A survey of Tutte-Whitney polynomials

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 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) 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 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 x ρ(e) ρ(x ) y X ρ(x ).

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 (G 1 G 2 ) = f (G 1 ) f (G 2 )),... then f is essentially a (partial) evaluation of the Tutte-Whitney polynomial.

y 2 1 2 1 1 2 x -1-2

chromatic y 2 1 2 1 1 2 x -1-2

chromatic flow y 2 1 2 1 1 2 x -1-2

chromatic flow reliability y 2 1 2 1 1 2 x -1-2

chromatic flow reliability weight enumerator y 2 1 2 1 1 2 x -1-2

chromatic flow reliability weight enumerator Potts model y 2 1 2 1 1 2 x -1-2

chromatic flow reliability weight enumerator Potts model Jones y 2 1 2 1 1 2 x -1-2

chromatic flow reliability weight enumerator Potts model Jones easy y 2 1 2 1 1 2 x -1-2

History

History Graphs: Chrom. poly

History Graphs: Chrom. poly Birkhoff 1912 P(G; q)

History Graphs: Chrom. poly Birkhoff 1912 P(G; q) Whitney, 1935 Tutte, 1947 R(G; x, y)

History Graphs: Chrom. poly Birkhoff 1912 P(G; q) Whitney, 1935 Tutte Tutte, 1947 R(G; x, y) 1954 T (G; x, y)

History Stat Mech: partition functions Graphs: Chrom. poly Birkhoff 1912 P(G; q) Whitney, 1935 Tutte Tutte, 1947 R(G; x, y) 1954 T (G; x, y)

History Stat Mech: partition functions Ising 1925 Ising model (q = 2) Graphs: Chrom. poly Birkhoff 1912 P(G; q) Whitney, 1935 Tutte Tutte, 1947 R(G; x, y) 1954 T (G; x, y)

History Stat Mech: partition functions Ising 1925 Ising model (q = 2) Potts 1952 Potts model (all q) Graphs: Chrom. poly Birkhoff 1912 P(G; q) Whitney, 1935 Tutte Tutte, 1947 R(G; x, y) 1954 T (G; x, y)

History Stat Mech: partition functions A. & T., 1943 Ashkin-Teller model (q = 4) Ising Potts 1925 1952 Ising Potts model model (q = 2) (all q) Graphs: Chrom. poly Birkhoff 1912 P(G; q) Whitney, 1935 Tutte Tutte, 1947 R(G; x, y) 1954 T (G; x, y)

History Stat Mech: partition functions Graphs: Chrom. poly A. & T., 1943 Ashkin-Teller model (q = 4) Ising Potts 1925 1952 Ising Potts model model (q = 2) (all q) Fortuin & Birkhoff Whitney, 1935 Tutte Kasteleyn 1912 P(G; q) Tutte, 1947 R(G; x, y) 1954 1972 T (G; x, y)

History Stat Mech: partition functions Graphs: Chrom. poly Linear codes: weight enumerator A. & T., 1943 Ashkin-Teller model (q = 4) Ising Potts 1925 1952 Ising Potts model model (q = 2) (all q) Fortuin & Birkhoff Whitney, 1935 Tutte Kasteleyn 1912 P(G; q) Tutte, 1947 R(G; x, y) 1954 1972 T (G; x, y)

History Stat Mech: partition functions Graphs: Chrom. poly Linear codes: weight enumerator A. & T., 1943 Network reliability: Ashkin-Teller model (q = 4) Ising Potts 1925 1952 Ising Potts model model (q = 2) (all q) Fortuin & Birkhoff Whitney, 1935 Tutte Kasteleyn 1912 P(G; q) Tutte, 1947 R(G; x, y) 1954 1972 T (G; x, y) MacWilliams 1963 Greene 1974 A C(z)

History Stat Mech: partition functions Graphs: Chrom. poly Linear codes: weight enumerator A. & T., 1943 Network reliability: Ashkin-Teller van Slyke & model (q = 4) Ising Frank, Potts 1971 1925 1952 Π(G; p) Ising Potts model model (q = 2) (all q) Fortuin & Birkhoff Whitney, 1935 Tutte Kasteleyn 1912 P(G; q) Tutte, 1947 R(G; x, y) 1954 1972 T (G; x, y) MacWilliams 1963 Greene 1974 A C(z)

