Counting matchings in graphs
with applications to the monomer-dimer models
Shmuel Friedland
Univ. Illinois at Chicago & Berlin Mathematical School
KTH, 16 April, 2008
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
1 / 38
Overview
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
2 / 38
Overview
Matchings in graphs
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
2 / 38
Overview
Matchings in graphs
Number of k -matchings in bipartite graphs as permanents
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
2 / 38
Overview
Matchings in graphs
Number of k -matchings in bipartite graphs as permanents
Lower and upper bounds on permanents
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
2 / 38
Overview
Matchings in graphs
Number of k -matchings in bipartite graphs as permanents
Lower and upper bounds on permanents
Exact lower and upper bounds on k -matchings in 2-regular graphs
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
2 / 38
Overview
Matchings in graphs
Number of k -matchings in bipartite graphs as permanents
Lower and upper bounds on permanents
Exact lower and upper bounds on k -matchings in 2-regular graphs
Expected number of k -matchings in r -regular bipartite graphs
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
2 / 38
Overview
Matchings in graphs
Number of k -matchings in bipartite graphs as permanents
Lower and upper bounds on permanents
Exact lower and upper bounds on k -matchings in 2-regular graphs
Expected number of k -matchings in r -regular bipartite graphs
p-matching and total matching entropies in infinite graphs
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
2 / 38
Overview
Matchings in graphs
Number of k -matchings in bipartite graphs as permanents
Lower and upper bounds on permanents
Exact lower and upper bounds on k -matchings in 2-regular graphs
Expected number of k -matchings in r -regular bipartite graphs
p-matching and total matching entropies in infinite graphs
Theoretical and computational results for Z2 , Z3
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
2 / 38
Overview
Matchings in graphs
Number of k -matchings in bipartite graphs as permanents
Lower and upper bounds on permanents
Exact lower and upper bounds on k -matchings in 2-regular graphs
Expected number of k -matchings in r -regular bipartite graphs
p-matching and total matching entropies in infinite graphs
Theoretical and computational results for Z2 , Z3
Asymptotic lower and upper matching conjectures
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
2 / 38
Overview
Matchings in graphs
Number of k -matchings in bipartite graphs as permanents
Lower and upper bounds on permanents
Exact lower and upper bounds on k -matchings in 2-regular graphs
Expected number of k -matchings in r -regular bipartite graphs
p-matching and total matching entropies in infinite graphs
Theoretical and computational results for Z2 , Z3
Asymptotic lower and upper matching conjectures
Plots and results
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
2 / 38
Overview
Matchings in graphs
Number of k -matchings in bipartite graphs as permanents
Lower and upper bounds on permanents
Exact lower and upper bounds on k -matchings in 2-regular graphs
Expected number of k -matchings in r -regular bipartite graphs
p-matching and total matching entropies in infinite graphs
Theoretical and computational results for Z2 , Z3
Asymptotic lower and upper matching conjectures
Plots and results
Summary and open problems
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
2 / 38
Figure: Matching on the two dimensional grid: Bipartite graph on 60 vertices,
101 edges, 24 dimers, 12 monomers
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
3 / 38
Matchings
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
4 / 38
Matchings
G = (V , E ) undirected graph with vertices V , edges E .
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
4 / 38
Matchings
G = (V , E ) undirected graph with vertices V , edges E .
matching in G: M ⊆ E
no two edges in M share a common endpoint.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
4 / 38
Matchings
G = (V , E ) undirected graph with vertices V , edges E .
matching in G: M ⊆ E
no two edges in M share a common endpoint.
e = (u, v) ∈ M is dimer
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
4 / 38
Matchings
G = (V , E ) undirected graph with vertices V , edges E .
matching in G: M ⊆ E
no two edges in M share a common endpoint.
e = (u, v) ∈ M is dimer
v not covered by M is monomer.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
4 / 38
Matchings
G = (V , E ) undirected graph with vertices V , edges E .
matching in G: M ⊆ E
no two edges in M share a common endpoint.
e = (u, v) ∈ M is dimer
v not covered by M is monomer.
M called monomer-dimer cover of G.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
4 / 38
Matchings
G = (V , E ) undirected graph with vertices V , edges E .
matching in G: M ⊆ E
no two edges in M share a common endpoint.
e = (u, v) ∈ M is dimer
v not covered by M is monomer.
M called monomer-dimer cover of G.
M is perfect matching ⇐⇒ no monomers.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
4 / 38
Matchings
G = (V , E ) undirected graph with vertices V , edges E .
matching in G: M ⊆ E
no two edges in M share a common endpoint.
e = (u, v) ∈ M is dimer
v not covered by M is monomer.
M called monomer-dimer cover of G.
M is perfect matching ⇐⇒ no monomers.
M is k-matching ⇐⇒ #M = k.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
4 / 38
Generating matching polynomial
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
5 / 38
Generating matching polynomial
φ(k, G) number of k-matchings in G, φ(0, G) := 1
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
5 / 38
Generating matching polynomial
φ(k, G) number of k-matchings in G, φ(0, G) := 1
P
ΦG (x) := k φ(k, G)x k matching generating polyn.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
5 / 38
Generating matching polynomial
φ(k, G) number of k-matchings in G, φ(0, G) := 1
P
ΦG (x) := k φ(k, G)x k matching generating polyn.
roots of ΦG (x) nonpositive Heilmann-Lieb 1972.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
5 / 38
Generating matching polynomial
φ(k, G) number of k-matchings in G, φ(0, G) := 1
P
ΦG (x) := k φ(k, G)x k matching generating polyn.
roots of ΦG (x) nonpositive Heilmann-Lieb 1972.
ΦG1 ∪G2 (x) = ΦG1 (x)ΦG2 (x)
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
5 / 38
Generating matching polynomial
φ(k, G) number of k-matchings in G, φ(0, G) := 1
P
ΦG (x) := k φ(k, G)x k matching generating polyn.
roots of ΦG (x) nonpositive Heilmann-Lieb 1972.
ΦG1 ∪G2 (x) = ΦG1 (x)ΦG2 (x)
Example: Kr ,r complete bipartite graph on 2r vertices.
ΦKr ,r (x) =
r 2
X
r
k =0
k
k!x k
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
5 / 38
Generating matching polynomial
φ(k, G) number of k-matchings in G, φ(0, G) := 1
P
ΦG (x) := k φ(k, G)x k matching generating polyn.
roots of ΦG (x) nonpositive Heilmann-Lieb 1972.
ΦG1 ∪G2 (x) = ΦG1 (x)ΦG2 (x)
Example: Kr ,r complete bipartite graph on 2r vertices.
ΦKr ,r (x) =
r 2
X
r
k =0
k
k!x k
G(r , 2n) set of r -regular bipartite graphs on 2n vertices
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
5 / 38
Generating matching polynomial
φ(k, G) number of k-matchings in G, φ(0, G) := 1
P
ΦG (x) := k φ(k, G)x k matching generating polyn.
roots of ΦG (x) nonpositive Heilmann-Lieb 1972.
ΦG1 ∪G2 (x) = ΦG1 (x)ΦG2 (x)
Example: Kr ,r complete bipartite graph on 2r vertices.
ΦKr ,r (x) =
r 2
X
r
k =0
k
k!x k
G(r , 2n) set of r -regular bipartite graphs on 2n vertices
qKr ,r ∈ G(r , 2rq) a union of q copies of Kr ,r .
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
5 / 38
Generating matching polynomial
φ(k, G) number of k-matchings in G, φ(0, G) := 1
P
ΦG (x) := k φ(k, G)x k matching generating polyn.
roots of ΦG (x) nonpositive Heilmann-Lieb 1972.
ΦG1 ∪G2 (x) = ΦG1 (x)ΦG2 (x)
Example: Kr ,r complete bipartite graph on 2r vertices.
