CSM Workshop 1: Zeros of Graph Polynomials
Enumeration of Spanning Subgraphs
with Degree Constraints
Dave Wagner
University of Waterloo
I. The Set-Up
graph notation
G=(V,E)
a (big) finite graph
graph notation
G=(V,E)
H  E (G )
a (big) finite graph
a set of edges, i.e. a spanning subgraph
graph notation
G=(V,E)
H  E (G )
a (big) finite graph
a set of edges, i.e. a spanning subgraph
deg( H ) : V  N
the degree function of H
graph notation
G=(V,E)
H  E (G )
a (big) finite graph
a set of edges, i.e. a spanning subgraph
deg( H ) : V  N
the degree function of H
the set of vertices of degree k in H is
Vk ( H )  v V : deg( H , v)  k 
energy of a subgraph
J
the energy of a single edge
energy of a subgraph
J
the energy of a single edge
k
the “chemical potential” of a vertex of degree k
energy of a subgraph
J
the energy of a single edge
k
the “chemical potential” of a vertex of degree k
the energy of a (spanning) subgraph H is
U ( H )  J # H   k #Vk ( H )
k
partition function
T
the absolute temperature
partition function
T
the absolute temperature
1

k BT
the inverse temperature
partition function
T
the absolute temperature
1

k BT
the inverse temperature
the Boltzmann weight of a subgraph H is
e
 U ( H )
partition function
T
the absolute temperature
1

k BT
the inverse temperature
e
the Boltzmann weight of a subgraph H is
the partition function is
ZG 
e
 U ( H )
 U ( H )
H  E(G)
polynomial expression
let
ye
 J
and
uk  e
  k
polynomial expression
let
ye
 J
and
for a subgraph H let
uk  e
  k
u deg(H )   udeg(H ,v )
vV
polynomial expression
let
ye
 J
and
for a subgraph H let
uk  e
  k
u deg(H )   udeg(H ,v )
vV
the partition function is
Z G (u, y ) 
u
H  E(G)
deg( H )
y
#H
multivariate version
let
x  xv : v V 
and
x
deg( H )
x
deg( H ,v )
v
vV
multivariate version
let
x  xv : v V 
and
x
deg( H )
x
deg( H ,v )
v
vV
the multivariate partition function is
~
ZG (u, x) 
u
H  E(G)
deg( H )
x
deg( H )
multivariate version
let
x  xv : v V 
and
x
deg( H )
x
deg( H ,v )
v
vV
the multivariate partition function is
~
ZG (u, x) 
then
u
H  E(G)
deg( H )
~
1/ 2
ZG (u, y)  ZG (u, y )
x
deg( H )
example
let
u0  u1  1
and
uk  0
for all k>=2
example
let
u0  u1  1
and
u deg(H )   udeg(H ,v )
vV
uk  0
for all k>=2
1 if H is a matching

otherwise
0
example
let
u0  u1  1
and
u deg(H )   udeg(H ,v )
vV
~
ZG (u, x)
uk  0
for all k>=2
1 if H is a matching

otherwise
0
and
Z G (u, y )
are, respectively, the multivariate and univariate
matching polynomials of G
vertex-dependent activities
the chemical potentials can vary from vertex to vertex:
vertex-dependent activities
the chemical potentials can vary from vertex to vertex:
let
u
(v)

 u , u ,..., u
(v)
0
(v)
1
(v)
d

where d  deg( G, v)
vertex-dependent activities
the chemical potentials can vary from vertex to vertex:
let
u
(v)

 u , u ,..., u
and redefine
(v)
0
(v)
1
(v)
d
u deg(H )   u
vV

where d  deg( G, v)
(v)
deg( H ,v )
vertex-dependent activities
the chemical potentials can vary from vertex to vertex:
let
u
(v)

 u , u ,..., u
and redefine
(v)
0
(v)
1
(v)
d
u deg(H )   u

where d  deg( G, v)
(v)
deg( H ,v )
vV
the multivariate partition function is still
~
ZG (u, x) 
u
H  E(G)
deg( H )
x
deg( H )
II. The Results
the key polynomials
for each vertex v of G form the key polynomial
 d  (v) k
K v ( z )     uk z
k 0  k 
d
in which d  deg( G, v)
the key polynomials
for each vertex v of G form the key polynomial
 d  (v) k
K v ( z )     uk z
k 0  k 
d
Since
u
(v)
k
e
  k( v )
in which d  deg( G, v)
this polynomial depends on T
the key polynomials
for each vertex v of G form the key polynomial
 d  (v) k
K v ( z )     uk z
k 0  k 
d
Since
u
(v)
k
e
except when all
  k( v )

(v )
k
in which d  deg( G, v)
this polynomial depends on T
 0, 
the key polynomials
for each vertex v of G form the key polynomial
 d  (v) k
K v ( z )     uk z
k 0  k 
d
Since
u
(v)
k
e
except when all
that is, when all
  k( v )

in which d  deg( G, v)
this polynomial depends on T
(v )
k
 0, 
(v)
k
 0,1
u
first theorem
Assume that all zeros of all the keys are within an angle
 of the negative real axis 0     / 2. Then…
first theorem
Assume that all zeros of all the keys are within an angle
 of the negative real axis 0     / 2. Then…
1. If arg( xv ) 

