Graph polynomials and their applications I: The tutte polynomial

Structural Analysis of Complex Networks

Volumn , Issue , 2011, Pages 219-255

  Ellis Monaghan, Joanna A ^a,b   Merino, Criel ^c  

^a SAINT MICHAEL'S COLLEGE   (United States)

^b University of Vermont ^*  (United States)

^c UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO   (Mexico)

Author keywords

Abelian sandpile model; Beta invariant; Chromatic polynomial; Flow polynomial; Graph polynomial; Reliability polynomial; Shelling polynomial; Spanning tree; Tutte polynomial

Indexed keywords

EID: 84892823098     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/978-0-8176-4789-6_9     Document Type: Chapter

References (116)

1. Aigner M, Ziegler GM (2001) Proofs from the book. Springer, Berlin, Heidelberg
2. Alon N, Frieze AM, Welsh D J A (1995) Polynomial time randomized approximation schemes for Tutte-Gröthendieck invariants: the dense case. Random Struct Algorithm 6:459-478
3. Andrzejak A (1998) An algorithm for the Tutte polynomials of graphs of bounded treewidth. Discrete Math 190:39-54
4. Bak P, Tang C, Wiesenfeld K (1988) Self-organized criticality. Phys Rev A 38:364-374
5. Benashski J, Martin R, Moore J, Traldi L (1995) On the β-invariant for graphs. Congr Numer 109:211-221
6. Biggs N (1996) Algebraic graph theory, 2nd edn. Cambridge University Press, Cambridge
7. Biggs N (1996) Chip firing and the critical group of a graph. Research report, London school of economics, London
8. Birkhoff GD (1912) A determinant formula for the number of ways of coloring a map. Ann Math 14:42-46
9. Björner A (1992) Homology and shellability of matroids and geometric lattices. In: White N (ed) Matroid applications, encyclopedia of mathematics and its applications. Cambridge University Press, Cambridge
10. Bodlaender HL (1993) A tourist guide through treewidth. Acta Cybernet 11:1-21
11. Bollobás B (1998) Modern graph theory. Graduate texts in mathematics. Springer, New York
12. Borgs C (2006) Absence of zeros for the chromatic polynomial on bounded degree graphs. Combinator Probab Comput 15:63-74
13. Brown JI, Hickman CA, Sokal AD, Wagner DG (2001) On the chromatic roots of generalized theta graphs. J Combin Theory B 83:272-297
14. Brylawski T (1971) A combinatorial model for series-parallel networks. Trans Am Math Soc 154:1-22
15. Brylawski T (1972) A decomposition for combinatorial geometries. Trans Am Math Soc 171:235-282
16. Brylawski T (1982) The Tutte polynomial, Part 1: general theory. In: Barlotti A (ed) Matroid theory and its applications. Proceedings of the 3rd international mathematical summer center (C. I. M. E. 1980)
17. Brylawski T, Oxley J (1992) The Tutte polynomial and its applications. In: White N (ed) Matroid applications, encyclopedia of mathematics and its applications. Cambridge University Press, Cambridge
18. Chang SC, Shrock R (2001) Exact Potts model partition functions on wider arbitrary-length strips of the square lattice. Physica A 296:234-288
19. Chang SC, Jacobsen J, Salas J, Shrock R (2004) Exact Potts model partition functions for strips of the triangular lattice. J Stat Phys 114:768-823
20. Chari MK, Colbourn CJ (1997) Reliability polynomials: a survey. J Combin Inf System Sci 22:177-193
21. Chia GL (1997) A bibliography on chromatic polynomials. Discrete Math 172:175-191
22. Choe YB, Oxley JG, Sokal AD, Wagner DG (2004) Homogeneous multivariate polynomials with the half-plane property. Adv Appl Math 32:88-187
23. Crapo HH (1967) A higher invariant for matroids. J Combin Theory 2:406-417
24. Crapo HH (1969) The Tutte polynomial. Aeq Math 3:211-229
25. Dhar D (1990) Self-organized critical state of sandpile automaton models. Phys Rev Lett 64:1613-1616
26. Diestel R (2000) Graph theory. Graduate texts in mathematics. Springer, New York
27. Dong FM, Koh KM (2004) On upper bounds for real roots of chromatic polynomials. Discrete Math 282:95-101
28. Dong FM, Koh KM (2007) Bounds for the coefficients of flow polynomials. J Combin Theory B 97:413-420
29. Dong FM, Koh KM, Teo KL (2005) Chromatic polynomials and chromaticity of graphs. World Scientific, Hackensack, NJ
30. Duffin RJ (1965) Topology of series-parallel networks. J Math Anal Appl 10:303-318
31. Ellis-Monaghan J (2004) Exploring the Tutte-Martin connection. Discrete Math 281:173-187
32. Ellis-Monaghan J (2004) Identities for the circuit partition polynomials, with applications to the diagonal Tutte polynomial. Adv Appl Math 32:188-197
