✗ ✔ A. & T., 1943 History Ashkin-Teller ✖ ✕ model ( q = 4) Ising ✬ ✩ ★ ✥ Potts ✕ ✁ 1925 1952 ✁ Stat Mech: Ising Potts ✲ partition model model ✧ ✦ ✫ ❏ ✪ (all q ) ( q = 2) functions ❏ ❏ Fortuin & ❏ Whitney, 1935 Graphs: Birkhoff Tutte ✓ ✏ ✓ ✏ ✓ ✏ Kasteleyn ❏ 1912 Tutte, 1947 1954 Chrom. 1972 ❏ ❏ ❫ ✒ ✑ ✲ ✒ ✑ ✒ ✑ poly ✲ ❛ ❛ P ( G ; q ) R ( G ; x , y ) ✲ T ( G ; x , y ) ❛ ✲ ❛ ✲ ✦ ✦ ✦ ✦ ✣ ✡ ✡ Greene Linear codes: MacWilliams ✓ ✏ ✡ 1974 1963 weight ✡ ✒ ✑ ✡ enumerator A C ( z )
✗ ✔ A. & T., 1943 Network reliability: History Ashkin-Teller ✖ ✕ model ( q = 4) Ising ✬ ✩ ★ ✥ Potts ✕ ✁ 1925 1952 ✁ Stat Mech: Ising Potts ✲ partition model model ✧ ✦ ✫ ❏ ✪ (all q ) ( q = 2) functions ❏ ❏ Fortuin & ❏ Whitney, 1935 Graphs: Birkhoff Tutte ✓ ✏ ✓ ✏ ✓ ✏ Kasteleyn ❏ 1912 Tutte, 1947 1954 Chrom. 1972 ❏ ❏ ❫ ✒ ✑ ✲ ✒ ✑ ✒ ✑ poly ✲ ❛ ❛ P ( G ; q ) R ( G ; x , y ) ✲ T ( G ; x , y ) ❛ ✲ ❛ ✲ ✦ ✦ ✦ ✦ ✣ ✡ ✡ Greene Linear codes: MacWilliams ✓ ✏ ✡ 1974 1963 weight ✡ ✒ ✑ ✡ enumerator A C ( z )
✗ ✔ A. & T., 1943 Network reliability: History Ashkin-Teller ✖ ✕ van Slyke & ✓ ✏ model ( q = 4) Frank, 1971 Ising ✬ ✩ ★ ✥ Potts ✕ ✁ ✒ ✑ 1925 1952 Π( G ; p ) ✁ Stat Mech: Ising Potts ✲ partition model model ✧ ✦ ✫ ❏ ✪ (all q ) ( q = 2) functions ❏ ❏ Fortuin & ❏ Whitney, 1935 Graphs: Birkhoff Tutte ✓ ✏ ✓ ✏ ✓ ✏ Kasteleyn ❏ 1912 Tutte, 1947 1954 Chrom. 1972 ❏ ❫ ❏ ✒ ✑ ✲ ✒ ✑ ✒ ✑ poly ✲ ❛ ❛ P ( G ; q ) R ( G ; x , y ) ✲ T ( G ; x , y ) ❛ ❛ ✲ ✲ ✦ ✦ ✦ ✦ ✣ ✡ ✡ Greene Linear codes: MacWilliams ✓ ✏ ✡ 1974 1963 weight ✡ ✒ ✡ ✑ enumerator A C ( z )
✗ ✔ A. & T., 1943 Network reliability: History Ashkin-Teller ✖ ✕ van Slyke & ✓ ✏ model ( q = 4) Frank, 1971 Ising ✬ ✩ ★ ✥ Potts ✁ ✕ ✒ ✑ 1925 1952 Π( G ; p ) ✁ ❇ Stat Mech: Ising Potts ✲ ❇ partition model model ✧ ✦ ✫ ❏ ✪ ❇ (all q ) ( q = 2) functions ❏ ❇ ❏ ❇ Fortuin & Oxley & ❏ ❇ Whitney, 1935 Graphs: Birkhoff Tutte ✓ ✏ ✓ ✏ ✓ ✏ Kasteleyn Welsh ❏ ❇ 1912 Tutte, 1947 1954 Chrom. 1972 1979 ❏ ❇ ◆ ❇ ❏ ❫ ✒ ✑ ✲ ✒ ✑ ✒ ✑ poly ✲ ❛ ❛ P ( G ; q ) R ( G ; x , y ) ✲ T ( G ; x , y ) ❛ ✲ ❛ ✲ ✦ ✦ ✦ ✦ ✡ ✣ ✡ Greene Linear codes: MacWilliams ✓ ✏ ✡ 1974 1963 weight ✡ ✒ ✑ ✡ enumerator A C ( z )