2
~
  for all v then ZG (u, x)  0.
first theorem
Assume that all zeros of all the keys are within an angle
 of the negative real axis 0     / 2. Then…
1. If arg( xv ) 
2. If

2
~
  for all v then ZG (u, x)  0.
arg( y)    2 then Z G (u, y )  0.
first theorem
Assume that all zeros of all the keys are within an angle
 of the negative real axis 0     / 2. Then…
1. If arg( xv ) 
2. If

2
~
  for all v then ZG (u, x)  0.
arg( y)    2 then Z G (u, y )  0.
This statement is independent of the size of the graph….
first theorem
Assume that all zeros of all the keys are within an angle
 of the negative real axis 0     / 2. Then…
1. If arg( xv ) 
2. If

2
~
  for all v then ZG (u, x)  0.
arg( y)    2 then Z G (u, y )  0.
This statement is independent of the size of the graph….
so it can be used for thermodynamic limits.
first theorem
first theorem
first theorem
Consider the case
  0:
first theorem
Consider the case
  0:
Assume that all zeros of all the keys are nonpositive real
numbers. Then…
first theorem
Consider the case
  0:
Assume that all zeros of all the keys are nonpositive real
numbers. Then…
1. If arg( xv ) 

2
~
for all v then ZG (u, x)  0.
first theorem
Consider the case
  0:
Assume that all zeros of all the keys are nonpositive real
numbers. Then…
1. If arg( xv ) 

2
~
for all v then ZG (u, x)  0.
(the half-plane property)
first theorem
Consider the case
  0:
Assume that all zeros of all the keys are nonpositive real
numbers. Then…
1. If arg( xv ) 

2
~
for all v then ZG (u, x)  0.
(the half-plane property)
2. All zeros of Z G (u, y ) are nonpositive real numbers.
the Heilmann-Lieb (1972) theorem
let
u0  u1  1
and
uk  0
for all k>=2
the Heilmann-Lieb (1972) theorem
let
u0  u1  1
for each vertex v,
and
uk  0
for all k>=2
K v ( z )  1  dz
has only real nonpositive zeros….
d  deg( G, v)
the Heilmann-Lieb (1972) theorem
let
u0  u1  1
for each vertex v,
and
uk  0
for all k>=2
K v ( z )  1  dz
d  deg( G, v)
has only real nonpositive zeros.…
1. The multivariate matching polynomial
has the half-plane property.
~
ZG (u, x)
the Heilmann-Lieb (1972) theorem
let
u0  u1  1
for each vertex v,
and
uk  0
for all k>=2
d  deg( G, v)
K v ( z )  1  dz
has only real nonpositive zeros….
1. The multivariate matching polynomial
has the half-plane property.
2. The univariate matching polynomial
has only real nonpositive zeros.
~
ZG (u, x)
Z G (u, y )
a generalization
fix functions f , g : V (G )  N such that f  g  f  1
(at every vertex)
a generalization
fix functions f , g : V (G )  N such that f  g  f  1
(at every vertex)
choose vertex chemical potentials so that
u
(v)
k
1 if f (v)  k  g (v)

otherwise
0
a generalization
fix functions f , g : V (G )  N such that f  g  f  1
(at every vertex)
choose vertex chemical potentials so that
u
(v)
k
1 if f (v)  k  g (v)

otherwise
0
Then every key has only real nonpositive zeros, so that
1.
~
ZG (u, x)
has the half-plane property
(new)
2.
Z G (u, y )
has only real nonpositive zeros
(W. 1996)
a theorem of Ruelle (1999)
let
u0  u1  u2  1
and
uk  0
for all k>=3
a theorem of Ruelle (1999)
let
u0  u1  u2  1
for each vertex v,
and
uk  0
for all k>=3
d  2
K v ( z )  1  dz    z
2
has all its zeros within
  /4
of the negative real axis
a theorem of Ruelle (1999)
let
u0  u1  u2  1
uk  0
for all k>=3
d  2
K v ( z )  1  dz    z
2
for each vertex v,
has all its zeros within
1. If arg( xv ) 
and

4
  /4
of the negative real axis
~
for all v then ZG (u, x)  0. (new)
a theorem of Ruelle (1999)
let
u0  u1  u2  1
has all its zeros within
2. If
arg( y ) 
uk  0
for all k>=3
d  2
K v ( z )  1  dz    z
2
for each vertex v,
1. If arg( xv ) 
and

4

2
  /4
of the negative real axis
~
for all v then ZG (u, x)  0. (new)
then Z G (u, y )  0.
a theorem of Ruelle (1999)
let
u0  u1  u2  1
has all its zeros within
2. If
arg( y ) 
uk  0
for all k>=3
d  2
K v ( z )  1  dz    z
2
for each vertex v,
1. If arg( xv ) 
and