ΦKr ,r (x) =
r 2
X
r
k =0
k
k!x k
G(r , 2n) set of r -regular bipartite graphs on 2n vertices
qKr ,r ∈ G(r , 2rq) a union of q copies of Kr ,r .
q
ΦqKr ,r = ΦKr ,r
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
5 / 38
Notations and definitions
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
6 / 38
Notations and definitions
hni := {1, 2, . . . , n − 1, n}
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
6 / 38
Notations and definitions
hni := {1, 2, . . . , n − 1, n}
For A = [aij ]ni,j ∈ Rn×n permanent of A:
X
perm A =
Y
aiσ(i)
all permutations σ on hni i=1
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
6 / 38
Notations and definitions
hni := {1, 2, . . . , n − 1, n}
For A = [aij ]ni,j ∈ Rn×n permanent of A:
X
perm A =
Y
aiσ(i)
all permutations σ on hni i=1
For C ∈ Rm×n and k ∈ hmin(m, n)i
permk C is the sum of the permanents of all k × k submatrices of
C
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
6 / 38
Notations and definitions
hni := {1, 2, . . . , n − 1, n}
For A = [aij ]ni,j ∈ Rn×n permanent of A:
X
perm A =
Y
aiσ(i)
all permutations σ on hni i=1
For C ∈ Rm×n and k ∈ hmin(m, n)i
permk C is the sum of the permanents of all k × k submatrices of
C
A
[aij ] ∈ Rn×n
doubly stochastic if
+ P
P=
n
n
a
=
1
=
i = 1, . . . , n
j=1 ij
j=1 aji ,
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
6 / 38
Notations and definitions
hni := {1, 2, . . . , n − 1, n}
For A = [aij ]ni,j ∈ Rn×n permanent of A:
X
perm A =
Y
aiσ(i)
all permutations σ on hni i=1
For C ∈ Rm×n and k ∈ hmin(m, n)i
permk C is the sum of the permanents of all k × k submatrices of
C
A
[aij ] ∈ Rn×n
doubly stochastic if
+ P
P=
n
n
a
=
1
=
i = 1, . . . , n
j=1 ij
j=1 aji ,
Ωn ⊂ Rn×n
is the set of doubly stochastic matrices
+
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
6 / 38
Notations and definitions
hni := {1, 2, . . . , n − 1, n}
For A = [aij ]ni,j ∈ Rn×n permanent of A:
X
perm A =
Y
aiσ(i)
all permutations σ on hni i=1
For C ∈ Rm×n and k ∈ hmin(m, n)i
permk C is the sum of the permanents of all k × k submatrices of
C
A
[aij ] ∈ Rn×n
doubly stochastic if
+ P
P=
n
n
a
=
1
=
i = 1, . . . , n
j=1 ij
j=1 aji ,
Ωn ⊂ Rn×n
is the set of doubly stochastic matrices
+
Pn ⊂ Ωn the set of permutation matrices
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
6 / 38
Notations and definitions
hni := {1, 2, . . . , n − 1, n}
For A = [aij ]ni,j ∈ Rn×n permanent of A:
X
perm A =
Y
aiσ(i)
all permutations σ on hni i=1
For C ∈ Rm×n and k ∈ hmin(m, n)i
permk C is the sum of the permanents of all k × k submatrices of
C
A
[aij ] ∈ Rn×n
doubly stochastic if
+ P
P=
n
n
a
=
1
=
i = 1, . . . , n
j=1 ij
j=1 aji ,
Ωn ⊂ Rn×n
is the set of doubly stochastic matrices
+
Pn ⊂ Ωn the set of permutation matrices
is the set of the extreme points of Ωn
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
6 / 38
Notations and definitions
hni := {1, 2, . . . , n − 1, n}
For A = [aij ]ni,j ∈ Rn×n permanent of A:
X
perm A =
Y
aiσ(i)
all permutations σ on hni i=1
For C ∈ Rm×n and k ∈ hmin(m, n)i
permk C is the sum of the permanents of all k × k submatrices of
C
A
[aij ] ∈ Rn×n
doubly stochastic if
+ P
P=
n
n
a
=
1
=
i = 1, . . . , n
j=1 ij
j=1 aji ,
Ωn ⊂ Rn×n
is the set of doubly stochastic matrices
+
Pn ⊂ Ωn the set of permutation matrices
is the set of the extreme points of Ωn
Birkhoff-Egerváry-König theorem (1946-1931-1916)
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
6 / 38
Bipartite graphs
Figure: An example of a bipartite graph
v1
v2
v3
v4
w1
w2
w3

1
0 

0 
0
w4

1 1
 1 1
Incidence matrix 
 1 1
1 0
1
0
0
0
1
0
0
0
w5
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
7 / 38
Formulas for k -matchings in bipartite graphs
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
8 / 38
Formulas for k -matchings in bipartite graphs
G = (V , E ) bipartite V = V1 ∪ V2 , E ⊂ V1 × V2 ,
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
8 / 38
Formulas for k -matchings in bipartite graphs
G = (V , E ) bipartite V = V1 ∪ V2 , E ⊂ V1 × V2 ,
m×n , #V = m, V = n.
represented by B(G) = B = [bij ]m×n
1
2
i,j=1 ∈ {0, 1}
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
8 / 38
Formulas for k -matchings in bipartite graphs
G = (V , E ) bipartite V = V1 ∪ V2 , E ⊂ V1 × V2 ,
m×n , #V = m, V = n.
represented by B(G) = B = [bij ]m×n
1
2
i,j=1 ∈ {0, 1}
Example: Any subgraph of Zd is bipartite
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
8 / 38
Formulas for k -matchings in bipartite graphs
G = (V , E ) bipartite V = V1 ∪ V2 , E ⊂ V1 × V2 ,
m×n , #V = m, V = n.
represented by B(G) = B = [bij ]m×n
1
2
i,j=1 ∈ {0, 1}
Example: Any subgraph of Zd is bipartite
CLAIM: φ(k, G) = permk (B(G)).
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
8 / 38
Formulas for k -matchings in bipartite graphs
G = (V , E ) bipartite V = V1 ∪ V2 , E ⊂ V1 × V2 ,
m×n , #V = m, V = n.
represented by B(G) = B = [bij ]m×n
1
2
i,j=1 ∈ {0, 1}
Example: Any subgraph of Zd is bipartite
CLAIM: φ(k, G) = permk (B(G)).
Prf: Suppose n = #V1 = #V2 .
Q
Then permutation σ : hni → hni is a perfect match iff ni=1 biσ(i) = 1.
The number of perfect matchings in G is φ(n, G) = perm B(G).
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
8 / 38
Formulas for k -matchings in bipartite graphs
G = (V , E ) bipartite V = V1 ∪ V2 , E ⊂ V1 × V2 ,
m×n , #V = m, V = n.
represented by B(G) = B = [bij ]m×n
1
2
i,j=1 ∈ {0, 1}
Example: Any subgraph of Zd is bipartite
CLAIM: φ(k, G) = permk (B(G)).
Prf: Suppose n = #V1 = #V2 .
Q
Then permutation σ : hni → hni is a perfect match iff ni=1 biσ(i) = 1.
The number of perfect matchings in G is φ(n, G) = perm B(G).
For G = (h2ni, E ) bipartite G ∈ G(r , 2n) ⇐⇒ 1r B(G) ∈ Ωn ⇐⇒
G is a disjoint (edge) union of r perfect matchings
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
8 / 38
Formulas for k -matchings in bipartite graphs
G = (V , E ) bipartite V = V1 ∪ V2 , E ⊂ V1 × V2 ,
m×n , #V = m, V = n.
represented by B(G) = B = [bij ]m×n
1
2
i,j=1 ∈ {0, 1}
Example: Any subgraph of Zd is bipartite
CLAIM: φ(k, G) = permk (B(G)).
Prf: Suppose n = #V1 = #V2 .
Q
Then permutation σ : hni → hni is a perfect match iff ni=1 biσ(i) = 1.
The number of perfect matchings in G is φ(n, G) = perm B(G).
For G = (h2ni, E ) bipartite G ∈ G(r , 2n) ⇐⇒ 1r B(G) ∈ Ωn ⇐⇒
G is a disjoint (edge) union of r perfect matchings
r k minC∈Ωn permk C ≤ φ(k, G) for any G ∈ G(r , 2n)
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
8 / 38
van der Waerden and Tverberg conjectures
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
9 / 38
van der Waerden and Tverberg conjectures
Jn = B(Kn,n ) = [1] the incidence matrix of the complete bipartite graph
Kn,n on 2n vertices
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
9 / 38
van der Waerden and Tverberg conjectures
Jn = B(Kn,n ) = [1] the incidence matrix of the complete bipartite graph
Kn,n on 2n vertices
van der Waerden permanent conjecture 1926:
1
min perm C = perm Jn
n
C∈Ωn
=
√
n!
−n
≈
2πn
e
nn
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,
models
2008
9 / 38
van der Waerden and Tverberg conjectures
Jn = B(Kn,n ) = [1] the incidence matrix of the complete bipartite graph
Kn,n on 2n vertices
van der Waerden permanent conjecture 1926:
1
min perm C = perm Jn
n
C∈Ωn
=
√
n!
−n
≈
2πn
e
nn
Tverberg permanent conjecture 1963:
1
min permk C = permk Jn
n
C∈Ωn
2
n k! =
k nk
for all k = 1, . . . , n.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
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() with applications to the monomer-dimer
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models
2008
9 / 38
History
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
10 / 38
History
In 1979 Friedland showed the lower bound perm C ≥ e−n for any
C ∈ Ωn following T. Bang’s announcement 1976.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
10 / 38
History
In 1979 Friedland showed the lower bound perm C ≥ e−n for any
C ∈ Ωn following T. Bang’s announcement 1976.
This settled the conjecture of Erdös-Rényi on the exponential
growth of the number of perfect matchings in d ≥ 3-regular
bipartite graphs 1968.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
10 / 38
History
In 1979 Friedland showed the lower bound perm C ≥ e−n for any
C ∈ Ωn following T. Bang’s announcement 1976.
