A survey of Tutte-Whitney polynomials Graham Farr Faculty of IT - - PowerPoint PPT Presentation

A survey of Tutte-Whitney polynomials

Graham Farr

Faculty of IT Monash University Graham.Farr@infotech.monash.edu.au

July 2007

Counting colourings

◮ proper colourings

Counting colourings

◮ proper colourings

Counting colourings

◮ proper colourings

Adjacent vertices receive different colours

Counting colourings

◮ proper colourings

Adjacent vertices receive different colours

◮ chromatic polynomial:

P(G; q) = # q-colourings of G

Deletion-contraction

For any edge e: P(G; q) = P(G \ e; q) − P(G/e; q) e u v u v u = v

Partition functions: Potts models

◮ general q-colourings (may be improper)

Partition functions: Potts models

◮ general q-colourings (may be improper)

Partition functions: Potts models

◮ general q-colourings (may be improper)

Partition functions: Potts models

◮ general q-colourings (may be improper)

Good and bad edges

Partition functions: Potts models

◮ general q-colourings (may be improper)

Good and bad edges

◮ Partition function:

Z(G; K, q) =

• all q-colourings

(not just proper) e−K·(# good edges)

All-terminal reliability

◮ Choose edges randomly: Pr(edge) = p

All-terminal reliability

◮ Choose edges randomly: Pr(edge) = p ◮ Want chosen edges to contain a spanning tree

All-terminal reliability

◮ Choose edges randomly: Pr(edge) = p ◮ Want chosen edges to contain a spanning tree

All-terminal reliability

◮ Choose edges randomly: Pr(edge) = p ◮ Want chosen edges to contain a spanning tree

chosen edges

All-terminal reliability

◮ Choose edges randomly: Pr(edge) = p ◮ Want chosen edges to contain a spanning tree

chosen edges

◮ Reliability:

Π(G, p) = Pr(chosen edges contain a spanning tree)

. . . etc

. . . etc

◮ flow polynomial

. . . etc

◮ flow polynomial ◮ # spanning trees, forests, spanning subgraphs

. . . etc

◮ flow polynomial ◮ # spanning trees, forests, spanning subgraphs ◮ weight enumerator of a linear code

. . . etc

◮ flow polynomial ◮ # spanning trees, forests, spanning subgraphs ◮ weight enumerator of a linear code ◮ Jones polynomial of an alternating link

. . . etc

◮ flow polynomial ◮ # spanning trees, forests, spanning subgraphs ◮ weight enumerator of a linear code ◮ Jones polynomial of an alternating link ◮ . . .

Tutte-Whitney polynomials

◮ The rank function of a graph:

for all X ⊆ E: ρ(X) := (# vertices that meet X) − (# components of X).

Tutte-Whitney polynomials

◮ The rank function of a graph:

for all X ⊆ E: ρ(X) := (# vertices that meet X) − (# components of X).

◮ Whitney rank generating function:

R(G; x, y) =

• X⊆E

xρ(E)−ρ(X)y|X|−ρ(X).

Tutte-Whitney polynomials

◮ The rank function of a graph:

for all X ⊆ E: ρ(X) := (# vertices that meet X) − (# components of X).

◮ Whitney rank generating function:

R(G; x, y) =

• X⊆E

xρ(E)−ρ(X)y|X|−ρ(X).

◮ Tutte polynomial:

T(G; x, y) = R(G; x − 1, y − 1).

The “Recipe Theorem”

Theorem

(Tutte 1947 → Brylawski 1972 → Oxley & Welsh 1979) If a function f on graphs . . .

◮ is invariant under isomorphism, ◮ satisfies a deletion-contraction relation, ◮ is multiplicative over components

(i.e., f (G1 ∪ G2) = f (G1) · f (G2)), . . . then f is essentially a (partial) evaluation of the Tutte-Whitney polynomial.

The “Recipe Theorem”

Theorem

(Tutte 1947 → Brylawski 1972 → Oxley & Welsh 1979) If a function f on graphs . . .

◮ is invariant under isomorphism, ◮ satisfies a deletion-contraction relation, ◮ is multiplicative over components

(i.e., f (G1 ∪ G2) = f (G1) · f (G2)), . . . then f is essentially a (partial) evaluation of the Tutte-Whitney polynomial.

Example

P(G; q) = (−1)ρ(E)qk(G)R(G; −q, −1)