33. Etienne G, Las Vergnas M (1998) External and internal elements of a matroid basis. Discrete Math 179:111-119
34. Farr GE (2006) The complexity of counting colourings of subgraphs of the grid. Combinator Probab Comput 15:377-383
35. Farr GE (2007) Tutte-Whitney polynomials: some history and generalizations. In: Grimmett GR, McDiarmid CJH (eds) Combinatorics, complexity, and chance: a tribute to Dominic Welsh. Oxford University Press, Oxford
36. Fernandez R, Procacci A (2008) Regions without complex zeros for chromatic polynomials on graphs with bounded degree. Combinator Probab Comput 17:225-238
37. Gabrielov A (1993) Abelian avalanches and the Tutte polynomials. Physica A 195:253-274
38. Garey MR, Johnson DS (1979) Computers and intractability - a guide to the theory of NP-completeness. W. H. Freeman, San Francisco
39. Gioan E (2007) Enumerating degree sequences in digraphs and a cycle-cocycle reversing system. Eur J Combinator 28:1351-1366
40. Gioan E, Las Vergnas M (2005) Activity preserving bijections between spanning trees and orientations in graphs. Discrete Math 298:169-188
41. Gioan E, Las Vergnas M (2007) On the evaluation at.j; j 2/of the Tutte polynomial of a ternary matroid. J Algebr Combinator 25:1-6
42. Goldberg LA, Jerrum MR (2007) Inapproximability of the Tutte polynomial. In STOC'07: Proceedings of the 39th annual ACM symposium on theory of computing. ACM Press, New York
43. Green C, Zaslavsky T (1983) On the interpretation of whitney numbers through arrangements of hyperplanes, zonotopes, non-radon partitions and orientations of graphs. Trans Am Math Soc 280:97-126
44. Jackson B (1993) A zero-free interval for chromatic polynomials of graphs. Combinator Probab Comput 2:325-336
45. Jackson B (2003) Zeros of chromatic and flow polynomials of graphs. J Geom. 76:95-109
46. Jackson B (2007) Zero-free intervals for flow polynomials of near-cubic graphs. Combinator Probab Comput 16:85-108
47. Jaeger F (1976) On nowhere-zero flows in multigraphs. In: Nash-Williams Crispin St JA, Sheehan J (eds) Proceedings of the 5th British combinatorial conference, Winnipeg
48. Jaeger F (1988) Nowhere-zero flow problems. In: Beineke LW, Wilson RJ (eds) Selected topics in graph theory, vol 3. Academic, New York
49. Jaeger F, Vertigan DL, Welsh D J A (1990) On the computational complexity of the Jones and Tutte polynomials. Math Proc Cambridge Philos Soc 108:35-53
50. Jerrum MR, Sinclair A (1993) Polynomial time approximation algorithms for the Ising model. SIAM J Comput 22:1087-1116
51. Kasteleyn PW (1961) The statistics of dimers on a lattice. Physica 27:1209-1225
52. Kleitman DJ, Winston KJ (1981) Forests and score vectors. Combinatorica 1:49-54
53. Kook W, Reiner V, Stanton D (1999) A convolution formula for the Tutte polynomial. J Combin Theory B 76:297-300
54. Las Vergnas M (1977) Acyclic and totally cyclic orientations of combinatorial geometries. Discrete Math 20:51-61
55. Las Vergnas M (1984) The Tutte polynomial of a morphism of matroids II. Activities of orientations. In: Bondy JA, Murty USR (eds) Progress in graph theory, Proceedings of Waterloo Silver Jubilee Combinatorial Conference 1982. Academic, Toronto
56. Las Vergnas M (1988) On the evaluation at (3, 3) of the Tutte polynomial of a graph. J Combin Theory B 44:367-372
57. Las Vergnas M. The Tutte polynomial of a morphism of matroids V. Derivatives as generating functions, Preprint
58. Lieb EH (1967) Residual entropy of square ice. Phys Rev 162:162-172
59. Makowsky JA (2005) Colored Tutte polynomials and Kauffman brackets for graphs of bounded tree width. Discrete Appl Math 145:276-290
60. Makowsky JA, Rotics U, Averbouch I, Godlin B (2006) Computing graph polynomials on graphs of bounded clique-width. In: Lecture Notes in Computer Science 4271. Springer, New York
61. Martin P (1977) Enumérations eulériennes dans le multigraphs et invariants de Tutte-Gröthendieck. PhD thesis, Grenoble
62. Martin P (1978) Remarkable valuation of the dichromatic polynomial of planar multigraphs. J Combin Theory B 24:318-324
63. McKee TA (2001) Recognizing dual-chordal graphs. Congr Numer 150:97-103
64. Merino C (1997) Chip firing and the Tutte polynomial. Ann Combin 1:253-259
65. Merino C (2001) The chip firing game and matroid complex. Discrete Mathematics and Theoretical Computer Science, Proceedings, vol AA, pp 245-256
66. Merino C, De Mier A, Noy M (2001) Irreducibility of the Tutte polynomial of a connected matroid. J Combin Theory B 83:298-304