History Stat Mech: partition functions Graphs: Chrom. poly Linear codes: weight enumerator A. & T., 1943 Network reliability: Ashkin-Teller van Slyke & model (q = 4) Ising Frank, Potts 1971 1925 1952 Π(G; p) Ising Potts model model (q = 2) (all q) Fortuin & Birkhoff Whitney, 1935 Oxley & Tutte Kasteleyn 1912 P(G; q) Tutte, 1947 R(G; x, y) 1954 Welsh 1972 1979 T (G; x, y) MacWilliams 1963 Greene 1974 A C(z)

History Stat Mech: partition functions Graphs: Chrom. poly Linear codes: weight enumerator A. & T., 1943 Network reliability: Ashkin-Teller van Slyke & model (q = 4) Ising Frank, Potts 1971 1925 1952 Π(G; p) Ising Potts model model (q = 2) (all q) Fortuin & Birkhoff Whitney, 1935 Oxley & Tutte Kasteleyn 1912 P(G; q) Tutte, 1947 R(G; x, y) 1954 Welsh 1972 1979 T (G; x, y) MacWilliams 1963 Greene 1974 A C(z) Knots: Jones poly

History Stat Mech: partition functions Graphs: Chrom. poly Linear codes: weight enumerator Knots: Jones poly A. & T., 1943 Network reliability: Ashkin-Teller van Slyke & model (q = 4) Ising Frank, Potts 1971 1925 1952 Π(G; p) Ising Potts model model (q = 2) (all q) Fortuin & Birkhoff Whitney, 1935 Oxley & Tutte Kasteleyn 1912 P(G; q) Tutte, 1947 R(G; x, y) 1954 Welsh 1972 1979 T (G; x, y) MacWilliams 1963 Greene 1974 A C(z) Jones 1985 V L (t)

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) 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 #P 1 ) 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 #P 1 ) 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),...... or extend the polynomials:

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 E (edge set) {}}{ E \ W {}}{ 0/1 entries.. W {}}{

Rank rowspace Incidence matrix echelon form vertices E (edge set) {}}{ E \ W {}}{ 0/1 entries.. W {}}{

Rank rowspace Incidence matrix echelon form E (edge set) {}}{ E \ W {}}{ W {}}{ 0 I 0 0 I 0.

Rank rowspace Incidence matrix echelon form ρ(e \ W ) ρ(e) E (edge set) {}}{ E \ W {}}{ W {}}{ 0 I 0 0 I 0.

Rank rowspace Incidence matrix echelon form ρ(e \ W ) ρ(e) E (edge set) {}}{ E \ W {}}{ W {}}{ 0 I 0 0 I 0. Count rowspace members that are 0 outside W : 2 ρ(e) ρ(e\w ) = ind Rowspace (X ) X W

Count rowspace members that are 0 outside W : 2 ρ(e) ρ(e\w ) = ind Rowspace (X ) X W

Count rowspace members that are 0 outside W : 2 ρ(e) ρ(e\w ) = ind Rowspace (X ) X W ( X E ρ(w ) = log ind(x ) ) 2 X E\W ind(x )

Count rowspace members that are 0 outside W : 2 ρ(e) ρ(e\w ) = ind Rowspace (X ) X W ( X E ρ(w ) = log ind(x ) ) 2 X E\W ind(x ) Generalise to other functions (not necessarily indicator functions of rowspaces) (GF, 1993): For any f : 2 E {0, 1}... or... R... : Define Qf by: ( X E (Qf )(W ) = log f (X ) ) 2 X E\W f (X )

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: ˆf (W ) := 1 2 n ( 1) W X f (X ) X E

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: ˆf (W ) := 1 2 n ( 1) W X f (X ) X E f Hadamard transform ˆf Q Q Qf matroid-style dual (Qf ) = Qˆf

Extending the Whitney rank generating function R(f ; x, y) = X E x Qf (E) Qf (X ) X Qf (X ) y

Extending the Whitney rank generating function R(f ; x, y) = X E R 1 (f ; x, y) = x Qf (E) X E x Qf (E) Qf (X ) X Qf (X ) y (xy) Qf (X ) y X

Extending the Whitney rank generating function R(f ; x, y) = X E R 1 (f ; x, y) = x Qf (E) X E x Qf (E) Qf (X ) X Qf (X ) y (xy) Qf (X ) y X Example: E = {1, 2} X f 1 {1} 1 {2} 1 {1, 2} 0

Extending the Whitney rank generating function R(f ; x, y) = X E R 1 (f ; x, y) = x Qf (E) X E x Qf (E) Qf (X ) X Qf (X ) y (xy) Qf (X ) y X Example: E = {1, 2} X f 1 {1} 1 {2} 1 {1, 2} 0 R(f ; x, y) = x log 2 3 + 2xy 2 log 2 3 + y 2 log 23

Deletion-contraction For e E, X E \ {e}: Deletion (f \e)(x ) = Contraction f (X ) + f (X {e}) ; (f //e)(x ) = f (X ) f ( ) + f ({e}) f ( ).