x 2 1 −1 −2 2 1

• 1
• 2

y

x 2 1 −1 −2 2 1

• 1
• 2

y chromatic

x 2 1 −1 −2 2 1

• 1
• 2

y chromatic flow

x 2 1 −1 −2 2 1

• 1
• 2

y chromatic flow reliability

x 2 1 −1 −2 2 1

• 1
• 2

y chromatic flow reliability weight enumerator

x 2 1 −1 −2 2 1

• 1
• 2

y chromatic flow reliability weight enumerator Potts model

x 2 1 −1 −2 2 1

• 1
• 2

y chromatic flow reliability weight enumerator Potts model Jones

x 2 1 −1 −2 2 1

• 1
• 2

y chromatic flow reliability weight enumerator Potts model Jones easy

x 2 1 −1 −2 2 1

• 1
• 2

y chromatic flow reliability weight enumerator Potts model Jones easy

History

History

Graphs: Chrom. poly

History

Graphs: Chrom. poly

✓ ✒ ✏ ✑

P(G; q) Birkhoff 1912

History

Graphs: Chrom. poly

✓ ✒ ✏ ✑

P(G; q) Birkhoff 1912

✲ ✓ ✒ ✏ ✑

R(G; x, y) Whitney, 1935 Tutte, 1947

History

Graphs: Chrom. poly

✓ ✒ ✏ ✑

P(G; q) Birkhoff 1912

✲ ✓ ✒ ✏ ✑

R(G; x, y) Whitney, 1935 Tutte, 1947

✲ ✓ ✒ ✏ ✑

T(G; x, y) Tutte 1954

History

Graphs: Chrom. poly

✓ ✒ ✏ ✑

P(G; q) Birkhoff 1912

✲ ✓ ✒ ✏ ✑

R(G; x, y) Whitney, 1935 Tutte, 1947

✲ ✓ ✒ ✏ ✑

T(G; x, y) Tutte 1954

✲ ❛ ❛ ❛ ❛ ✦ ✦ ✦ ✦

History

Graphs: Chrom. poly

✓ ✒ ✏ ✑

P(G; q) Birkhoff 1912

✲ ✓ ✒ ✏ ✑

R(G; x, y) Whitney, 1935 Tutte, 1947

✲ ✓ ✒ ✏ ✑

T(G; x, y) Tutte 1954

✲ ❛ ❛ ❛ ❛ ✦ ✦ ✦ ✦

Stat Mech: partition functions

History

Graphs: Chrom. poly

✓ ✒ ✏ ✑

P(G; q) Birkhoff 1912

✲ ✓ ✒ ✏ ✑

R(G; x, y) Whitney, 1935 Tutte, 1947

✲ ✓ ✒ ✏ ✑

T(G; x, y) Tutte 1954

✲ ❛ ❛ ❛ ❛ ✦ ✦ ✦ ✦

Stat Mech: partition functions

★ ✧ ✥ ✦

Ising model (q = 2) Ising 1925

History

Graphs: Chrom. poly

✓ ✒ ✏ ✑

P(G; q) Birkhoff 1912

✲ ✓ ✒ ✏ ✑

R(G; x, y) Whitney, 1935 Tutte, 1947

✲ ✓ ✒ ✏ ✑

T(G; x, y) Tutte 1954

✲ ❛ ❛ ❛ ❛ ✦ ✦ ✦ ✦

Stat Mech: partition functions

★ ✧ ✥ ✦

Ising model (q = 2) Ising 1925

✲ ✬ ✫ ✩ ✪

Potts model (all q) Potts 1952

History

Graphs: Chrom. poly

✓ ✒ ✏ ✑

P(G; q) Birkhoff 1912

✲ ✓ ✒ ✏ ✑

R(G; x, y) Whitney, 1935 Tutte, 1947

✲ ✓ ✒ ✏ ✑

T(G; x, y) Tutte 1954

✲ ❛ ❛ ❛ ❛ ✦ ✦ ✦ ✦

Stat Mech: partition functions

★ ✧ ✥ ✦

Ising model (q = 2) Ising 1925

✲ ✬ ✫ ✩ ✪

Potts model (all q) Potts 1952

✁ ✁ ✕ ✗ ✖ ✔ ✕

Ashkin-Teller model (q = 4)

• A. & T., 1943

History

Graphs: Chrom. poly

✓ ✒ ✏ ✑

P(G; q) Birkhoff 1912

✲ ✓ ✒ ✏ ✑

R(G; x, y) Whitney, 1935 Tutte, 1947

✲ ✓ ✒ ✏ ✑

T(G; x, y) Tutte 1954

✲ ❛ ❛ ❛ ❛ ✦ ✦ ✦ ✦

Stat Mech: partition functions

★ ✧ ✥ ✦

Ising model (q = 2) Ising 1925

✲ ✬ ✫ ✩ ✪

Potts model (all q) Potts 1952

✁ ✁ ✕ ✗ ✖ ✔ ✕

Ashkin-Teller model (q = 4)

• A. & T., 1943

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❫

Fortuin & Kasteleyn 1972

✲

History

Graphs: Chrom. poly

✓ ✒ ✏ ✑

P(G; q) Birkhoff 1912

✲ ✓ ✒ ✏ ✑

R(G; x, y) Whitney, 1935 Tutte, 1947

✲ ✓ ✒ ✏ ✑

T(G; x, y) Tutte 1954

✲ ❛ ❛ ❛ ❛ ✦ ✦ ✦ ✦

Stat Mech: partition functions

★ ✧ ✥ ✦

Ising model (q = 2) Ising 1925

✲ ✬ ✫ ✩ ✪

Potts model (all q) Potts 1952

✁ ✁ ✕ ✗ ✖ ✔ ✕

Ashkin-Teller model (q = 4)