✗ ✔ A. & T., 1943 Network reliability: History Ashkin-Teller ✖ ✕ van Slyke & ✓ ✏ model ( q = 4) Frank, 1971 Ising ✬ ✩ ★ ✥ Potts ✕ ✁ ✒ ✑ 1925 1952 Π( G ; p ) ✁ ❇ Stat Mech: Ising Potts ✲ ❇ partition model model ✧ ✦ ✫ ❏ ✪ ❇ (all q ) ( q = 2) functions ❏ ❇ ❏ ❇ Fortuin & Oxley & ❏ ❇ Whitney, 1935 Graphs: Birkhoff Tutte ✓ ✏ ✓ ✏ ✓ ✏ Kasteleyn Welsh ❏ ❇ 1912 Tutte, 1947 1954 Chrom. 1972 1979 ❏ ❇ ❇ ◆ ❏ ❫ ✒ ✑ ✲ ✒ ✑ ✒ ✑ poly ✲ ❛ ❛ P ( G ; q ) R ( G ; x , y ) ✲ T ( G ; x , y ) ❛ ✲ ❛ ✲ ✦ ✦ ✦ ✦ ✡ ✣ ✡ Greene Linear codes: MacWilliams ✓ ✏ ✡ 1974 1963 weight ✡ ✒ ✡ ✑ enumerator A C ( z ) Knots: Jones poly
✗ ✔ A. & T., 1943 Network reliability: History Ashkin-Teller ✖ ✕ van Slyke & ✓ ✏ model ( q = 4) Frank, 1971 Ising ✬ ✩ ★ ✥ Potts ✕ ✁ ✒ ✑ 1925 1952 Π( G ; p ) ✁ ❇ Stat Mech: Ising Potts ✲ ❇ partition model model ✧ ✦ ✫ ❏ ✪ ❇ (all q ) ( q = 2) functions ❏ ❇ ❏ ❇ Fortuin & Oxley & ❏ ❇ Whitney, 1935 Graphs: Birkhoff Tutte ✓ ✏ ✓ ✏ ✓ ✏ Kasteleyn Welsh ❏ ❇ 1912 Tutte, 1947 1954 Chrom. 1972 1979 ❏ ❇ ❇ ◆ ❏ ❫ ✒ ✑ ✲ ✒ ✑ ✒ ✑ poly ✲ ❛ ❛ P ( G ; q ) R ( G ; x , y ) ✲ T ( G ; x , y ) ❛ ✲ ❛ ✲ ✦ ✦ ✦ ✦ ✣ ✡ ✡ Greene Linear codes: MacWilliams ✓ ✏ ✡ 1974 1963 weight ✡ ✒ ✑ ✡ enumerator A C ( z ) ✓ ✏ Jones 1985 ✒ ✑ Knots: V L ( t ) Jones poly
✗ ✔ A. & T., 1943 Network reliability: History Ashkin-Teller ✖ ✕ van Slyke & ✓ ✏ model ( q = 4) Frank, 1971 Ising ✬ ✩ ★ ✥ Potts ✕ ✁ ✒ ✑ 1925 1952 Π( G ; p ) ✁ ❇ Stat Mech: Ising Potts ✲ ❇ partition model model ✧ ✦ ✫ ✪ ❏ ❇ (all q ) ( q = 2) functions ❏ ❇ ❏ ❇ Fortuin & Oxley & ❏ ❇ Whitney, 1935 Graphs: Birkhoff Tutte ✓ ✏ ✓ ✏ ✓ ✏ Kasteleyn Welsh ❏ ❇ 1912 Tutte, 1947 1954 Chrom. 1972 1979 ❏ ❇ ❇ ◆ ❏ ❫ ✒ ✑ ✲ ✒ ✑ ✒ ✑ poly ✲ ❛ ❛ P ( G ; q ) R ( G ; x , y ) ✲ T ( G ; x , y ) ❛ ❛ ✲ ✲ ✦ ✦ ✦ ✦ ✣ ✡ ✆ ✡ Thistle- ✆ Greene thwaite Linear codes: MacWilliams ✓ ✏ ✡ ✆ 1974 1987 1963 weight ✡ ✆ ✒ ✑ ✡ ✆ enumerator A C ( z ) ✆ ✓ ✆ ✏ Jones 1985 ✆ ✒ ✑ Knots: V L ( t ) Jones poly
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 #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), . . .
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 E (edge set) � �� � Incidence matrix E \ W W � �� � � �� � . . . vertices · · · 0/1 entries · · · . . .
Rank ↔ rowspace E (edge set) � �� � Incidence matrix E \ W W − → echelon form � �� � � �� � . . . vertices · · · 0/1 entries · · · . . .
