4/10/08 Slide 1 Jo Ellis-Monaghan St. Michaels College, Colchester, VT USA e-mail: [email protected] website: http://academics.smcvt.edu/jellis-monaghan
4/10/08 Slide 1
Jo Ellis-MonaghanSt. Michaels College, Colchester, VT USAe-mail: [email protected]: http://academics.smcvt.edu/jellis-monaghan
4/10/08 Slide 2
Bridges and Loops
A bridge is an edge whose deletion separates the graph
A loop is an edge with both ends incident to the same vertex
bridgesNot a bridge
A loop
4/10/08 Slide 3
e
Delete e
Contract eG
G-e
G/e
Deletion and contraction
Circuit vs cycle…
4/10/08 Slide 4
Classical Dichromatic Polynomiala.k.a. Potts Model Partition Function
The classical dichromatic polynomial:
Potts Model Partition Function with q spins, and Hamiltonian
All edges have the same interaction energy in this model.If q and v are indeterminates rather than physical values, these
polynomials are the same (Fortuin & Kastelyn ’72).
( ) ( )
( )
( ); , A n k Ak A
A E G
Z G u v u v − +
⊆
= ∑
exp( ) 1v J Tκ= −
( )( ) ( , )i j
ij E Gh Jω δ σ σ
∈
= − ∑
( ) ( )
( )
( )
( ); , expA n k Ak A
A E G states
hZ G q v q vTω
ωκ
− +
⊆
−⎛ ⎞= = ⎜ ⎟⎝ ⎠
∑ ∑
4/10/08 Slide 5
Essential Properties• if e is not a loop.
• if GH is the disjoint union of G and H.
• if G consists of a single vertex and nloops.
(Whitney) Tutte ’67
( )( ; , ) ( ; , ) / ; ,Z G u v Z G e u v Z G e u v= − +
( ) ( ) ( ); , ; , ; ,Z GH u v Z G u v Z H u v=
( ; , ) ( 1)nZ G u v u v= +
These capture essential (local) physical properties, and are (defining) theoretical tools.
4/10/08 Slide 6
Some Multivariable Dichromatic PolynomialsTraldi ‘89:
The interaction energy may depend on the edge
Fortuin & Kastelyn ‘72:
Doubly weighted, but we+ ve = 1, so equivalent to above.
( )( )
( ) ( )
( )
( )( ); , , ; ,A n k A k Ak Ae e
A E G A E Ge A e A
Z G t q q t v Z G q q v− +
⊆ ⊆∈ ∈
= → =∑ ∑∏ ∏v v
( ) ( )
( ); , ,
c
k Ae e
A E G e A e A
Z G q q v w⊆ ∈ ∈
= ∑ ∏ ∏v w
( )( ; , ) ( ; , ) / ; ,eZ G u Z G e u v Z G e u= − +v v v
( )( ; , , ) ( ; , , ) / ; , ,e eZ G u w Z G e u v Z G e u= − +v w v w v w
4/10/08 Slide 7
A slight shift…Classical Tutte polynomial:
( ) ( ); , ( ; , ) \ ; ,T G x y T G e x y T G e x y= − +
( ); , i jT G x y x y=
Let e be an edge of G that is neither a bridge nor a loop. Then,
And if G consists of i bridges and j loops, then
( ) ( ) ( ); , 1 ; ,V k Gk G u vu v T G v Z G u vv
− +⎛ ⎞+ =⎜ ⎟⎝ ⎠
The Tutte polynomial is a translation of the dichromatic/Potts
4/10/08 Slide 8
Fully ‘multivariablized’ Tutte polynomialZaslavsky ’92, Bollobás & Riordan ’99, (E-M &Traldi ’06)
( ),n nW E c α= nE
( )( )( )( )
/ , if is a bridge
, , if is a loop
/ , ( , ) else
e
e
e e
X W G e c e
W G c Y W G e c e
x W G e c y W G e c
⎧⎪
= −⎨⎪ + −⎩
Let , where is the edgeless graph on n vertices.
Each edge has four parameters associated with it: two ‘interaction energies’, a loop value, and a bridge value.