• A. & T., 1943

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❫

Fortuin & Kasteleyn 1972

✲

Linear codes: weight enumerator

History

Graphs: Chrom. poly

✓ ✒ ✏ ✑

P(G; q) Birkhoff 1912

✲ ✓ ✒ ✏ ✑

R(G; x, y) Whitney, 1935 Tutte, 1947

✲ ✓ ✒ ✏ ✑

T(G; x, y) Tutte 1954

✲ ❛ ❛ ❛ ❛ ✦ ✦ ✦ ✦

Stat Mech: partition functions

★ ✧ ✥ ✦

Ising model (q = 2) Ising 1925

✲ ✬ ✫ ✩ ✪

Potts model (all q) Potts 1952

✁ ✁ ✕ ✗ ✖ ✔ ✕

Ashkin-Teller model (q = 4)

• A. & T., 1943

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❫

Fortuin & Kasteleyn 1972

✲

Linear codes: weight enumerator

✓ ✒ ✏ ✑

AC(z) MacWilliams 1963

History

Graphs: Chrom. poly

✓ ✒ ✏ ✑

P(G; q) Birkhoff 1912

✲ ✓ ✒ ✏ ✑

R(G; x, y) Whitney, 1935 Tutte, 1947

✲ ✓ ✒ ✏ ✑

T(G; x, y) Tutte 1954

✲ ❛ ❛ ❛ ❛ ✦ ✦ ✦ ✦

Stat Mech: partition functions

★ ✧ ✥ ✦

Ising model (q = 2) Ising 1925

✲ ✬ ✫ ✩ ✪

Potts model (all q) Potts 1952

✁ ✁ ✕ ✗ ✖ ✔ ✕

Ashkin-Teller model (q = 4)

• A. & T., 1943

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❫

Fortuin & Kasteleyn 1972

✲

Linear codes: weight enumerator

✓ ✒ ✏ ✑

AC(z) MacWilliams 1963

✡ ✡ ✡ ✡ ✡ ✣

Greene 1974

✲

History

Graphs: Chrom. poly

✓ ✒ ✏ ✑

P(G; q) Birkhoff 1912

✲ ✓ ✒ ✏ ✑

R(G; x, y) Whitney, 1935 Tutte, 1947

✲ ✓ ✒ ✏ ✑

T(G; x, y) Tutte 1954

✲ ❛ ❛ ❛ ❛ ✦ ✦ ✦ ✦

Stat Mech: partition functions

★ ✧ ✥ ✦

Ising model (q = 2) Ising 1925

✲ ✬ ✫ ✩ ✪

Potts model (all q) Potts 1952

✁ ✁ ✕ ✗ ✖ ✔ ✕

Ashkin-Teller model (q = 4)

• A. & T., 1943

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❫

Fortuin & Kasteleyn 1972

✲

Linear codes: weight enumerator

✓ ✒ ✏ ✑

AC(z) MacWilliams 1963

✡ ✡ ✡ ✡ ✡ ✣

Greene 1974

✲

Network reliability:

History

Graphs: Chrom. poly

✓ ✒ ✏ ✑

P(G; q) Birkhoff 1912

✲ ✓ ✒ ✏ ✑

R(G; x, y) Whitney, 1935 Tutte, 1947

✲ ✓ ✒ ✏ ✑

T(G; x, y) Tutte 1954

✲ ❛ ❛ ❛ ❛ ✦ ✦ ✦ ✦

Stat Mech: partition functions

★ ✧ ✥ ✦

Ising model (q = 2) Ising 1925

✲ ✬ ✫ ✩ ✪

Potts model (all q) Potts 1952

✁ ✁ ✕ ✗ ✖ ✔ ✕

Ashkin-Teller model (q = 4)

• A. & T., 1943

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❫

Fortuin & Kasteleyn 1972

✲

Linear codes: weight enumerator

✓ ✒ ✏ ✑

AC(z) MacWilliams 1963

✡ ✡ ✡ ✡ ✡ ✣

Greene 1974

✲

Network reliability:

✓ ✒ ✏ ✑

Π(G; p) van Slyke & Frank, 1971

History

Graphs: Chrom. poly

✓ ✒ ✏ ✑

P(G; q) Birkhoff 1912

✲ ✓ ✒ ✏ ✑

R(G; x, y) Whitney, 1935 Tutte, 1947

✲ ✓ ✒ ✏ ✑

T(G; x, y) Tutte 1954

✲ ❛ ❛ ❛ ❛ ✦ ✦ ✦ ✦

Stat Mech: partition functions

★ ✧ ✥ ✦

Ising model (q = 2) Ising 1925

✲ ✬ ✫ ✩ ✪

Potts model (all q) Potts 1952

✁ ✁ ✕ ✗ ✖ ✔ ✕

Ashkin-Teller model (q = 4)

• A. & T., 1943

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❫

Fortuin & Kasteleyn 1972

✲

Linear codes: weight enumerator

✓ ✒ ✏ ✑

AC(z) MacWilliams 1963

✡ ✡ ✡ ✡ ✡ ✣

Greene 1974

✲

Network reliability:

✓ ✒ ✏ ✑

Π(G; p) van Slyke & Frank, 1971

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ◆

Oxley & Welsh 1979

History

Graphs: Chrom. poly

✓ ✒ ✏ ✑

P(G; q) Birkhoff 1912

✲ ✓ ✒ ✏ ✑

R(G; x, y) Whitney, 1935 Tutte, 1947

✲ ✓ ✒ ✏ ✑

T(G; x, y) Tutte 1954

✲ ❛ ❛ ❛ ❛ ✦ ✦ ✦ ✦

Stat Mech: partition functions

★ ✧ ✥ ✦

Ising model (q = 2) Ising 1925

✲ ✬ ✫ ✩ ✪

Potts model (all q) Potts 1952

✁ ✁ ✕ ✗ ✖ ✔ ✕

Ashkin-Teller model (q = 4)