Rank ↔ rowspace E (edge set) � �� � Incidence matrix E \ W W − → echelon form � �� � � �� � 0 I · · · · · · · · · . . 0 0 I . 0
Rank ↔ rowspace E (edge set) � �� � Incidence matrix E \ W W − → echelon form � �� � � �� � ✻ ✻ ρ ( E \ W ) 0 I · · · · · · · · · ❄ ρ ( E ) . . 0 0 I . ❄ 0
Rank ↔ rowspace E (edge set) � �� � Incidence matrix E \ W W − → echelon form � �� � � �� � ✻ ✻ ρ ( E \ W ) 0 I · · · · · · · · · ❄ ρ ( E ) . . 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 ind( X ) ρ ( W ) = log 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 ind( X ) ρ ( W ) = log 2 � 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 : � 2 ρ ( E ) − ρ ( E \ W ) = ind Rowspace ( X ) X ⊆ W � � � X ⊆ E ind( X ) ρ ( W ) = log 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 f ( X ) ( Qf )( W ) = log 2 � X ⊆ E \ W f ( X )
Count rowspace members that are 0 outside W : � 2 ρ ( E ) − ρ ( E \ W ) = ind Rowspace ( X ) X ⊆ W � � � X ⊆ E ind( X ) ρ ( W ) = log 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 f ( X ) ( Qf )( W ) = log 2 � X ⊆ E \ W f ( X ) Inversion:
Count rowspace members that are 0 outside W : � 2 ρ ( E ) − ρ ( E \ W ) = ind Rowspace ( X ) X ⊆ W � � � X ⊆ E ind( X ) ρ ( W ) = log 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 f ( X ) ( Qf )( W ) = log 2 � X ⊆ E \ W f ( X ) Inversion: if ρ : 2 E → { 0 , 1 } then define Q † ρ by ( Q † ρ )( V ) = ( − 1) | V | � ( − 1) | W | 2 ρ ( E ) − ρ ( E \ W ) W ⊆ V
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: f ( W ) := 1 � ˆ ( − 1) | W ∩ X | f ( X ) 2 n 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: f ( W ) := 1 � ˆ ( − 1) | W ∩ X | f ( X ) 2 n X ⊆ E ✲ Q f Qf Hadamard transform ↓ ↓ matroid-style dual Q ✲ ( Qf ) ∗ = Q ˆ ˆ f f
Extending the Whitney rank generating function � x Qf ( E ) − Qf ( X ) y | X |− Qf ( X ) R ( f ; x , y ) = X ⊆ E
Extending the Whitney rank generating function � x Qf ( E ) − Qf ( X ) y | X |− Qf ( X ) R ( f ; x , y ) = X ⊆ E x Qf ( E ) � ( xy ) − Qf ( X ) y | X | R 1 ( f ; x , y ) = X ⊆ E
Extending the Whitney rank generating function � x Qf ( E ) − Qf ( X ) y | X |− Qf ( X ) R ( f ; x , y ) = X ⊆ E x Qf ( E ) � ( xy ) − Qf ( X ) y | X | R 1 ( f ; x , y ) = X ⊆ E Example: X f ∅ 1 E = { 1 , 2 } { 1 } 1 { 2 } 1 { 1 , 2 } 0
Extending the Whitney rank generating function � x Qf ( E ) − Qf ( X ) y | X |− Qf ( X ) R ( f ; x , y ) = X ⊆ E x Qf ( E ) � ( xy ) − Qf ( X ) y | X | R 1 ( f ; x , y ) = X ⊆ E Example: X f ∅ 1 E = { 1 , 2 } { 1 } 1 { 2 } 1 { 1 , 2 } 0 R ( f ; x , y ) = x log 2 3 + 2 xy 2 − log 2 3 + y 2 − log 2 3
Deletion-contraction For e ∈ E , X ⊆ E \ { e } : Deletion Contraction \ e )( X ) = f ( X ) + f ( X ∪ { e } ) / e )( X ) = f ( X ) ( f \ ; ( f / f ( ∅ ) . f ( ∅ ) + f ( { e } )
Deletion-contraction For e ∈ E , X ⊆ E \ { e } : Deletion Contraction \ e )( X ) = f ( X ) + f ( X ∪ { e } ) / e )( X ) = f ( X ) ( f \ ; ( f / f ( ∅ ) . f ( ∅ ) + f ( { e } ) Deletion-contraction rule: ˆ “ ” “ ” 1+ f ( { e } f ( { e } log 2 1+ R ( f ; x , y ) = x log 2 ˆ f ( ∅ ) R ( f \ \ e ; x , y )+ y R ( f / / e ; x , y ) f ( ∅ )
Generalised Tutte-Whitney plane y 2 1 − 2 − 1 1 2 x -1 -2
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