This settled the conjecture of Erdös-Rényi on the exponential
growth of the number of perfect matchings in d ≥ 3-regular
bipartite graphs 1968.
van der Waerden permanent conjecture was proved by Egorichev
and Falikman 1981.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
10 / 38
History
In 1979 Friedland showed the lower bound perm C ≥ e−n for any
C ∈ Ωn following T. Bang’s announcement 1976.
This settled the conjecture of Erdös-Rényi on the exponential
growth of the number of perfect matchings in d ≥ 3-regular
bipartite graphs 1968.
van der Waerden permanent conjecture was proved by Egorichev
and Falikman 1981.
Tverberg conjecture was proved by Friedland 1982
————————————————————————–
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
10 / 38
History
In 1979 Friedland showed the lower bound perm C ≥ e−n for any
C ∈ Ωn following T. Bang’s announcement 1976.
This settled the conjecture of Erdös-Rényi on the exponential
growth of the number of perfect matchings in d ≥ 3-regular
bipartite graphs 1968.
van der Waerden permanent conjecture was proved by Egorichev
and Falikman 1981.
Tverberg conjecture was proved by Friedland 1982
————————————————————————–
79 proof is tour de force according to Bang
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
10 / 38
History
In 1979 Friedland showed the lower bound perm C ≥ e−n for any
C ∈ Ωn following T. Bang’s announcement 1976.
This settled the conjecture of Erdös-Rényi on the exponential
growth of the number of perfect matchings in d ≥ 3-regular
bipartite graphs 1968.
van der Waerden permanent conjecture was proved by Egorichev
and Falikman 1981.
Tverberg conjecture was proved by Friedland 1982
————————————————————————–
79 proof is tour de force according to Bang
81 proofs involve directly (Egorichev) and indirectly (Falikman) use
of Alexandroff mixed volume inequalities with the conditions for
the extremal matrix
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
10 / 38
History
In 1979 Friedland showed the lower bound perm C ≥ e−n for any
C ∈ Ωn following T. Bang’s announcement 1976.
This settled the conjecture of Erdös-Rényi on the exponential
growth of the number of perfect matchings in d ≥ 3-regular
bipartite graphs 1968.
van der Waerden permanent conjecture was proved by Egorichev
and Falikman 1981.
Tverberg conjecture was proved by Friedland 1982
————————————————————————–
79 proof is tour de force according to Bang
81 proofs involve directly (Egorichev) and indirectly (Falikman) use
of Alexandroff mixed volume inequalities with the conditions for
the extremal matrix
82 proof uses methods of 81 proofs with extra ingredients
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
10 / 38
History
In 1979 Friedland showed the lower bound perm C ≥ e−n for any
C ∈ Ωn following T. Bang’s announcement 1976.
This settled the conjecture of Erdös-Rényi on the exponential
growth of the number of perfect matchings in d ≥ 3-regular
bipartite graphs 1968.
van der Waerden permanent conjecture was proved by Egorichev
and Falikman 1981.
Tverberg conjecture was proved by Friedland 1982
————————————————————————–
79 proof is tour de force according to Bang
81 proofs involve directly (Egorichev) and indirectly (Falikman) use
of Alexandroff mixed volume inequalities with the conditions for
the extremal matrix
82 proof uses methods of 81 proofs with extra ingredients
There are new simple proofs using nonnegative hyperbolic
polynomials e.g. Friedland-Gurvits 2008
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
10 / 38
Lower matching bounds for 0 − 1 matrices
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
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in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
11 / 38
Lower matching bounds for 0 − 1 matrices
Voorhoeve-1979 (d = 3) Schrijver-1998
φ(n, G) ≥ (
(r − 1)r −1 n
)
r r −2
for
G ∈ G(r , 2n)
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
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() with applications to the monomer-dimer
KTH, 16 April,models
2008
11 / 38
Lower matching bounds for 0 − 1 matrices
Voorhoeve-1979 (d = 3) Schrijver-1998
φ(n, G) ≥ (
(r − 1)r −1 n
)
r r −2
for
G ∈ G(r , 2n)
Gurvits 2006: A ∈ Ωn , each column has at most r nonzero entries:
perm A ≥
r!
r r (r −1) r − 1 (r −1)n
.
rr r − 1
r
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
11 / 38
Lower matching bounds for 0 − 1 matrices
Voorhoeve-1979 (d = 3) Schrijver-1998
φ(n, G) ≥ (
(r − 1)r −1 n
)
r r −2
for
G ∈ G(r , 2n)
Gurvits 2006: A ∈ Ωn , each column has at most r nonzero entries:
perm A ≥
r!
r r (r −1) r − 1 (r −1)n
.
rr r − 1
r
Cor : φ(n, G) ≥
r r (r −1) (r − 1)r −1 n
r!
(
)
rr r − 1
r r −2
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
11 / 38
Lower matching bounds for 0 − 1 matrices
Voorhoeve-1979 (d = 3) Schrijver-1998
φ(n, G) ≥ (
(r − 1)r −1 n
)
r r −2
for
G ∈ G(r , 2n)
Gurvits 2006: A ∈ Ωn , each column has at most r nonzero entries:
perm A ≥
r!
r r (r −1) r − 1 (r −1)n
.
rr r − 1
r
r r (r −1) (r − 1)r −1 n
r!
(
)
rr r − 1
r r −2
2
n
nr − k nr −k kr k
Con FKM 2006 : φ(k, G) ≥
(
)
( ) , G ∈ G(r , 2n)
k
nr
n
Cor : φ(n, G) ≥
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
11 / 38
Lower matching bounds for 0 − 1 matrices
Voorhoeve-1979 (d = 3) Schrijver-1998
φ(n, G) ≥ (
(r − 1)r −1 n
)
r r −2
for
G ∈ G(r , 2n)
Gurvits 2006: A ∈ Ωn , each column has at most r nonzero entries:
perm A ≥
r!
r r (r −1) r − 1 (r −1)n
.
rr r − 1
r
r r (r −1) (r − 1)r −1 n
r!
(
)
rr r − 1
r r −2
2
n
nr − k nr −k kr k
Con FKM 2006 : φ(k, G) ≥
(
)
( ) , G ∈ G(r , 2n)
k
nr
n
Cor : φ(n, G) ≥
F-G 2008 showed weaker inequalities
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
11 / 38
Upper matching bounds for 0 − 1 matrices
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
12 / 38
Upper matching bounds for 0 − 1 matrices
Assume A ∈ {0, 1}n×n .
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
12 / 38
Upper matching bounds for 0 − 1 matrices
Assume A ∈ {0, 1}n×n .
ri is i − th row sum of A
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
12 / 38
Upper matching bounds for 0 − 1 matrices
Assume A ∈ {0, 1}n×n .
ri is i − th row sum of A
Bregman 1973: perm A ≤
Qn
i=1 (ri !)
1
ri
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
12 / 38
Upper matching bounds for 0 − 1 matrices
Assume A ∈ {0, 1}n×n .
ri is i − th row sum of A
1
Q
Bregman 1973: perm A ≤ ni=1 (ri !) ri
φ(qr , G) ≤ φ(qr , qKr ,r ) for any G ∈ G(r , 2qr )
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
12 / 38
Upper matching bounds for 0 − 1 matrices
Assume A ∈ {0, 1}n×n .
ri is i − th row sum of A
1
Q
Bregman 1973: perm A ≤ ni=1 (ri !) ri
φ(qr , G) ≤ φ(qr , qKr ,r ) for any G ∈ G(r , 2qr )
Con FKM 2006: φ(k, G) ≤ φ(k, qKr ,r ) for any G ∈ G(r , 2qr ) and
k = 1, . . . , qr
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
12 / 38
Upper matching bounds for 0 − 1 matrices
Assume A ∈ {0, 1}n×n .
ri is i − th row sum of A
1
Q
Bregman 1973: perm A ≤ ni=1 (ri !) ri
φ(qr , G) ≤ φ(qr , qKr ,r ) for any G ∈ G(r , 2qr )
Con FKM 2006: φ(k, G) ≤ φ(k, qKr ,r ) for any G ∈ G(r , 2qr ) and
k = 1, . . . , qr
c4 (G) - The number of 4-cycles in G
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
12 / 38
Upper matching bounds for 0 − 1 matrices
Assume A ∈ {0, 1}n×n .
ri is i − th row sum of A
1
Q
Bregman 1973: perm A ≤ ni=1 (ri !) ri
φ(qr , G) ≤ φ(qr , qKr ,r ) for any G ∈ G(r , 2qr )
Con FKM 2006: φ(k, G) ≤ φ(k, qKr ,r ) for any G ∈ G(r , 2qr ) and
k = 1, . . . , qr
c4 (G) - The number of 4-cycles in G
Thm: For any r -regular graph G = (V , E ),
c4 (G) ≤
r #V (r − 1)2
2
4
Equality iff G = qKr ,r
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
12 / 38
Upper matching bounds for 0 − 1 matrices
Assume A ∈ {0, 1}n×n .