• A. & T., 1943

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❫

Fortuin & Kasteleyn 1972

✲

Linear codes: weight enumerator

✓ ✒ ✏ ✑

AC(z) MacWilliams 1963

✡ ✡ ✡ ✡ ✡ ✣

Greene 1974

✲

Network reliability:

✓ ✒ ✏ ✑

Π(G; p) van Slyke & Frank, 1971

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ◆

Oxley & Welsh 1979

Knots: Jones poly

History

Graphs: Chrom. poly

✓ ✒ ✏ ✑

P(G; q) Birkhoff 1912

✲ ✓ ✒ ✏ ✑

R(G; x, y) Whitney, 1935 Tutte, 1947

✲ ✓ ✒ ✏ ✑

T(G; x, y) Tutte 1954

✲ ❛ ❛ ❛ ❛ ✦ ✦ ✦ ✦

Stat Mech: partition functions

★ ✧ ✥ ✦

Ising model (q = 2) Ising 1925

✲ ✬ ✫ ✩ ✪

Potts model (all q) Potts 1952

✁ ✁ ✕ ✗ ✖ ✔ ✕

Ashkin-Teller model (q = 4)

• A. & T., 1943

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❫

Fortuin & Kasteleyn 1972

✲

Linear codes: weight enumerator

✓ ✒ ✏ ✑

AC(z) MacWilliams 1963

✡ ✡ ✡ ✡ ✡ ✣

Greene 1974

✲

Network reliability:

✓ ✒ ✏ ✑

Π(G; p) van Slyke & Frank, 1971

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ◆

Oxley & Welsh 1979

Knots: Jones poly

✓ ✒ ✏ ✑

VL(t) Jones 1985

History

Graphs: Chrom. poly

✓ ✒ ✏ ✑

P(G; q) Birkhoff 1912

✲ ✓ ✒ ✏ ✑

R(G; x, y) Whitney, 1935 Tutte, 1947

✲ ✓ ✒ ✏ ✑

T(G; x, y) Tutte 1954

✲ ❛ ❛ ❛ ❛ ✦ ✦ ✦ ✦

Stat Mech: partition functions

★ ✧ ✥ ✦

Ising model (q = 2) Ising 1925

✲ ✬ ✫ ✩ ✪

Potts model (all q) Potts 1952

✁ ✁ ✕ ✗ ✖ ✔ ✕

Ashkin-Teller model (q = 4)

• A. & T., 1943

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❫

Fortuin & Kasteleyn 1972

✲

Linear codes: weight enumerator

✓ ✒ ✏ ✑

AC(z) MacWilliams 1963

✡ ✡ ✡ ✡ ✡ ✣

Greene 1974

✲

Network reliability:

✓ ✒ ✏ ✑

Π(G; p) van Slyke & Frank, 1971

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ◆

Oxley & Welsh 1979

Knots: Jones poly

✓ ✒ ✏ ✑

VL(t) Jones 1985 ✆

✆ ✆ ✆ ✆ ✆ ✆ ✆

Thistle- thwaite 1987

Complexity of computing all of R(G; x, y)

◮ Graphs:

#P-hard (Linial, 1986)

Complexity of computing all of R(G; x, y)

◮ Graphs:

#P-hard (Linial, 1986)

◮ Bipartite graphs:

#P-hard (Linial, 1986)

Complexity of computing all of R(G; x, y)

◮ Graphs:

#P-hard (Linial, 1986)

◮ Bipartite graphs:

#P-hard (Linial, 1986)

◮ Bipartite planar graphs:

#P-hard (Vertigan & Welsh, 1992)

Complexity of computing all of R(G; x, y)

◮ Graphs:

#P-hard (Linial, 1986)

◮ Bipartite graphs:

#P-hard (Linial, 1986)

◮ Bipartite planar graphs:

#P-hard (Vertigan & Welsh, 1992)

◮ Planar graphs, max degree 3:

#P-hard (Vertigan, 1990)

Complexity of computing all of R(G; x, y)

◮ Graphs:

#P-hard (Linial, 1986)

◮ Bipartite graphs:

#P-hard (Linial, 1986)

◮ Bipartite planar graphs:

#P-hard (Vertigan & Welsh, 1992)

◮ Planar graphs, max degree 3:

#P-hard (Vertigan, 1990)

◮ Bounded tree-width:

p-time (Noble, 1998; Andrzejak, 1998)

Complexity of computing all of R(G; x, y)

◮ Graphs:

#P-hard (Linial, 1986)

◮ Bipartite graphs:

#P-hard (Linial, 1986)

◮ Bipartite planar graphs:

#P-hard (Vertigan & Welsh, 1992)

◮ Planar graphs, max degree 3:

#P-hard (Vertigan, 1990)

◮ Square grid graphs:

Open (in #P1)

◮ Bounded tree-width:

p-time (Noble, 1998; Andrzejak, 1998)