Deletion-contraction For e E, X E \ {e}: Deletion (f \e)(x ) = Contraction f (X ) + f (X {e}) ; (f //e)(x ) = f (X ) f ( ) + f ({e}) f ( ). Deletion-contraction rule: R(f ; x, y) = x log 2 1+ f ({e} f ( ) R(f \e; x, y)+y log 2 1+ ˆf ({e} ˆf ( ) R(f //e; x, y)

Generalised Tutte-Whitney plane y 2 1 2 1 1 2 x -1-2

Generalised Tutte-Whitney plane chromatic y 2 1 2 1 1 2 x -1-2

Generalised Tutte-Whitney plane chromatic clutter reliability y 2 1 2 1 1 2 x -1-2

Generalised Tutte-Whitney plane chromatic clutter reliability weight enum. nonlinear code y 2 1 2 1 1 2 x -1-2

Generalised Tutte-Whitney plane chromatic y clutter reliability weight enum. nonlinear code gen. Potts model 2 1 2 1 1 2 x -1-2

Generalised Tutte-Whitney plane chromatic y clutter reliability weight enum. nonlinear code gen. Potts model easy 2 1 2 1 1 2 x -1-2

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 λ e)(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 λ e)(x ) (f \e)(x ) f (X ) f ( ) f (X ) + λf (X {e}) f ( ) + λf ({e}) f (X ) + f (X {e}) f ( ) + f ({e})

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 λ e = ˆf λ e

Duality Duality between deletion and contraction can be extended. Define λ := 1 λ 1 + λ Then Fixed points: f λ e = ˆf λ e λ = ± 2 1

λ-rank functions Define Q (λ) f by: ( (1 + λ (Q (λ) ) V ) X E f )(W ) = log λ X f (X ) 2 X E\W λ W V (λ ) W V f (X )

λ-rank functions Define Q (λ) f by: ( (1 + λ (Q (λ) ) V ) X E f )(W ) = log λ X f (X ) 2 X E\W λ W V (λ ) W V f (X ) Duality: (Q (λ) ) = Q (λ )

λ-rank functions Define Q (λ) f by: ( (1 + λ (Q (λ) ) V ) X E f )(W ) = log λ X f (X ) 2 X E\W λ W V (λ ) W V f (X ) Duality: Inversion: (Q (λ) ) = Q (λ ) (Q (λ) ρ)(v ) = ( 1) V (λ λ ) S ( 1) W (1 + λ ) W (λ ) W V λ W V 2 ρ(e) ρ(e\w ) W V

A continuum of λ-whitney functions R (λ) (f ; x, y) = X E x Q(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 x Q(λ) f (E) Q (λ) f (X ) y X Q(λ) f (X ) R (λ) 1 (f ; x, y) = x Q(λ) f (E) (xy) Q(λ) f (X ) y X X E

A continuum of λ-whitney functions R (λ) (f ; x, y) = X E x Q(λ) f (E) Q (λ) f (X ) y X Q(λ) f (X ) R (λ) 1 (f ; x, y) = x Q(λ) f (E) (xy) Q(λ) f (X ) y X X E R(ˆf ; x, y) ( x + y) E R (λ) (f ; x, y) R(f ; x, y) 0 2 1 λ 1

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; contains the partition function of the Ashkin-Teller model on a graph G

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 Left colours: Good and bad edges +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

Ashkin-Teller model (1943) 4-colourings (may be improper): colours are (±1, ±1) +1 +1 1 +1 Right colours: Good and bad edges +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

Ashkin-Teller model (1943) 4-colourings (may be improper): colours are (±1, ±1) +1 +1 1 +1 Product colours: Good and bad edges +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 Partition function (symmetric Ashkin-Teller): Z AT (G; K, K, q) = e (2K+K ) E e 0 B @ Product colours: Good and bad edges 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, Z AT (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, Z AT (G) is a specialisation of R(G : x, y). In general, Z AT (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 Z AT (G) (even in symmetric case).

λ-tutte-whitney plane y 2 1 2 1 1 2 x -1-2

λ-tutte-whitney plane y weight enum. nonlinear code 2 1 2 1 1 2 x -1-2

λ-tutte-whitney plane y weight enum. nonlinear code Ashkin-Teller model 2 1 2 1 1 2 x -1-2

λ-tutte-whitney plane y weight enum. nonlinear code Ashkin-Teller model easy 2 1 2 1 1 2 x -1-2