ri is i − th row sum of A
1
Q
Bregman 1973: perm A ≤ ni=1 (ri !) ri
φ(qr , G) ≤ φ(qr , qKr ,r ) for any G ∈ G(r , 2qr )
Con FKM 2006: φ(k, G) ≤ φ(k, qKr ,r ) for any G ∈ G(r , 2qr ) and
k = 1, . . . , qr
c4 (G) - The number of 4-cycles in G
Thm: For any r -regular graph G = (V , E ),
c4 (G) ≤
r #V (r − 1)2
2
4
Equality iff G = qKr ,r
Prf: Any edge in e ∈ E can be in at most (r − 1)2 different
4-cycles.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
12 / 38
An example
Figure: Edge neighborhood of V2 W2 of 4- regular graph on 8 vertices
v1
v2
v3
v4
w1
w2
w3
w4
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
13 / 38
Upper perfect matching bounds for general graphs
G = (V , E ) Non-bipartite graph on 2n vertices
φ(n, G) ≤
Y
1
((deg v)!) 2 deg v
v ∈V
If deg v > 0, ∀v ∈ V equality holds iff G is a disjoint union of complete
balanced bipartite graphs
Kahn-Lóvasz unpublished, Friedland 2008-arXiv, Alon-Friedland
2008-arXiv.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
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in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
14 / 38
Exact values for small matchings
For G ∈ G(r , 2n)
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
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() with applications to the monomer-dimer
KTH, 16 April,models
2008
15 / 38
Exact values for small matchings
For G ∈ G(r , 2n)
1
φ(1, G) = nr
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
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() with applications to the monomer-dimer
KTH, 16 April,models
2008
15 / 38
Exact values for small matchings
For G ∈ G(r , 2n)
1
φ(1, G) = nr
2
φ(2, G) =
nr 2 −
2n
r
2
=
nr (nr −(2r −1))
2
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
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() with applications to the monomer-dimer
KTH, 16 April,models
2008
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Exact values for small matchings
For G ∈ G(r , 2n)
1
φ(1, G) = nr
2
φ(2, G) =
3
φ(3, G) =
nr 2−
nr
3 −
2n
2n
nr (nr −(2r −1))
r
2 =
2
r
−
nr
(r
−
1)2 − 2n 2r (nr
3
− 2r − (r − 2))
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
15 / 38
Exact values for small matchings
For G ∈ G(r , 2n)
1
φ(1, G) = nr
2
φ(2, G) =
3
φ(3, G) =
4
φ(4, G) = p1 (n, r ) + c4 (G)
p1 (n, r ) =
n4 r 4 n3 r 3
n2 r 2
2 +nr 5 − 5r + 7r 2 −
+
(1−2r
)+
19
−
60r
+
52r
24
4
24
4
nr 2−
nr
3 −
2n
2n
nr (nr −(2r −1))
r
2 =
2
r
−
nr
(r
−
1)2 − 2n 2r (nr
3
− 2r − (r − 2))
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
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() with applications to the monomer-dimer
KTH, 16 April,models
2008
7r 3
2
15 / 38
Exact values for small matchings
For G ∈ G(r , 2n)
1
φ(1, G) = nr
2
φ(2, G) =
3
φ(3, G) =
4
φ(4, G) = p1 (n, r ) + c4 (G)
p1 (n, r ) =
n4 r 4 n3 r 3
n2 r 2
2 +nr 5 − 5r + 7r 2 −
+
(1−2r
)+
19
−
60r
+
52r
24
4
24
4
nr 2−
nr
3 −
2n
2n
nr (nr −(2r −1))
r
2 =
2
r
−
nr
(r
−
1)2 − 2n 2r (nr
3
− 2r − (r − 2))
Notation:
f (x) =
N
X
i=0
i
ai x g(x) =
N
X
i=0
7r 3
2
bi x i ⇐⇒
ai ≤ bi for i = 1, . . . , N.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
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() with applications to the monomer-dimer
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2-regular graphs
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
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() with applications to the monomer-dimer
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2-regular graphs
Γ(r , n) the set of r -regular graphs on n-vertices
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
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() with applications to the monomer-dimer
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2-regular graphs
Γ(r , n) the set of r -regular graphs on n-vertices
A connected G ∈ Γ(2, n) is cycle Cn : 1 → 2 → · · · → n → 1
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
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() with applications to the monomer-dimer
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2008
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2-regular graphs
Γ(r , n) the set of r -regular graphs on n-vertices
A connected G ∈ Γ(2, n) is cycle Cn : 1 → 2 → · · · → n → 1
K2,2 = C4
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
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() with applications to the monomer-dimer
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2-regular graphs
Γ(r , n) the set of r -regular graphs on n-vertices
A connected G ∈ Γ(2, n) is cycle Cn : 1 → 2 → · · · → n → 1
K2,2 = C4
G ∈ Γ(2, n) iff G a union of cycles
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
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matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
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2-regular graphs
Γ(r , n) the set of r -regular graphs on n-vertices
A connected G ∈ Γ(2, n) is cycle Cn : 1 → 2 → · · · → n → 1
K2,2 = C4
G ∈ Γ(2, n) iff G a union of cycles
G ∈ G(2, 2n) iff G union of even cycles
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
16 / 38
2-regular graphs
Γ(r , n) the set of r -regular graphs on n-vertices
A connected G ∈ Γ(2, n) is cycle Cn : 1 → 2 → · · · → n → 1
K2,2 = C4
G ∈ Γ(2, n) iff G a union of cycles
G ∈ G(2, 2n) iff G union of even cycles
For G ∈ Γ(2, n):
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
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in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
16 / 38
2-regular graphs
Γ(r , n) the set of r -regular graphs on n-vertices
A connected G ∈ Γ(2, n) is cycle Cn : 1 → 2 → · · · → n → 1
K2,2 = C4
G ∈ Γ(2, n) iff G a union of cycles
G ∈ G(2, 2n) iff G union of even cycles
For G ∈ Γ(2, n):
n
ΦG (x) Φ n K2,2 (x) = ΦC4 (x) 4 if 4|n
4
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
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() with applications to the monomer-dimer
KTH, 16 April,models
2008
16 / 38
2-regular graphs
Γ(r , n) the set of r -regular graphs on n-vertices
A connected G ∈ Γ(2, n) is cycle Cn : 1 → 2 → · · · → n → 1
K2,2 = C4
G ∈ Γ(2, n) iff G a union of cycles
G ∈ G(2, 2n) iff G union of even cycles
For G ∈ Γ(2, n):
n
ΦG (x) Φ n K2,2 (x) = ΦC4 (x) 4 if 4|n
4
ΦG (x) Φ n−5 K2,2 ∪C5 (x) = ΦC4 (x)
4
n−5
4
ΦC5 (x) if 4|n − 1
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
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matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
16 / 38
2-regular graphs
Γ(r , n) the set of r -regular graphs on n-vertices
A connected G ∈ Γ(2, n) is cycle Cn : 1 → 2 → · · · → n → 1
K2,2 = C4
G ∈ Γ(2, n) iff G a union of cycles
G ∈ G(2, 2n) iff G union of even cycles
For G ∈ Γ(2, n):
n
ΦG (x) Φ n K2,2 (x) = ΦC4 (x) 4 if 4|n
4
ΦG (x) Φ n−5 K2,2 ∪C5 (x) = ΦC4 (x)
4
ΦG (x) Φ n−6 K
4
2,2 ∪C6
(x) = ΦC4 (x)
n−5
4
n−6
4
ΦC5 (x) if 4|n − 1
ΦC6 (x) if 4|n − 2
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
