have to show that both andare acyclic for exactly pairs of with vu

21 xeG ,/~ ,

Let z denote the vertex identifying {u,v} ineG /

Acyclic Orientations of Graphs

ProofZeG /

some acyclic orientation compatible with

uv

impossible to add a circle by the new edge {u,v}

G

two acyclic orientations, compatible with

Acyclic Orientations of Graphs

ProofZeG /

exactly one, necessarily acyclic, compatible with

uv

Some two acyclic orientations, compatible with

G

All other vertices of G remains the same

Acyclic Orientations of Graphs

ProofAnd so both and are acyclic for exactly pairs of with

And so –

vu

21 xeG ,/~ ,

xeGxeGxG ,/~,~,~

Acyclic Orientations of Graphs

ProofIt is obvious that for x = 1 every orientationis compatible with

And so the expression count the number ofacyclic orientations in G

Q.E.D

1: V