4/10/08 Slide 9
Need to be careful about order…
( )( )
0
0
0
X y y X x Y Y x
Y x Y Y x x y y x
X x Y Y x x y y x
λ μ λ μ λ μ λ μ
ν λ μ λ μ λ μ λ μ
ν λ μ λ μ λ μ λ μ
− − + =
− − + =
− − + =
Need to have:
Necessary and sufficient to assure the function is well-defined, i.e. independent of the order of deletion and contraction.
vs.
4/10/08 Slide 10
Dichromatic analogDon’t always get something analogous to the dichromatic, but do in all ‘reasonable’ situations, e.g. parameters in a field and not all zero, domain = all graphs.
In this case, with
( ) ( ) ( ) ( )
( ); , , ,
c
S n k Sk S k Ge e
S E G e S e S
Z G q r q r x y− +−
⊆ ∈ ∈
= ∑ ∏ ∏x y
, e e e e e eX x qy Y x ry= + = +
This gives a model with two degrees of freedom in the weights.
( )
( ) c
k Se e
S E G e S e S
q x y⊆ ∈ ∈
→ ∑ ∏ ∏
4/10/08 Slide 11
The Martin, or circuit partition polynomial
These polynomials are generating functions for Eulerian k-partitions in a graph.
Martin ’77, cf Las Vergnas
Let G be an Eulerian graph (all vertices of even degree).
Let G be an Eulerian digraph (directed edges and Kirchhoff laws).
( ) ( )1 1
; ( ) , ; ( )k kk k
k k
J G x f G x j G x f G x= =
= =∑ ∑
G = An Euleriandigraph G :
An Eulerian 2-partition of G :
1-partition 2-Partition 3-Partition
4/10/08 Slide 12
State Model and Recursive Forms
= xv → 1 + 1 + 1
v → 1 + 0 + 1
The Circuit Partition Polynomial for Oriented Graphs: (The weight is 1 for coherentstates, 0 else)
The Circuit Partition Polynomial
= x
( )( )
( )( )
( ) ( ); , ;c S c S
S states G S states G
J G x x j G x x∈ ∈
= =∑ ∑A graph state with 2 components
Multiple formulations facilitate theoretical techniques, e.g. induction.
4/10/08 Slide 13
v → a + b + c
= x
The general form:
A multivariable, or weighted, version“On Transition Polynomials of 4-Regular Graphs” --F. Jaeger, 1987
( ) 3 2; , 2Q G x x xA x= + +
v → 1 + 1 + 0
= x
, get
+
+ + +
Then for G =
So
For example, if the weight system is:
A is the weight system specifying the coefficient for each transition.
4/10/08 Slide 14
, where G is an arbitrary Eulerian graph, and W is a weight system which assigns a value in R (a ring with unit) to every pair of adjacent half edges in G. Then,
( ); ,N G W x
if pair bounds the same black face
if pair bounds the same white face else
b
W w
c
⎧⎪⎪= ⎨⎪⎪⎩
+ defabc + … = lmnx
E.g.
(New edge pairs have weight 1)
G =
4 3 2 2 2 2 2 2 ...b x b w x b c x= + + +
+ w2
+ …
+ c2b2
=
b2 b2 + w2 + c2
The generalized transition polynomialE-M 98, E-M & Sarmiento 02
4/10/08 Slide 15
Overview of relationships
Parameterized TuttePolynomials
The ClassicalTutte Polynomial
The GeneralizedTransition Polynomial
For planar graphs via the medial graph
The Circuit Partition Polynomial
4/10/08 Slide 16
The Tutte-transition connection
A Planar graph G Gm with the vertex faces colored black
Orient Gm so that black faces are to the left of each edge.
e
delete
contract
Then, with this orientation of Gm, ( ) ( ); ( ; 1, 1)k Gmj G x x t G x x= + +
Martin 77
4/10/08 Slide 17
Same construction for multivariable case
v → √a √a + √b √b + 0
ewith weights a and b
The edge weights of G induce a weight system on mG
G a planar graphs with oriented medial graph mG
Then ( ) ( ); , ( ; , , , )k GmN G W x x Z G x x= v w
The multivariable Tutte & transition polynomials are alsorelated for planar graphs via the medial construction.
4/10/08 Slide 18
Beyond Tutte: The Penrose Polynomial
N assimilates polynomials the Tutte polynomial doesn’t.