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() with applications to the monomer-dimer
KTH, 16 April,models
2008
16 / 38
2-regular graphs
Γ(r , n) the set of r -regular graphs on n-vertices
A connected G ∈ Γ(2, n) is cycle Cn : 1 → 2 → · · · → n → 1
K2,2 = C4
G ∈ Γ(2, n) iff G a union of cycles
G ∈ G(2, 2n) iff G union of even cycles
For G ∈ Γ(2, n):
n
ΦG (x) Φ n K2,2 (x) = ΦC4 (x) 4 if 4|n
4
ΦG (x) Φ n−5 K2,2 ∪C5 (x) = ΦC4 (x)
4
ΦG (x) Φ n−6 K
4
2,2 ∪C6
(x) = ΦC4 (x)
ΦG (x) Φ n−7 K2,2 ∪C7 (x) = ΦC4 (x)
4
n−5
4
n−6
4
n−7
4
ΦC5 (x) if 4|n − 1
ΦC6 (x) if 4|n − 2
ΦC7 (x) if 4|n − 3,
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
16 / 38
2-regular graphs
Γ(r , n) the set of r -regular graphs on n-vertices
A connected G ∈ Γ(2, n) is cycle Cn : 1 → 2 → · · · → n → 1
K2,2 = C4
G ∈ Γ(2, n) iff G a union of cycles
G ∈ G(2, 2n) iff G union of even cycles
For G ∈ Γ(2, n):
n
ΦG (x) Φ n K2,2 (x) = ΦC4 (x) 4 if 4|n
4
ΦG (x) Φ n−5 K2,2 ∪C5 (x) = ΦC4 (x)
4
ΦG (x) Φ n−6 K
4
2,2 ∪C6
(x) = ΦC4 (x)
ΦG (x) Φ n−7 K2,2 ∪C7 (x) = ΦC4 (x)
4
n
3
n−5
4
n−6
4
n−7
4
ΦG (x) Φ n K3 (x) = ΦC3 (x) if 3|n
ΦC5 (x) if 4|n − 1
ΦC6 (x) if 4|n − 2
ΦC7 (x) if 4|n − 3,
3
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
16 / 38
2-regular graphs
Γ(r , n) the set of r -regular graphs on n-vertices
A connected G ∈ Γ(2, n) is cycle Cn : 1 → 2 → · · · → n → 1
K2,2 = C4
G ∈ Γ(2, n) iff G a union of cycles
G ∈ G(2, 2n) iff G union of even cycles
For G ∈ Γ(2, n):
n
ΦG (x) Φ n K2,2 (x) = ΦC4 (x) 4 if 4|n
4
ΦG (x) Φ n−5 K2,2 ∪C5 (x) = ΦC4 (x)
4
ΦG (x) Φ n−6 K
4
2,2 ∪C6
(x) = ΦC4 (x)
ΦG (x) Φ n−7 K2,2 ∪C7 (x) = ΦC4 (x)
4
n−5
4
n−6
4
n−7
4
n
3
ΦG (x) Φ n K3 (x) = ΦC3 (x) if 3|n
3
ΦG (x) Φ n−4 K3 ∪C4 (x) = ΦC3 (x)
3
n−4
3
ΦC5 (x) if 4|n − 1
ΦC6 (x) if 4|n − 2
ΦC7 (x) if 4|n − 3,
ΦC4 (x) if 3|n − 1,
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
16 / 38
2-regular graphs
Γ(r , n) the set of r -regular graphs on n-vertices
A connected G ∈ Γ(2, n) is cycle Cn : 1 → 2 → · · · → n → 1
K2,2 = C4
G ∈ Γ(2, n) iff G a union of cycles
G ∈ G(2, 2n) iff G union of even cycles
For G ∈ Γ(2, n):
n
ΦG (x) Φ n K2,2 (x) = ΦC4 (x) 4 if 4|n
4
ΦG (x) Φ n−5 K2,2 ∪C5 (x) = ΦC4 (x)
4
ΦG (x) Φ n−6 K
4
2,2 ∪C6
(x) = ΦC4 (x)
ΦG (x) Φ n−7 K2,2 ∪C7 (x) = ΦC4 (x)
4
n−5
4
n−6
4
n−7
4
n
3
ΦG (x) Φ n K3 (x) = ΦC3 (x) if 3|n
3
ΦG (x) Φ n−4 K3 ∪C4 (x) = ΦC3 (x)
3
ΦG (x) Φ n−5 K3 ∪C5 (x) = ΦC3 (x)
3
n−4
3
n−5
3
ΦC5 (x) if 4|n − 1
ΦC6 (x) if 4|n − 2
ΦC7 (x) if 4|n − 3,
ΦC4 (x) if 3|n − 1,
ΦC5 (x) if 3|n − 2
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
16 / 38
2-regular graphs
Γ(r , n) the set of r -regular graphs on n-vertices
A connected G ∈ Γ(2, n) is cycle Cn : 1 → 2 → · · · → n → 1
K2,2 = C4
G ∈ Γ(2, n) iff G a union of cycles
G ∈ G(2, 2n) iff G union of even cycles
For G ∈ Γ(2, n):
n
ΦG (x) Φ n K2,2 (x) = ΦC4 (x) 4 if 4|n
4
ΦG (x) Φ n−5 K2,2 ∪C5 (x) = ΦC4 (x)
4
ΦG (x) Φ n−6 K
4
2,2 ∪C6
(x) = ΦC4 (x)
ΦG (x) Φ n−7 K2,2 ∪C7 (x) = ΦC4 (x)
4
n−5
4
n−6
4
n−7
4
n
3
ΦG (x) Φ n K3 (x) = ΦC3 (x) if 3|n
3
ΦG (x) Φ n−4 K3 ∪C4 (x) = ΦC3 (x)
3
n−4
3
n−5
3
ΦC5 (x) if 4|n − 1
ΦC6 (x) if 4|n − 2
ΦC7 (x) if 4|n − 3,
ΦC4 (x) if 3|n − 1,
ΦG (x) Φ n−5 K3 ∪C5 (x) = ΦC3 (x) ΦC5 (x) if 3|n − 2
3
If n even G multi-bipartite 2-regular graph then ΦG (x) ΦCn (x).
Shmuel Friedland
Univ. Illinois
Chicago
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
16 / 38
Equality
iffatG
= C& nBerlin
Relations between matching polynomials
For 0 ≤ i ≤ j
ΦCi (x)ΦCj (x) − ΦCi+j (x) = (−1)i x i ΦCj−i (x)
Pn path 1 → 2 → . . . → n.
pn (x) := ΦPn (x), qn (x) := ΦCn (x)
pk (x) = pk −1 (x) + xpk −2 (x)
qk (x) = pk (x) + xpk −2 (x)
If n = 0, 1 mod 4
pn−1 = p1 pn−1 ≺ p3 pn−3 ≺ · · · ≺ p2⌊ n ⌋−1 pn−2⌊ n ⌋+1 ≺
4
4
p2⌊ n ⌋ pn−2⌊ n ⌋ ≺ p2⌊ n ⌋−2 pn−2⌊ n ⌋+2 ≺ · · · ≺ p2 pn−2 ≺ p0 pn = pn
4
4
4
4
qn−1 = q1 qn−1 ≺ q3 qn−3 ≺ · · · ≺ q2⌊ n ⌋−1 qn−2⌊ n ⌋+1 ≺
4
4
q2⌊ n ⌋ qn−2⌊ n ⌋ ≺ q2⌊ n ⌋−2 qn−2⌊ n ⌋+2 ≺ · · · ≺ q2 qn−2 ≺ qn+1
4
4
4
4
Characterization of maximal and minimal matching polynomial
graphs in family of graphs with given number of vertices of
degrees one and two
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
17 / 38
Cubic bipartite graphs
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
18 / 38
Cubic bipartite graphs
G(3, 6) = {K3,3 }
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
18 / 38
Cubic bipartite graphs
G(3, 6) = {K3,3 }
G(3, 8) = {Q3 } three dimensional cube
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
18 / 38
Cubic bipartite graphs
G(3, 6) = {K3,3 }
G(3, 8) = {Q3 } three dimensional cube
G(3, 10) = {G1 , M10 } have incomparable matching polynomials
ψ(x, G1 ) := 1 + 15x + 75x 2 + 145x 3 + 96x 4 + 12x 5
ψ(x, M10 ) := 1 + 15x + 75x 2 + 145x 3 + 95x 4 + 13x 5
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
18 / 38
Cubic bipartite graphs
G(3, 6) = {K3,3 }
G(3, 8) = {Q3 } three dimensional cube
G(3, 10) = {G1 , M10 } have incomparable matching polynomials
ψ(x, G1 ) := 1 + 15x + 75x 2 + 145x 3 + 96x 4 + 12x 5
ψ(x, M10 ) := 1 + 15x + 75x 2 + 145x 3 + 95x 4 + 13x 5
For 2n from 12 to 24 the extremal graphs, with the maximal
φ(l, G):
2n
6 K3,3
S
2n−8
6 K3,3 SQ3
2n−10
6 K3,3 (G1
if 6|2n
if 6|(2n − 2)
or M10 ) if 6|(2n − 4)
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
18 / 38
Two bipartite 3-regular graphs on 10 vertices
M10
G1
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
19 / 38
Expected values of k -matchings
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
20 / 38
Expected values of k -matchings
Permutation σ : hnr i → hnr i induces G(σ) ∈ Gmult (r , 2n)
and vice versa
+j)
G(σ) = {(i, ⌈ σ((i−1)r
⌉), j = 1, . . . , r , i = 1, . . . , n} ⊂ hni × hni
r
number of different σ inducing the same simple G is (r !)n
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
20 / 38
Expected values of k -matchings
Permutation σ : hnr i → hnr i induces G(σ) ∈ Gmult (r , 2n)
and vice versa
+j)
G(σ) = {(i, ⌈ σ((i−1)r
⌉), j = 1, . . . , r , i = 1, . . . , n} ⊂ hni × hni
r
number of different σ inducing the same simple G is (r !)n
µ probability measure on Gmult (r , 2n):
µ(G(σ)) = ((nr )!)−1
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
20 / 38
Expected values of k -matchings
Permutation σ : hnr i → hnr i induces G(σ) ∈ Gmult (r , 2n)
and vice versa
+j)
G(σ) = {(i, ⌈ σ((i−1)r
⌉), j = 1, . . . , r , i = 1, . . . , n} ⊂ hni × hni
r
number of different σ inducing the same simple G is (r !)n
µ probability measure on Gmult (r , 2n):
µ(G(σ)) = ((nr )!)−1
FKM 06:
2
E (k, n, r ) := E(φ(k, G)) = kn r 2k k!(nr − k)!)(nr )!)−1 ,
k = 1, . . . , n
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
20 / 38
Expected values of k -matchings
Permutation σ : hnr i → hnr i induces G(σ) ∈ Gmult (r , 2n)
and vice versa
+j)
G(σ) = {(i, ⌈ σ((i−1)r
⌉), j = 1, . . . , r , i = 1, . . . , n} ⊂ hni × hni
r
number of different σ inducing the same simple G is (r !)n
µ probability measure on Gmult (r , 2n):
µ(G(σ)) = ((nr )!)−1
FKM 06:
2
E (k, n, r ) := E(φ(k, G)) = kn r 2k k!(nr − k)!)(nr )!)−1 ,
k = 1, . . . , n
1 ≤ kl ≤ nl , l = 1, ..., increasing sequences of integers s.t.