Complexity of computing all of R(G; x, y)

◮ Graphs:

#P-hard (Linial, 1986)

◮ Bipartite graphs:

#P-hard (Linial, 1986)

◮ Bipartite planar graphs:

#P-hard (Vertigan & Welsh, 1992)

◮ Planar graphs, max degree 3:

#P-hard (Vertigan, 1990)

◮ Square grid subgraphs, max deg 3:

#P-hard (GF, 2006)

◮ Square grid graphs:

Open (in #P1)

◮ Bounded tree-width:

p-time (Noble, 1998; Andrzejak, 1998)

Complexity of evaluating at specific points

Theorem

(Jaeger, Vertigan and Welsh, 1990) The problem of determining R(G; x, y), given a graph G, is #P-hard at all points (x, y) except those where xy = 1 and (x, y) = (0, 0), (−1, −2), (−2, −1), (−2, −2).

Generalisations

Extensions from graphs to:

Generalisations

Extensions from graphs to:

◮ representable matroids (Smith), matroids (Tutte, Crapo),

greedoids (Gordon & McMahon), Boolean functions or set systems (GF), hyperplane arrangements (Welsh & Whittle, Ardila), semimatroids (Ardila), signed graphs (Murasugi), rooted graphs (Wu, King & Lu), K-terminal graphs (Traldi), biased graphs (Zaslavsky), matroid perspectives (Las Vergnas), matroid pairs (Welsh & Kayibi), bimatroids (Kung), graphic polymatroids (Borzacchini), general polymatroids (Oxley & Whittle), . . .

Generalisations

Extensions from graphs to:

◮ representable matroids (Smith), matroids (Tutte, Crapo),

greedoids (Gordon & McMahon), Boolean functions or set systems (GF), hyperplane arrangements (Welsh & Whittle, Ardila), semimatroids (Ardila), signed graphs (Murasugi), rooted graphs (Wu, King & Lu), K-terminal graphs (Traldi), biased graphs (Zaslavsky), matroid perspectives (Las Vergnas), matroid pairs (Welsh & Kayibi), bimatroids (Kung), graphic polymatroids (Borzacchini), general polymatroids (Oxley & Whittle), . . . . . . or extend the polynomials:

Generalisations

Extensions from graphs to:

◮ representable matroids (Smith), matroids (Tutte, Crapo),

greedoids (Gordon & McMahon), Boolean functions or set systems (GF), hyperplane arrangements (Welsh & Whittle, Ardila), semimatroids (Ardila), signed graphs (Murasugi), rooted graphs (Wu, King & Lu), K-terminal graphs (Traldi), biased graphs (Zaslavsky), matroid perspectives (Las Vergnas), matroid pairs (Welsh & Kayibi), bimatroids (Kung), graphic polymatroids (Borzacchini), general polymatroids (Oxley & Whittle), . . . . . . or extend the polynomials:

◮ multivariate polynomials of various kinds: variables at each

vertex (Noble & Welsh), or edge (Fortuin & Kasteleyn, Traldi, Kung, Sokal, Bollob´ as & Riordan, Zaslavsky, Ellis-Monaghan & Riordan, Britz).

Generalisations

Generalisations

Common themes:

Generalisations

Common themes:

◮ interesting partial evaluations

Generalisations

Common themes:

◮ interesting partial evaluations ◮ deletion-contraction relations

Generalisations

Common themes:

◮ interesting partial evaluations ◮ deletion-contraction relations ◮ Recipe Theorems

Generalisations

Common themes:

◮ interesting partial evaluations ◮ deletion-contraction relations ◮ Recipe Theorems ◮ easier proofs

Generalisations

Common themes:

◮ interesting partial evaluations ◮ deletion-contraction relations ◮ Recipe Theorems ◮ easier proofs ◮ roots

Generalisations

Common themes:

◮ interesting partial evaluations ◮ deletion-contraction relations ◮ Recipe Theorems ◮ easier proofs ◮ roots ◮ how much of the graph is determined by the polynomial?

Generalisations

Common themes:

◮ interesting partial evaluations ◮ deletion-contraction relations ◮ Recipe Theorems ◮ easier proofs ◮ roots ◮ how much of the graph is determined by the polynomial?

We now look at a generalisation to Boolean functions . . .

Rank ↔ rowspace

Incidence matrix edges vertices 0/1 entries · · · · · · . . . . . .                        

Rank ↔ rowspace

Incidence matrix vertices 0/1 entries · · · · · · . . . . . .                         E \ W

• W
• E

(edge set)

Rank ↔ rowspace

Incidence matrix vertices 0/1 entries · · · · · · . . . . . .                         E \ W

• W
• E

(edge set)

• −

→ echelon form

Rank ↔ rowspace

Incidence matrix                         E \ W

• W
• E

(edge set)

• −

→ echelon form I · · · · · · · · · I . . .

Rank ↔ rowspace

Incidence matrix                         E \ W

• W
• E

(edge set)

• −

→ echelon form I · · · · · · · · · I . . .

✻ ❄

ρ(E \ W )

✻ ❄

ρ(E)

Rank ↔ rowspace

Incidence matrix                         E \ W

• W
• E

(edge set)

• −

→ echelon form I · · · · · · · · · I . . .