4

2
  /4
of the negative real axis
~
for all v then ZG (u, x)  0. (new)
then Z G (u, y )  0.
2
(Ruelle proves that for 2. it suffices that Re( y ) 
(  1) 2
for a graph with maximum degree  .)
second theorem
Assume that all zeros of all the keys have
modulus at least  . Then…
~
1. If xv   for all v then ZG (u, x)  0.
2. If
y   2 then Z G (u, y )  0.
third theorem
Assume that all zeros of all the keys have
modulus at most  , and that the degree of each key
equals the degree of the corresponding vertex. Then…
~
1. If xv   for all v then ZG (u, x)  0.
2. If
y 
2
then Z G (u, y )  0.
corollary
If all zeros of all keys are on the unit circle, and all keys
have the same degree as the corresponding vertex, then
every zero of Z G (u, y ) is on the unit circle.
corollary
If all zeros of all keys are on the unit circle, and all keys
have the same degree as the corresponding vertex, then
every zero of Z G (u, y ) is on the unit circle.
For any graph G, every zero of
y
H  E (G )
is on the unit circle.
#H
 deg( G, v) 



vV  deg( H , v) 
1
application
consider a sequence of graphs G whose union is
an infinite graph 
application
consider a sequence of graphs G whose union is
an infinite graph 
assume that each graph G is d-regular
application
consider a sequence of graphs G whose union is
an infinite graph 
assume that each graph G is d-regular
that all keys are the same K (  , z )
application
consider a sequence of graphs G whose union is
an infinite graph 
assume that each graph G is d-regular
that all keys are the same K (  , z )
and that the thermodynamic limit free energy exists:
1
f  (  , J , μ)  lim
log Z G (u, y )
G  # V (G )
application
consider a sequence of graphs G whose union is
an infinite graph 
assume that each graph G is d-regular
that all keys are the same K (  , z )
and that the thermodynamic limit free energy exists:
1
f  (  , J , μ)  lim
log Z G (u, y )
G  # V (G )
If the free energy is non-analytic at a nonnegative real  *
then K (* , z ) has a zero not at the origin with nonnegative
real part.
example 1.
let
u0  1
and
uk  0
for all k>=3
example 1.
let
u0  1
and
uk  0
for all k>=3
d 
2
the key is K ( z )  1  du1 z    u2 z
 2
example 1.
let
u0  1
and
uk  0
for all k>=3
d 
2
the key is K ( z )  1  du1 z    u2 z
 2
if
u1  e
 1
0
then the zeros of K(z) have negative
real part…. No phase transitions for any physical (J,T)
example 1.
let
u0  1
and
uk  0
for all k>=3
d 
2
the key is K ( z )  1  du1 z    u2 z
 2
if
u1  e
 1
0
then the zeros of K(z) have negative
real part…. No phase transitions for any physical (J,T)
from the second theorem it follows that when
there is no phase transition for
J  2
k BT 
log d2

u1  0
example 2.
fix functions f , g : V (G )  N such that
(at every vertex)
f  g  f 3
example 2.
fix functions f , g : V (G )  N such that
(at every vertex)
choose vertex chemical potentials so that
u
(v)
k
1 if f (v)  k  g (v)

otherwise
0
f  g  f 3
example 2.
fix functions f , g : V (G )  N such that
(at every vertex)
f  g  f 3
choose vertex chemical potentials so that
u
(v)
k
1 if f (v)  k  g (v)

otherwise
0
When the thermodynamic limit f  ( , J , μ) exists
it is analytic for all physical values of (J,T).
(no phase transitions)
example 3.
in a 2d-regular graph, consider the key
 2d  d
K ( z )  1    z  u z 2 d
d 
example 3.
in a 2d-regular graph, consider the key
 2d  d
K ( z )  1    z  u z 2 d
d 
for a thermodynamic limit
a phase transition with
can only happen at
f  (  , J , μ)
of these

k BT 
 2d 
2 log    log 4
d 
2
2
d
1 
  (dJ  2 )  log    e  
2 d 
III. Summary
summary
* very general set-up, but it records no global structure
summary
* very general set-up, but it records no global structure
* unifies a number of previously considered things
summary
* very general set-up, but it records no global structure
* unifies a number of previously considered things
* very mild hypotheses, but similarly weak conclusions
about absence of phase transitions:
summary
* very general set-up, but it records no global structure
* unifies a number of previously considered things
* very mild hypotheses, but similarly weak conclusions
about absence of phase transitions:
* many general “soft” results
summary
* very general set-up, but it records no global structure
* unifies a number of previously considered things
* very mild hypotheses, but similarly weak conclusions
about absence of phase transitions:
* many general “soft” results
* some quantitative “hard” versions of qualitatively
intuitive results
summary
* very general set-up, but it records no global structure
* unifies a number of previously considered things
* very mild hypotheses, but similarly weak conclusions
about absence of phase transitions:
* many general “soft” results
* some quantitative “hard” versions of qualitatively
intuitive results
* proofs are short and easy:
(half-plane property/polarize & Grace-Walsh-Szego/
“monkey business”/diagonalize)