liml→∞ nkll = p ∈ [0, 1]. Then
lim
l→∞
log E (kl , nl , r )
= f (p, r )
2nk
f (p, r ) := 12 (p log r − p log p − 2(1 − p) log(1 − p)+ (r − p) log(1 − pr ))
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
20 / 38
p-matching and total matching entropies
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
21 / 38
p-matching and total matching entropies
G = (V , E ) infinite, degree of each vertex bounded by N,
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
21 / 38
p-matching and total matching entropies
G = (V , E ) infinite, degree of each vertex bounded by N,
p ∈ [0, 1]-matching entropy, (p-dimer entropy) of G
hG (p) =
sup
lim sup
on all sequences
l→∞
log φ(kl , Gl )
#Vl
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
21 / 38
p-matching and total matching entropies
G = (V , E ) infinite, degree of each vertex bounded by N,
p ∈ [0, 1]-matching entropy, (p-dimer entropy) of G
hG (p) =
sup
lim sup
on all sequences
l→∞
log φ(kl , Gl )
#Vl
and total matching entropy, (monomer-dimer entropy)
hG =
sup
lim sup
on all sequences
l→∞
log
P0.5(#Vl )
k =0
#Vl
φ(k, Gl )
,
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
21 / 38
p-matching and total matching entropies
G = (V , E ) infinite, degree of each vertex bounded by N,
p ∈ [0, 1]-matching entropy, (p-dimer entropy) of G
hG (p) =
sup
lim sup
on all sequences
l→∞
log φ(kl , Gl )
#Vl
and total matching entropy, (monomer-dimer entropy)
hG =
sup
lim sup
on all sequences
l→∞
log
P0.5(#Vl )
k =0
#Vl
φ(k, Gl )
,
Gl = (El , Vl ), l ∈ N a sequence of finite graphs converging to G, and
2kl
=p
l→∞ #Vl
lim
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
21 / 38
p-matching and total matching entropies
G = (V , E ) infinite, degree of each vertex bounded by N,
p ∈ [0, 1]-matching entropy, (p-dimer entropy) of G
hG (p) =
sup
lim sup
on all sequences
l→∞
log φ(kl , Gl )
#Vl
and total matching entropy, (monomer-dimer entropy)
hG =
sup
lim sup
on all sequences
l→∞
log
P0.5(#Vl )
k =0
#Vl
φ(k, Gl )
,
Gl = (El , Vl ), l ∈ N a sequence of finite graphs converging to G, and
2kl
=p
l→∞ #Vl
lim
hG = maxp∈[0,1] hG (p)
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
21 / 38
Hammersley’s results
John Michael Hammersley 21.3.1920 - 2.5.2004
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
22 / 38
Hammersley’s results
John Michael Hammersley 21.3.1920 - 2.5.2004
Book: Monte Carlo Methods 1964
(MCM attributed to Stan Ulam - 1944)
Self Avoiding Walks and Percolation Theory
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
22 / 38
Hammersley’s results
John Michael Hammersley 21.3.1920 - 2.5.2004
Book: Monte Carlo Methods 1964
(MCM attributed to Stan Ulam - 1944)
Self Avoiding Walks and Percolation Theory
60’s: G := Zd infinite 2d -regular bipartite graph
Vl = hs1,l i × hs2,l i × . . . × hsd,l i,
lim si,l = ∞, i = 1, . . . , d
Then hd (p) := hG (p) same for all sequences
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
22 / 38
Hammersley’s results
John Michael Hammersley 21.3.1920 - 2.5.2004
Book: Monte Carlo Methods 1964
(MCM attributed to Stan Ulam - 1944)
Self Avoiding Walks and Percolation Theory
60’s: G := Zd infinite 2d -regular bipartite graph
Vl = hs1,l i × hs2,l i × . . . × hsd,l i,
lim si,l = ∞, i = 1, . . . , d
Then hd (p) := hG (p) same for all sequences
hp (d ) is concave
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
22 / 38
Facts
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
23 / 38
Facts
hd (p) - p-d -dimensional monomer-dimer entropy
L. Pauling 1935, Fowler and Rushbrooke 1937
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
23 / 38
Facts
hd (p) - p-d -dimensional monomer-dimer entropy
L. Pauling 1935, Fowler and Rushbrooke 1937
h1 (p) = f (p, 2)
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
23 / 38
Facts
hd (p) - p-d -dimensional monomer-dimer entropy
L. Pauling 1935, Fowler and Rushbrooke 1937
h1 (p) = f (p, 2)
hd (1) - d -dimer entropy
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
23 / 38
Facts
hd (p) - p-d -dimensional monomer-dimer entropy
L. Pauling 1935, Fowler and Rushbrooke 1937
h1 (p) = f (p, 2)
hd (1) - d -dimer entropy
hd = maxp∈[0,1] hd (p) - d -monomer-dimer entropy
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
23 / 38
Facts
hd (p) - p-d -dimensional monomer-dimer entropy
L. Pauling 1935, Fowler and Rushbrooke 1937
h1 (p) = f (p, 2)
hd (1) - d -dimer entropy
hd = maxp∈[0,1] hd (p) - d -monomer-dimer entropy
h2 (1) =
∞
1 X (−1)q
= 0.29156090 . . .
π
(2q + 1)2
q=0
Fisher,Kasteleyn 1961
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
23 / 38
Facts
hd (p) - p-d -dimensional monomer-dimer entropy
L. Pauling 1935, Fowler and Rushbrooke 1937
h1 (p) = f (p, 2)
hd (1) - d -dimer entropy
hd = maxp∈[0,1] hd (p) - d -monomer-dimer entropy
h2 (1) =
∞
1 X (−1)q
= 0.29156090 . . .
π
(2q + 1)2
q=0
Fisher,Kasteleyn 1961
h2 (p),
p ∈ [0, 1)
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
23 / 38
Facts
hd (p) - p-d -dimensional monomer-dimer entropy
L. Pauling 1935, Fowler and Rushbrooke 1937
h1 (p) = f (p, 2)
hd (1) - d -dimer entropy
hd = maxp∈[0,1] hd (p) - d -monomer-dimer entropy
h2 (1) =
∞
1 X (−1)q
= 0.29156090 . . .
π
(2q + 1)2
q=0
Fisher,Kasteleyn 1961
h2 (p),
p ∈ [0, 1)
Baxter 1968 heuristical high precision computations
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
23 / 38
Facts
hd (p) - p-d -dimensional monomer-dimer entropy
L. Pauling 1935, Fowler and Rushbrooke 1937
h1 (p) = f (p, 2)
hd (1) - d -dimer entropy
hd = maxp∈[0,1] hd (p) - d -monomer-dimer entropy
h2 (1) =
∞
1 X (−1)q
= 0.29156090 . . .
π
(2q + 1)2
q=0
Fisher,Kasteleyn 1961
h2 (p),
p ∈ [0, 1)
Baxter 1968 heuristical high precision computations
Hammersley 70: 2-digits precision using MC
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
23 / 38
Facts
hd (p) - p-d -dimensional monomer-dimer entropy
L. Pauling 1935, Fowler and Rushbrooke 1937
h1 (p) = f (p, 2)
hd (1) - d -dimer entropy
hd = maxp∈[0,1] hd (p) - d -monomer-dimer entropy
h2 (1) =
∞
1 X (−1)q
= 0.29156090 . . .
π
(2q + 1)2
q=0
Fisher,Kasteleyn 1961
h2 (p),
p ∈ [0, 1)
Baxter 1968 heuristical high precision computations
Hammersley 70: 2-digits precision using MC
.66279897(190) 6 h2 6 .66279897(2844913) F-P 2005
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
23 / 38
Facts
hd (p) - p-d -dimensional monomer-dimer entropy
L. Pauling 1935, Fowler and Rushbrooke 1937
h1 (p) = f (p, 2)
hd (1) - d -dimer entropy
hd = maxp∈[0,1] hd (p) - d -monomer-dimer entropy
h2 (1) =
∞
1 X (−1)q
= 0.29156090 . . .