✻ ❄

ρ(E \ W )

✻ ❄

ρ(E) Count rowspace members that are 0 outside W : 2ρ(E)−ρ(E\W ) =

• X⊆W

indRowspace(X)

Count rowspace members that are 0 outside W : 2ρ(E)−ρ(E\W ) =

• X⊆W

indRowspace(X)

Count rowspace members that are 0 outside W : 2ρ(E)−ρ(E\W ) =

• X⊆W

indRowspace(X) ρ(W ) = log2

X⊆E ind(X)

• X⊆E\W ind(X)

Count rowspace members that are 0 outside W : 2ρ(E)−ρ(E\W ) =

• X⊆W

indRowspace(X) ρ(W ) = log2

X⊆E ind(X)

• X⊆E\W ind(X)
• Generalise to other functions

(not necessarily indicator functions of rowspaces) (GF, 1993):

Count rowspace members that are 0 outside W : 2ρ(E)−ρ(E\W ) =

• X⊆W

indRowspace(X) ρ(W ) = log2

X⊆E ind(X)

• X⊆E\W ind(X)
• Generalise to other functions

(not necessarily indicator functions of rowspaces) (GF, 1993): For any f : 2E → {0, 1} . . . or . . . → R . . . : Define Qf by: (Qf )(W ) = log2

X⊆E f (X)

• X⊆E\W f (X)

Count rowspace members that are 0 outside W : 2ρ(E)−ρ(E\W ) =

• X⊆W

indRowspace(X) ρ(W ) = log2

X⊆E ind(X)

• X⊆E\W ind(X)
• Generalise to other functions

(not necessarily indicator functions of rowspaces) (GF, 1993): For any f : 2E → {0, 1} . . . or . . . → R . . . : Define Qf by: (Qf )(W ) = log2

X⊆E f (X)

• X⊆E\W f (X)
• Inversion:

Count rowspace members that are 0 outside W : 2ρ(E)−ρ(E\W ) =

• X⊆W

indRowspace(X) ρ(W ) = log2

X⊆E ind(X)

• X⊆E\W ind(X)
• Generalise to other functions

(not necessarily indicator functions of rowspaces) (GF, 1993): For any f : 2E → {0, 1} . . . or . . . → R . . . : Define Qf by: (Qf )(W ) = log2

X⊆E f (X)

• X⊆E\W f (X)
• Inversion: if ρ : 2E → {0, 1} then define Q†ρ by

(Q†ρ)(V ) = (−1)|V |

W ⊆V

(−1)|W |2ρ(E)−ρ(E\W )

Properties of the transform Q

Basic properties:

◮ (Q†Qf )(V ) = f (V )

f (∅)

◮ (QQ†ρ)(V ) = ρ(V ) − ρ(∅)

Properties of the transform Q

Basic properties:

◮ (Q†Qf )(V ) = f (V )

f (∅)

◮ (QQ†ρ)(V ) = ρ(V ) − ρ(∅) ◮ Q†QQ† = Q† ◮ QQ†Q = Q

Properties of the transform Q

Basic properties:

◮ (Q†Qf )(V ) = f (V )

f (∅)

◮ (QQ†ρ)(V ) = ρ(V ) − ρ(∅) ◮ Q†QQ† = Q† ◮ QQ†Q = Q

Relationship with the Hadamard transform: ˆ f (W ) := 1 2n

• X⊆E

(−1)|W ∩X|f (X)

Properties of the transform Q

Basic properties:

◮ (Q†Qf )(V ) = f (V )

f (∅)

◮ (QQ†ρ)(V ) = ρ(V ) − ρ(∅) ◮ Q†QQ† = Q† ◮ QQ†Q = Q

Relationship with the Hadamard transform: ˆ f (W ) := 1 2n

• X⊆E

(−1)|W ∩X|f (X) f

✲ Q

Qf Hadamard transform ↓ ↓ matroid-style dual ˆ f

✲ Q

(Qf )∗ = Qˆ f

Extending the Whitney rank generating function

R(f ; x, y) =

• X⊆E

xQf (E)−Qf (X)y|X|−Qf (X)

Extending the Whitney rank generating function

R(f ; x, y) =

• X⊆E

xQf (E)−Qf (X)y|X|−Qf (X) R1(f ; x, y) = xQf (E)

X⊆E

(xy)−Qf (X)y|X|

Extending the Whitney rank generating function

R(f ; x, y) =

• X⊆E

xQf (E)−Qf (X)y|X|−Qf (X) R1(f ; x, y) = xQf (E)

X⊆E

(xy)−Qf (X)y|X| Example: E = {1, 2} X f ∅ 1 {1} 1 {2} 1 {1, 2}

Extending the Whitney rank generating function

R(f ; x, y) =

• X⊆E

xQf (E)−Qf (X)y|X|−Qf (X) R1(f ; x, y) = xQf (E)

X⊆E

(xy)−Qf (X)y|X| Example: E = {1, 2} X f ∅ 1 {1} 1 {2} 1 {1, 2} R(f ; x, y) = xlog2 3 + 2xy2−log2 3 + y2−log23

Deletion-contraction

For e ∈ E, X ⊆ E \ {e}: Deletion Contraction (f \ \e)(X) = f (X) + f (X ∪ {e}) f (∅) + f ({e}) ; (f / /e)(X) = f (X) f (∅) .

