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) a (big) finite graph a set of edges, i.e. a spanning subgraph

graph notation G=(V,E) a (big) finite graph a set of edges, i.e. a spanning subgraph the degree function of H

graph notation G=(V,E) a (big) finite graph a set of edges, i.e. a spanning subgraph the degree function of H the set of vertices of degree k in H is

energy of a subgraph J the energy of a single edge

energy of a subgraph J the energy of a single edge the “chemical potential” of a vertex of degree k

energy of a subgraph J the energy of a single edge the “chemical potential” of a vertex of degree k the energy of a (spanning) subgraph H is

partition function T the absolute temperature

partition function T the absolute temperature the inverse temperature

partition function T the absolute temperature the inverse temperature the Boltzmann weight of a subgraph H is

partition function T the absolute temperature the inverse temperature the Boltzmann weight of a subgraph H is the partition function is

polynomial expression let and

polynomial expression let and for a subgraph H let

polynomial expression let and for a subgraph H let the partition function is

multivariate version let and

multivariate version let and the multivariate partition function is

multivariate version let and the multivariate partition function is then

example let and for all k>=2

example let and for all k>=2

example let and for all k>=2 and 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 where

vertex-dependent activities the chemical potentials can vary from vertex to vertex: let where and redefine

vertex-dependent activities the chemical potentials can vary from vertex to vertex: let where and redefine the multivariate partition function is still

II. The Results

the key polynomials for each vertex v of G form the key polynomial in which

the key polynomials for each vertex v of G form the key polynomial in which Since this polynomial depends on T

the key polynomials for each vertex v of G form the key polynomial in which Since this polynomial depends on T except when all

the key polynomials for each vertex v of G form the key polynomial in which Since this polynomial depends on T except when all that is, when all

first theorem Assume that all zeros of all the keys are within an angle of the negative real axis Then…

first theorem Assume that all zeros of all the keys are within an angle of the negative real axis Then… 1. If for all v then

first theorem Assume that all zeros of all the keys are within an angle of the negative real axis Then… 1. If for all v then 2. If then

first theorem Assume that all zeros of all the keys are within an angle of the negative real axis Then… 1. If for all v then 2. If then 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 Then… 1. If for all v then 2. If then This statement is independent of the size of the graph…. so it can be used for thermodynamic limits.

first theorem

Consider the case

first theorem Consider the case Assume that all zeros of all the keys are nonpositive real numbers. Then…

first theorem Consider the case Assume that all zeros of all the keys are nonpositive real numbers. Then… 1. If for all v then

first theorem Consider the case Assume that all zeros of all the keys are nonpositive real numbers. Then… 1. If for all v then (the half-plane property)

first theorem Consider the case Assume that all zeros of all the keys are nonpositive real numbers. Then… 1. If for all v then (the half-plane property) 2. All zeros of are nonpositive real numbers.

the Heilmann-Lieb (1972) theorem let and for all k>=2

the Heilmann-Lieb (1972) theorem let and for all k>=2 for each vertex v, has only real nonpositive zeros….

the Heilmann-Lieb (1972) theorem let and for all k>=2 for each vertex v, has only real nonpositive zeros.… 1. The multivariate matching polynomial has the half-plane property.

the Heilmann-Lieb (1972) theorem let and for all k>=2 for each vertex v, 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.

a generalization fix functions such that (at every vertex)

a generalization fix functions such that (at every vertex) choose vertex chemical potentials so that

a generalization fix functions such that (at every vertex) choose vertex chemical potentials so that Then every key has only real nonpositive zeros, so that 1. has the half-plane property (new) 2. has only real nonpositive zeros (W. 1996)

a theorem of Ruelle (1999) let and for all k>=3

a theorem of Ruelle (1999) let and for all k>=3 for each vertex v, has all its zeros within of the negative real axis

a theorem of Ruelle (1999) let and for all k>=3 for each vertex v, has all its zeros within of the negative real axis 1. If for all v then (new)

a theorem of Ruelle (1999) let and for all k>=3 for each vertex v, has all its zeros within of the negative real axis 1. If for all v then (new) 2. If then

a theorem of Ruelle (1999) let and for all k>=3 for each vertex v, has all its zeros within of the negative real axis 1. If for all v then (new) 2. If then (Ruelle proves that for 2. it suffices that for a graph with maximum degree.)

second theorem Assume that all zeros of all the keys have modulus at least. Then… 1. If for all v then 2. If then

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 for all v then 2. If then

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 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 is on the unit circle. For any graph G, every zero of is on the unit circle.

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

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 and that the thermodynamic limit free energy exists:

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 and that the thermodynamic limit free energy exists: If the free energy is non-analytic at a nonnegative real then has a zero not at the origin with nonnegative real part.

example 1. let and for all k>=3

example 1. let and for all k>=3 the key is

example 1. let and for all k>=3 the key is if then the zeros of K(z) have negative real part…. No phase transitions for any physical (J,T)

example 1. let and for all k>=3 the key is if 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

example 2. fix functions such that (at every vertex)

example 2. fix functions such that (at every vertex) choose vertex chemical potentials so that

example 2. fix functions such that (at every vertex) choose vertex chemical potentials so that When the thermodynamic limit exists it is analytic for all physical values of (J,T). (no phase transitions)

example 3. in a 2d-regular graph, consider the key

example 3. in a 2d-regular graph, consider the key for a thermodynamic limit of these a phase transition with can only happen at

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)