π
(2q + 1)2
q=0
Fisher,Kasteleyn 1961
h2 (p),
p ∈ [0, 1)
Baxter 1968 heuristical high precision computations
Hammersley 70: 2-digits precision using MC
.66279897(190) 6 h2 6 .66279897(2844913) F-P 2005
Friedland-Peled confirmed Baxter’s computations to be published
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
23 / 38
Computations of 3-dimensional entropies
Computational methods
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
24 / 38
Computations of 3-dimensional entropies
Computational methods
Effective computations done by transfer matrices
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
24 / 38
Computations of 3-dimensional entropies
Computational methods
Effective computations done by transfer matrices
Entropies are estimated by upper and lower bounds using the
spectral radii of transfer matrices
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
24 / 38
Computations of 3-dimensional entropies
Computational methods
Effective computations done by transfer matrices
Entropies are estimated by upper and lower bounds using the
spectral radii of transfer matrices
Group automorphisms of discrete tori speed up computations
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
24 / 38
Computations of 3-dimensional entropies
Computational methods
Effective computations done by transfer matrices
Entropies are estimated by upper and lower bounds using the
spectral radii of transfer matrices
Group automorphisms of discrete tori speed up computations
Estimate of 3-dimensional entropies
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
24 / 38
Computations of 3-dimensional entropies
Computational methods
Effective computations done by transfer matrices
Entropies are estimated by upper and lower bounds using the
spectral radii of transfer matrices
Group automorphisms of discrete tori speed up computations
Estimate of 3-dimensional entropies
h3 (1) > 0.440075 Schrijver’s lower bound 1998
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
24 / 38
Computations of 3-dimensional entropies
Computational methods
Effective computations done by transfer matrices
Entropies are estimated by upper and lower bounds using the
spectral radii of transfer matrices
Group automorphisms of discrete tori speed up computations
Estimate of 3-dimensional entropies
h3 (1) > 0.440075 Schrijver’s lower bound 1998
h3 (1) 6 0.463107 Ciucu 1998
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
24 / 38
Computations of 3-dimensional entropies
Computational methods
Effective computations done by transfer matrices
Entropies are estimated by upper and lower bounds using the
spectral radii of transfer matrices
Group automorphisms of discrete tori speed up computations
Estimate of 3-dimensional entropies
h3 (1) > 0.440075 Schrijver’s lower bound 1998
h3 (1) 6 0.463107 Ciucu 1998
h3 (1) 6 0.457547 Lundow 2001 (massive parallel computations)
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
24 / 38
Computations of 3-dimensional entropies
Computational methods
Effective computations done by transfer matrices
Entropies are estimated by upper and lower bounds using the
spectral radii of transfer matrices
Group automorphisms of discrete tori speed up computations
Estimate of 3-dimensional entropies
h3 (1) > 0.440075 Schrijver’s lower bound 1998
h3 (1) 6 0.463107 Ciucu 1998
h3 (1) 6 0.457547 Lundow 2001 (massive parallel computations)
h3 > 0.7652789557 Friedland-Peled 2005 (Tverberg conjecture)
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
24 / 38
Computations of 3-dimensional entropies
Computational methods
Effective computations done by transfer matrices
Entropies are estimated by upper and lower bounds using the
spectral radii of transfer matrices
Group automorphisms of discrete tori speed up computations
Estimate of 3-dimensional entropies
h3 (1) > 0.440075 Schrijver’s lower bound 1998
h3 (1) 6 0.463107 Ciucu 1998
h3 (1) 6 0.457547 Lundow 2001 (massive parallel computations)
h3 > 0.7652789557 Friedland-Peled 2005 (Tverberg conjecture)
h3 6 .7862023450 Friedland-Peled 2005
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
24 / 38
Computations of 3-dimensional entropies
Computational methods
Effective computations done by transfer matrices
Entropies are estimated by upper and lower bounds using the
spectral radii of transfer matrices
Group automorphisms of discrete tori speed up computations
Estimate of 3-dimensional entropies
h3 (1) > 0.440075 Schrijver’s lower bound 1998
h3 (1) 6 0.463107 Ciucu 1998
h3 (1) 6 0.457547 Lundow 2001 (massive parallel computations)
h3 > 0.7652789557 Friedland-Peled 2005 (Tverberg conjecture)
h3 6 .7862023450 Friedland-Peled 2005
h3 ≥ .7845241927 Friedland-Gurvits 2008
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
24 / 38
Computations of 3-dimensional entropies
Computational methods
Effective computations done by transfer matrices
Entropies are estimated by upper and lower bounds using the
spectral radii of transfer matrices
Group automorphisms of discrete tori speed up computations
Estimate of 3-dimensional entropies
h3 (1) > 0.440075 Schrijver’s lower bound 1998
h3 (1) 6 0.463107 Ciucu 1998
h3 (1) 6 0.457547 Lundow 2001 (massive parallel computations)
h3 > 0.7652789557 Friedland-Peled 2005 (Tverberg conjecture)
h3 6 .7862023450 Friedland-Peled 2005
h3 ≥ .7845241927 Friedland-Gurvits 2008
h3 ≥ .7849602275 Friedland-Krop-Lundow-Markström
accepted JOSS 2008
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
24 / 38
Asymptotic Lower and Upper Matching conjectures
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
25 / 38
Asymptotic Lower and Upper Matching conjectures
FKLM 06:
Gl = (El , Vl ) ∈ G(r , #Vl ), l = 1, 2, . . . , and liml→∞
2kl
#Vl
= p.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
25 / 38
Asymptotic Lower and Upper Matching conjectures
FKLM 06:
Gl = (El , Vl ) ∈ G(r , #Vl ), l = 1, 2, . . . , and liml→∞
lowr (p) :=
inf
lim inf
all allowable sequences l→∞
2kl
#Vl
= p.
log φ(kl , Gl )
#Vl
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
25 / 38
Asymptotic Lower and Upper Matching conjectures
FKLM 06:
Gl = (El , Vl ) ∈ G(r , #Vl ), l = 1, 2, . . . , and liml→∞
lowr (p) :=
inf
lim inf
all allowable sequences l→∞
2kl
#Vl
= p.
log φ(kl , Gl )
#Vl
ALMC: lowr (p) = f (p, r ) (For most of the sequences lim inf = f (p, r ))
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
25 / 38
Asymptotic Lower and Upper Matching conjectures
FKLM 06:
Gl = (El , Vl ) ∈ G(r , #Vl ), l = 1, 2, . . . , and liml→∞
lowr (p) :=
inf
lim inf
all allowable sequences l→∞
2kl
#Vl
= p.
log φ(kl , Gl )
#Vl
ALMC: lowr (p) = f (p, r ) (For most of the sequences lim inf = f (p, r ))
uppr (p) :=
sup
lim sup
all allowable sequences
l→∞
log φ(kl , Gl )
#Vl
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
25 / 38
Asymptotic Lower and Upper Matching conjectures
FKLM 06:
Gl = (El , Vl ) ∈ G(r , #Vl ), l = 1, 2, . . . , and liml→∞
lowr (p) :=
inf
lim inf
all allowable sequences l→∞
2kl
#Vl
= p.
log φ(kl , Gl )
#Vl
ALMC: lowr (p) = f (p, r ) (For most of the sequences lim inf = f (p, r ))
uppr (p) :=
sup
lim sup
all allowable sequences
l→∞
log φ(kl , Gl )
#Vl
AUMC: uppr (p) = hK (r ) (p), K (r ) countable union of Kr ,r
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
25 / 38
Asymptotic Lower and Upper Matching conjectures
FKLM 06:
Gl = (El , Vl ) ∈ G(r , #Vl ), l = 1, 2, . . . , and liml→∞
lowr (p) :=
inf
lim inf
all allowable sequences l→∞
2kl
#Vl
= p.
log φ(kl , Gl )
#Vl
ALMC: lowr (p) = f (p, r ) (For most of the sequences lim inf = f (p, r ))
uppr (p) :=
sup
lim sup
all allowable sequences
l→∞
log φ(kl , Gl )
#Vl
AUMC: uppr (p) = hK (r ) (p), K (r ) countable union of Kr ,r
Pr (t) :=
p(t) := Pr′ (t) ∈ (0, 1),
log
r 2
2kt
k =0 k k! e
Pr
, t ∈ R,
2r
hK (r ) (p(t)) := Pr (t) − tp(t)
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
25 / 38
r =4
h2
0.6
0.5
AUMC
0.4
B
0.3
ALMC
FT
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
26 / 38
r =6
h3High
0.8
h3Low
0.6
AUMC
ALMC
FT
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
inpgraphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
27 / 38
Lower asymptotic bounds Friedland-Gurvits 2008
Thm: r ≥ 3, s ≥ 1 integers,
Bn ∈ Ωn , n = 1, 2, . . . each column of Bn has at most r -nonzero entries.