( ); , ( ; )mN G A x P G x=
v → 0 + 1 - 1
Defined for planar graphs and computed via the medial graph. Motivated by diagrams from tensor analysis, and surprisingly, for 3-regular planar graphs,
# 3-edge colorings ↔ 4 color theorem( ) ( )21;3 ; 2
4
V
P G P G−⎛ ⎞= − =⎜ ⎟⎝ ⎠
“Applications of Negative Dimensional Tensors”—R. Penrose, 1969
4/10/08 Slide 19
The Kauffman bracket of knot theory
( )2 2 2 2; , ( ) [ ]LN G W a a a a K L− −+ = +
v+
→ 1/( √a √a) + √a √a + 0
v-
→ √a √a +1/( √a √a) + 0
The Kauffman bracket of a link L:
[ ] 1K =○[ ] 1K L∪ =○
Let GL be the signed, 2 face colored universe of a link L:
a + a-1
4/10/08 Slide 20
Hopf Algebras—definition by exampleThe Binomial Bialgebra B
B is an infinite dimensional vector space over C with basis
B is an algebra, with (associative) multiplication given by:by
B is a coalgebra, with (coassociative) comultiplication given by:
B is a bialgebra, since the multiplication and the comultiplicationare compatible, i.e. the comultiplication is an algebra map:
, where
B is a Hopf algebra, with antipode:
{ }21, ,x x …
:m B B B⊗ → ( )r s r sm x x x +⊗ =
0
nn n r r
r
nx x x
r−
=
⎛ ⎞Δ = ⊗⎜ ⎟
⎝ ⎠∑
( )r s r sx x x xΔ ⋅ = Δ ⋅Δ ( ) ( )a b c d ac bd⊗ ⋅ ⊗ = ⊗
( )S x x= −
4/10/08 Slide 21
is a locally finite partially ordered set with order relationΓ
( )( ) ( )( ), ,A W A G W G≤
( )W A ( )W G
Γ is a hereditary family
if A is an Eulerian subgraph of G, and if A has a weight system
that is inherited from
.
is also closed under disjoint union (direct product)
( ){ , }G WΓ =
Underlying Algebraic Structure
Γ
4/10/08 Slide 22
:m Γ⊗Γ→ Γ
( )( ) ( )( )( ) ( )( ), , ,m G W G H W H GH W GH⊗ =
: Rμ →Γ ( ) ( ),r r Wμ = E
:Δ Γ→ Γ⊗Γ
( ) ( )( ) ( )( )1 1 2 2, , ,G W A W A A W AΔ = ⊗∑
: Rε Γ → ( )1 if
,0 else
GG Wε
=⎧= ⎨⎩
E
( ) ( ) ( )( )1 1, 1 ,PP PG W A A W A Aζ = −∑ … …
P
is an incidence Hopf algebra (Schmitt, 94).
by
Unit: by
Comultiplication: by
Counit: by
Antipode:
sum over all ordered partitions P of G into edge-disjoint Eulerian subgraphs.
Multiplication:
( ){ , }Rspan G WΓ =
Incidence Hopf Algebra
4/10/08 Slide 23
Theorem: If we give the structure of a binomial
bialgebra, then is a Hopf algebra map.
The proof is straightforward combinatorics, but requires hereditary properties, and hence the pair weights and not just state weights.
[ ]R x
[ ]:N R xΓ→
N is a Hopf algebra map
4/10/08 Slide 24
Structural identities
( ) ( )( ) ( )( )1 1 2 2; , ; , ; ,N G W x y N A W A x N A W A y+ =∑
( ) ( )( ); , , ;N G W x N G W xζ− =
II. (From the antipode)
I. (From the comultiplication)
These are powerful theoretic tools, particularly for inductive arguments.
4/10/08 Slide 25
Identity I:Combinatorial interpretations for the Martin polynomials for allintegers (previously only known for –2, -1, 0, 1 in the oriented case, and –2, 0, 2 in the unoriented case) –E-M, also Bollobas.
Combinatorial interpretations for the diagonal Tutte polynomial (also derivatives) of a planar graph for all integers (previously –1, 3 were the only known non-trivial values) – E-M.
Identity II:Used to determine combinatorial interpretations for the Penrose polynomial for all negative integers (previously only known for positive integers)—Sarmiento, E-M&Sarmiento.
Some applications
4/10/08 Slide 26
Jo Ellis-MonaghanSt. Michaels College, Colchester, VT USAe-mail: [email protected]: http://academics.smcvt.edu/jellis-monaghan