Deletion-contraction

For e ∈ E, X ⊆ E \ {e}: Deletion Contraction (f \ \e)(X) = f (X) + f (X ∪ {e}) f (∅) + f ({e}) ; (f / /e)(X) = f (X) f (∅) . Deletion-contraction rule: R(f ; x, y) = xlog2

“ 1+ f ({e}

f (∅)

”

R(f \ \e; x, y)+y

log2 “ 1+

ˆ f ({e} ˆ f (∅)

”

R(f / /e; x, y)

Generalised Tutte-Whitney plane

x 2 1 −1 −2 2 1

• 1
• 2

y

Generalised Tutte-Whitney plane

x 2 1 −1 −2 2 1

• 1
• 2

y chromatic

Generalised Tutte-Whitney plane

x 2 1 −1 −2 2 1

• 1
• 2

y chromatic clutter reliability

Generalised Tutte-Whitney plane

x 2 1 −1 −2 2 1

• 1
• 2

y chromatic clutter reliability weight enum. nonlinear code

Generalised Tutte-Whitney plane

x 2 1 −1 −2 2 1

• 1
• 2

y chromatic clutter reliability weight enum. nonlinear code

• gen. Potts model

Generalised Tutte-Whitney plane

x 2 1 −1 −2 2 1

• 1
• 2

y chromatic clutter reliability weight enum. nonlinear code

• gen. Potts model

easy

Generalised Tutte-Whitney plane

x 2 1 −1 −2 2 1

• 1
• 2

y chromatic clutter reliability weight enum. nonlinear code

• gen. Potts model

easy

Generalised Tutte-Whitney plane

x 2 1 −1 −2 2 1

• 1
• 2

y chromatic clutter reliability weight enum. nonlinear code

• gen. Potts model

easy

Interpolating between contraction and deletion

For e ∈ E, X ⊆ E \ {e}: Contraction Deletion (f / /e)(X) (f \ \e)(X) f (X) f (∅) f (X) + f (X ∪ {e}) f (∅) + f ({e})

Interpolating between contraction and deletion

For e ∈ E, X ⊆ E \ {e}: Contraction λ-minor Deletion (f / /e)(X) (f λe)(X) (f \ \e)(X) f (X) f (∅) f (X) + λf (X ∪ {e}) f (∅) + λf ({e}) f (X) + f (X ∪ {e}) f (∅) + f ({e})

Interpolating between contraction and deletion

For e ∈ E, X ⊆ E \ {e}: Contraction λ-minor Deletion (λ = 0) (λ = 1) (f / /e)(X) (f λe)(X) (f \ \e)(X) f (X) f (∅) f (X) + λf (X ∪ {e}) f (∅) + λf ({e}) f (X) + f (X ∪ {e}) f (∅) + f ({e})

Interpolating between contraction and deletion

For e ∈ E, X ⊆ E \ {e}: Contraction λ-minor Deletion (λ = 0) (λ = 1) (f / /e)(X) (f λe)(X) (f \ \e)(X) f (X) f (∅) f (X) + λf (X ∪ {e}) f (∅) + λf ({e}) f (X) + f (X ∪ {e}) f (∅) + f ({e}) λ 1

Duality

Duality between deletion and contraction can be extended.

Duality

Duality between deletion and contraction can be extended. Define λ∗ := 1 − λ 1 + λ

Duality

Duality between deletion and contraction can be extended. Define λ∗ := 1 − λ 1 + λ Then

• f λe = ˆ

f λ∗e

Duality

Duality between deletion and contraction can be extended. Define λ∗ := 1 − λ 1 + λ Then

• f λe = ˆ

f λ∗e Fixed points: λ = ± √ 2 − 1

λ-rank functions

Define Q(λ)f by: (Q(λ)f )(W ) = log2

• (1 + λ∗)|V |

X⊆E λ|X|f (X)

• X⊆E\W λ|W ∩ ¯

V |(λ∗)|W ∩V |f (X)

λ-rank functions

Define Q(λ)f by: (Q(λ)f )(W ) = log2

• (1 + λ∗)|V |

X⊆E λ|X|f (X)

• X⊆E\W λ|W ∩ ¯

V |(λ∗)|W ∩V |f (X)

• Duality:

(Q(λ))∗ = Q(λ∗)

λ-rank functions

Define Q(λ)f by: (Q(λ)f )(W ) = log2

• (1 + λ∗)|V |

X⊆E λ|X|f (X)

• X⊆E\W λ|W ∩ ¯

V |(λ∗)|W ∩V |f (X)

• Duality:

(Q(λ))∗ = Q(λ∗) Inversion: (Q†(λ)ρ)(V ) = (−1)|V |(λ − λ∗)−|S| ×

• W ⊆V

(−1)|W |(1 + λ∗)−|W |(λ∗)|W ∩ ¯

V |λ| ¯ W ∩ ¯ V |2ρ(E)−ρ(E\W )

A continuum of λ-Whitney functions

R(λ)(f ; x, y) =

• X⊆E

xQ(λ)f (E)−Q(λ)f (X)y|X|−Q(λ)f (X)

