1. CSM Workshop 1: Zeros of Graph Polynomials Enumeration of Spanning Subgraphs with Degree Constraints Dave Wagner University of Waterloo

2. I. The Set-Up

3. graph notation G=(V,E) a (big) finite graph

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

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

6. 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

7. energy of a subgraph J the energy of a single edge

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

9. 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

10. partition function T the absolute temperature

11. partition function T the absolute temperature the inverse temperature

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

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

14. polynomial expression let and

15. polynomial expression let and for a subgraph H let

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

17. multivariate version let and

18. multivariate version let and the multivariate partition function is

19. multivariate version let and the multivariate partition function is then

20. example let and for all k>=2

21. example let and for all k>=2

22. example let and for all k>=2 and are, respectively, the multivariate and univariate matching polynomials of G

23. vertex-dependent activities the chemical potentials can vary from vertex to vertex:

24. vertex-dependent activities the chemical potentials can vary from vertex to vertex: let where

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

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

27. II. The Results

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

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

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

31. 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

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

33. 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

34. 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

35. 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….

36. 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.

37. first theorem

39. Consider the case

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

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

42. 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)

43. 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.

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

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

46. 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.

47. 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.

48. a generalization fix functions such that (at every vertex)

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

50. 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)

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

52. 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

53. 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)

54. 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

55. 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.)

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

57. 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

58. 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.

59. 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.

60. application consider a sequence of graphs G whose union is an infinite graph

61. application consider a sequence of graphs G whose union is an infinite graph assume that each graph G is d-regular

62. 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

63. 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:

64. 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.

65. example 1. let and for all k>=3

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

67. 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)

68. 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

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

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

71. 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)

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

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

74. III. Summary

75. summary * very general set-up, but it records no global structure

76. summary * very general set-up, but it records no global structure * unifies a number of previously considered things

77. 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:

78. 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

79. 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

80. 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)