kn ∈ [0, n] ∩ N, n = 1, 2, . . . , limn→∞ knn = p ∈ (0, 1] then
log permkn Bn
1
> (−p log p − 2(1 − p) log(1 − p)) +
2n
2
1
1
1
1−p
(r + s − 1) log(1 −
) − (s − 1 + p) log(1 −
)
2
r +s
2
s
lim inf
n→∞
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
28 / 38
Lower asymptotic bounds Friedland-Gurvits 2008
Thm: r ≥ 3, s ≥ 1 integers,
Bn ∈ Ωn , n = 1, 2, . . . each column of Bn has at most r -nonzero entries.
kn ∈ [0, n] ∩ N, n = 1, 2, . . . , limn→∞ knn = p ∈ (0, 1] then
log permkn Bn
1
> (−p log p − 2(1 − p) log(1 − p)) +
2n
2
1
1
1
1−p
(r + s − 1) log(1 −
) − (s − 1 + p) log(1 −
)
2
r +s
2
s
lim inf
n→∞
Prf combines properties positive hyperbolic polynomials, capacity and
the measure on G(r , 2n)
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
28 / 38
Lower asymptotic bounds Friedland-Gurvits 2008
Thm: r ≥ 3, s ≥ 1 integers,
Bn ∈ Ωn , n = 1, 2, . . . each column of Bn has at most r -nonzero entries.
kn ∈ [0, n] ∩ N, n = 1, 2, . . . , limn→∞ knn = p ∈ (0, 1] then
log permkn Bn
1
> (−p log p − 2(1 − p) log(1 − p)) +
2n
2
1
1
1
1−p
(r + s − 1) log(1 −
) − (s − 1 + p) log(1 −
)
2
r +s
2
s
lim inf
n→∞
Prf combines properties positive hyperbolic polynomials, capacity and
the measure on G(r , 2n)
r
Cor: r -ALMC holds for ps = r +s
, s = 0, 1, . . . ,
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
28 / 38
Lower asymptotic bounds Friedland-Gurvits 2008
Thm: r ≥ 3, s ≥ 1 integers,
Bn ∈ Ωn , n = 1, 2, . . . each column of Bn has at most r -nonzero entries.
kn ∈ [0, n] ∩ N, n = 1, 2, . . . , limn→∞ knn = p ∈ (0, 1] then
log permkn Bn
1
> (−p log p − 2(1 − p) log(1 − p)) +
2n
2
1
1
1
1−p
(r + s − 1) log(1 −
) − (s − 1 + p) log(1 −
)
2
r +s
2
s
lim inf
n→∞
Prf combines properties positive hyperbolic polynomials, capacity and
the measure on G(r , 2n)
r
Cor: r -ALMC holds for ps = r +s
, s = 0, 1, . . . ,
Con: under Thm assumptions
lim inf
n→∞
log permkn Bn
p
> f (r , p) − log r
2n
2
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
28 / 38
Lower asymptotic bounds Friedland-Gurvits 2008
Thm: r ≥ 3, s ≥ 1 integers,
Bn ∈ Ωn , n = 1, 2, . . . each column of Bn has at most r -nonzero entries.
kn ∈ [0, n] ∩ N, n = 1, 2, . . . , limn→∞ knn = p ∈ (0, 1] then
log permkn Bn
1
> (−p log p − 2(1 − p) log(1 − p)) +
2n
2
1
1
1
1−p
(r + s − 1) log(1 −
) − (s − 1 + p) log(1 −
)
2
r +s
2
s
lim inf
n→∞
Prf combines properties positive hyperbolic polynomials, capacity and
the measure on G(r , 2n)
r
Cor: r -ALMC holds for ps = r +s
, s = 0, 1, . . . ,
Con: under Thm assumptions
lim inf
n→∞
For ps =
r
r +s , s
log permkn Bn
p
> f (r , p) − log r
2n
2
= 0, 1, . . . , conjecture holds
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
28 / 38
Known lower and upper bounds for p-matchings
FKLM accepted JOSS 08:
lowr (p) ≥ max(lowr ,1 (p), lowr ,2 (p))
uppr (p) ≤ min(uppr ,1 (p), uppr ,2 (p))
Lower estimates are based on F-G inequalities
and Newton P
inequalities:
n
f (x) = x + ni=1 ai x n−i have nonpositive roots
−1
then kn
ak log concave sequence
Upper estimates are based on Bregman inequalities :
n−k
k
n (r !) n (n!) n
φ(k, G) ≤
k
(n − k)!
and
n k
max
φ(k, G) =
r
k
G∈Gmult (r ,2n)
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
29 / 38
Concavity results
1
hd (p) + (p log p + (1 − p) log(1 − p))
2
concave
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
30 / 38
Concavity results
1
hd (p) + (p log p + (1 − p) log(1 − p))
2
concave
Hence hd (p) concave - Hammersley
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
30 / 38
Concavity results
1
hd (p) + (p log p + (1 − p) log(1 − p))
2
concave
Hence hd (p) concave - Hammersley
1
lowr (p) + (p log p + (1 − p) log(1 − p))
2
concave
Hence lowr (p) concave -
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
30 / 38
Concavity results
1
hd (p) + (p log p + (1 − p) log(1 − p))
2
concave
Hence hd (p) concave - Hammersley
1
lowr (p) + (p log p + (1 − p) log(1 − p))
2
concave
Hence lowr (p) concave Prf: Newton inequalities
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
30 / 38
r = 4 lower bounds
0.6
0.5
0.4
0.3
0.2
0.1
0.2
0.4
0.6
0.8
1
Figure: fl(p, 4)-red, low4,1 (p)-blue, f (p, 4)-green
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
31 / 38
r = 4 lower bounds differences
0.2
0.4
0.6
0.8
1
-0.005
-0.01
-0.015
-0.02
-0.025
-0.03
-0.035
Figure: low4,1 (p) − f (p, 4)-black, low4,2 (p) − f (p, 4)-blue
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
32 / 38
r = 4 upper bounds
r=4
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1
Figure: hK (4) -green, upp4,1 -blue, upp4,2 -orange
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
33 / 38
Summary
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
34 / 38
Summary
Matches in graphs and their number is a basic concept in graphs.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
34 / 38
Summary
Matches in graphs and their number is a basic concept in graphs.
Counting k-matching in bipartite graphs is equivalent to computing
permanents of 0 − 1 matrices.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
34 / 38
Summary
Matches in graphs and their number is a basic concept in graphs.
Counting k-matching in bipartite graphs is equivalent to computing
permanents of 0 − 1 matrices.
Estimating matchings in regular bipartite graphs fuses
combinatorics and analysis.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
34 / 38
Summary
Matches in graphs and their number is a basic concept in graphs.
Counting k-matching in bipartite graphs is equivalent to computing
permanents of 0 − 1 matrices.
Estimating matchings in regular bipartite graphs fuses
combinatorics and analysis.
Generalizing matching concepts to infinite graphs brings in the
elements of statistical physics:
entropies, the grand partition function, pressure and probability.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
34 / 38
Summary
Matches in graphs and their number is a basic concept in graphs.
Counting k-matching in bipartite graphs is equivalent to computing
permanents of 0 − 1 matrices.
Estimating matchings in regular bipartite graphs fuses
combinatorics and analysis.
Generalizing matching concepts to infinite graphs brings in the
elements of statistical physics:
entropies, the grand partition function, pressure and probability.
Computation of these entropies to a good precision needs
massive memory and huge computational power.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
34 / 38
Open problems
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
35 / 38
Open problems
Lower and upper matching conjectures for regular bipartite
graphs.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
35 / 38
Open problems
Lower and upper matching conjectures for regular bipartite
graphs.
Asymptotic lower and upper matching conjectures.
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
35 / 38
Open problems
Lower and upper matching conjectures for regular bipartite
graphs.
Asymptotic lower and upper matching conjectures.
Closed formula or high precision values for d ≥ 3 dimer and
monomer-dimer entropies in Zd .
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
35 / 38
Open problems
Lower and upper matching conjectures for regular bipartite
graphs.
Asymptotic lower and upper matching conjectures.
Closed formula or high precision values for d ≥ 3 dimer and
monomer-dimer entropies in Zd .
Other lattices as Bethe lattices, i.e. infinite regular trees
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
35 / 38
Open problems
Lower and upper matching conjectures for regular bipartite
graphs.
Asymptotic lower and upper matching conjectures.
Closed formula or high precision values for d ≥ 3 dimer and
monomer-dimer entropies in Zd .
Other lattices as Bethe lattices, i.e. infinite regular trees
Non-bipartite graphs
Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
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2008
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Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
36 / 38
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Shmuel Friedland Univ. Illinois at Chicago & Berlin
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Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
37 / 38
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Shmuel Friedland Univ. Illinois at Chicago & Berlin
Counting
Mathematical
matchingsSchool
in graphs
() with applications to the monomer-dimer
KTH, 16 April,models
2008
38 / 38