A continuum of λ-Whitney functions

R(λ)(f ; x, y) =

• X⊆E

xQ(λ)f (E)−Q(λ)f (X)y|X|−Q(λ)f (X) R(λ)

1 (f ; x, y)

= xQ(λ)f (E)

X⊆E

(xy)−Q(λ)f (X)y|X|

A continuum of λ-Whitney functions

R(λ)(f ; x, y) =

• X⊆E

xQ(λ)f (E)−Q(λ)f (X)y|X|−Q(λ)f (X) R(λ)

1 (f ; x, y)

= xQ(λ)f (E)

X⊆E

(xy)−Q(λ)f (X)y|X| R(ˆ f ; x, y) √ 2 − 1 (√x + √y)|E| λ R(λ)(f ; x, y) 1 R(f ; x, y)

Properties of the λ-Whitney function:

R(λ)(f ; x, y)

◮ obeys a deletion-contraction-type relation

(with operations λ , λ∗ );

Properties of the λ-Whitney function:

R(λ)(f ; x, y)

◮ obeys a deletion-contraction-type relation

(with operations λ , λ∗ );

◮ contains the weight enumerator of a nonlinear code

Properties of the λ-Whitney function:

R(λ)(f ; x, y)

◮ obeys a deletion-contraction-type relation

(with operations λ , λ∗ );

◮ contains the weight enumerator of a nonlinear code

. . . but this is also in R(f ; x, y), i.e., don’t need λ;

Properties of the λ-Whitney function:

R(λ)(f ; x, y)

◮ obeys a deletion-contraction-type relation

(with operations λ , λ∗ );

◮ contains the weight enumerator of a nonlinear code

. . . but this is also in R(f ; x, y), i.e., don’t need λ;

◮ contains the partition function of the Ashkin-Teller model

• n a graph G

Properties of the λ-Whitney function:

R(λ)(f ; x, y)

◮ obeys a deletion-contraction-type relation

(with operations λ , λ∗ );

◮ contains the weight enumerator of a nonlinear code

. . . but this is also in R(f ; x, y), i.e., don’t need λ;

◮ contains the partition function of the Ashkin-Teller model

• n a graph G

. . . which is not determined by R(G; x, y), so do need λ.

Ashkin-Teller model (1943)

◮ 4-colourings (may be improper): colours are (±1, ±1)

Ashkin-Teller model (1943)

◮ 4-colourings (may be improper): colours are (±1, ±1)

+1 −1 −1 −1 +1 +1 −1 +1

Ashkin-Teller model (1943)

◮ 4-colourings (may be improper): colours are (±1, ±1)

+1 −1 −1 −1 +1 +1 −1 +1 Left colours: Good and bad edges

Ashkin-Teller model (1943)

◮ 4-colourings (may be improper): colours are (±1, ±1)

+1 −1 −1 −1 +1 +1 −1 +1

Ashkin-Teller model (1943)

◮ 4-colourings (may be improper): colours are (±1, ±1)

+1 −1 −1 −1 +1 +1 −1 +1 Right colours: Good and bad edges

Ashkin-Teller model (1943)

◮ 4-colourings (may be improper): colours are (±1, ±1)

+1 −1 −1 −1 +1 +1 −1 +1

Ashkin-Teller model (1943)

◮ 4-colourings (may be improper): colours are (±1, ±1)

+1 −1 −1 −1 +1 +1 −1 +1 Product colours: Good and bad edges

Ashkin-Teller model (1943)

◮ 4-colourings (may be improper): colours are (±1, ±1)

+1 −1 −1 −1 +1 +1 −1 +1 Product colours: Good and bad edges

◮ Partition function (symmetric Ashkin-Teller):

ZAT(G; K, K ′, q) = e(2K+K ′)|E| e

− B @

K · (# good “left” edges) + K · (# good “right” edges) + K ′ · (# good “product” edges)

1 C A

Ashkin-Teller model (1943)

Special cases:

◮ K = K ′: Potts model (up to a factor) ◮ K ′ = 0: product of two Ising models (each q = 2)

For these, ZAT(G) is a specialisation of R(G : x, y).

Ashkin-Teller model (1943)

Special cases:

◮ K = K ′: Potts model (up to a factor) ◮ K ′ = 0: product of two Ising models (each q = 2)

For these, ZAT(G) is a specialisation of R(G : x, y). In general, ZAT(G) is not a specialisation of R(G : x, y). Example (M. C. Gray; see Tutte (1974)): These graphs have same R(G; x, y), but different ZAT(G) (even in symmetric case).

λ-Tutte-Whitney plane

x 2 1 −1 −2 2 1

• 1
• 2

y

λ-Tutte-Whitney plane

x 2 1 −1 −2 2 1

• 1
• 2

y weight enum. nonlinear code

λ-Tutte-Whitney plane

x 2 1 −1 −2 2 1

• 1
• 2

y weight enum. nonlinear code Ashkin-Teller model

λ-Tutte-Whitney plane

x 2 1 −1 −2 2 1

• 1
• 2

y weight enum. nonlinear code Ashkin-Teller model easy

λ-Tutte-Whitney plane

x 2 1 −1 −2 2 1

• 1
• 2

y weight enum. nonlinear code Ashkin-Teller model easy