67. Noble SD (1998) Evaluating the Tutte polynomial for graphs of bounded tree-width. Combinator Probab Comput 7:307-321
68. Noble SD (2007) The complexity of graph polynomials. In: Grimmett GR, McDiarmid CJH (eds) Combinatorics, complexity, and chance: a tribute to Dominic Welsh. Oxford University Press, Oxford
69. Oum S, Seymour PD (2006) Approximating clique-width and branch-width. J Combin Theory B 96:514-528
70. Oxley J (1982) On Crapo's beta invariant for matroids. Stud Appl Math 66:267-277
71. Oxley J, Welsh D J A (1979) The Tutte polynomial and percolation. In: Bondy JA, Murty USR (eds) Graph theory and related topics. Academic, London
72. Pauling L (1935) The structure and entropy of ice and of other crystals with some randomness of atomic arrangement. J Am Chem Soc 57:2680-2684
73. Procacci A, Scoppola B, Gerasimov V (2003) Potts model on infinite graphs and the limit of chromatic polynomials. Commun Math Phys 235:215-231
74. Provan JS, Billera LJ (1980) Decompositions of simplicial complexes related to diameters of convex polyhedra. Math Oper Res 5:576-594
75. Read RC (1968) An introduction to chromatic polynomials. J Combin Theory B 4:52-71
76. Read RC, Rosenstiehl P (1978) On the principal edge tripartition of a graph. Ann Discrete Math 3:195-226
77. Read RC, Royle G (1991) Chromatic roots of families of graphs. In: Alavi Y et al (eds) Graph theory, combinatorics, and applications. Wiley, New York
78. Robertson N, Seymour PD (1984) Graph minors. I. Excluding a forest. J Combin Theory B 35:39-61
79. Robertson N, Seymour PD (1984) Graph minors. III. Planar tree-width. J Combin Theory B 36:49-64
80. Robertson N, Seymour PD (1986) Graph minors. II. Algorithmic aspects of tree-width. J Algorithm 7:309-322
81. Royle G (2007) Graphs with chromatic roots in the interval (1, 2). Electron J Combinator 14(1):N18
82. Royle G (2008) Planar triangulations with real chromatic roots arbitrarily close to four. Ann Combinator 12:195-210
83. Schwärzler W (1993) The coefficients of the Tutte polynomial are not unimodal. J Combin Theory B 58:240-242
84. Sekine K, Imai H, Tani S (1995) Computing the Tutte polynomial of a graph of moderate size. In: Lecture Notes in Computer Science. Springer, Berlin
85. Seymour PD (1981) Nowhere-zero 6-flows. J Combin Theory B, 30, 130-135
86. Seymour PD, Welsh D J A (1975) Combinatorial applications of an inequality of statistical mechanics. Math Proc Cambridge Philos Soc 77:485-495
87. Shrock R, Tsai SH (1997) Asymptotic limits and zeros of chromatic polynomials and ground state entropy of Potts antiferromagnets. Phys Rev E 55:5165-5179
88. Shrock R, Tsai SH (1997) Families of graphs with Wr. G; q/functions that are nonanalytic at 1=q D 0. Phys Rev E 56:3935-3943
89. Shrock R, Tsai SH (1998) Ground state entropy of Potts antiferromagnets: cases with noncompact W boundaries having multiple points at 1=q D 0. Physica A 259:315-348
90. Shrock R (2001) Chromatic polynomials and their zeros and asymptotic limits for families of graphs. Discrete Math 231:421-446
91. Sokal AD (2001) A personal list of unsolved problems concerning lattice gases and antiferromagnetic Potts models. Markov Process Relat Fields 7:21-38
92. Sokal AD (2001) Bounds on the complex zeros of (di) chromatic polynomials and Potts-model partition functions. Combinator Probab Comput 10:41-77
93. Sokal AD (2004) Chromatic roots are dense in the whole complex plane. Combinator Probab Comput 13:221-261
94. Stanley R (1973) Acyclic orientations of graphs. Discrete Math 5:171-178
95. Stanley R (1980) Decomposition of rational polytopes. Ann Discrete Math 6:333-342
96. Stanley R (1996) Combinatorics and commutative algebra, 2nd edn. Progress in Mathematics, vol 41. Birkhäuser, Boston
97. Stanley R (1996) Enumerative combinatorics, vol 1. Cambridge University Press, Cambridge
98. Stanley R (1999) Enumerative combinatorics, vol 2. Cambridge University Press, Cambridge
99. Thomassen C (1997) The zero-free intervals for chromatic polynomials of graphs. Combinator Probab Comput 6:497-506
100. Traldi L (2006) On the colored Tutte polynomial of a graph of bounded treewidth. Discrete Appl Math 154:1032-1036
101. Tutte WT (1947) A ring in graph theory. Proc Cambridge Philos Soc 43:26-40
102. Tutte WT (1948) An algebraic theory of graphs, PhD thesis, University of Cambridge
103. Tutte WT (1954) A contribution to the theory of chromatic polynomials. Can J Math 6:80-91
104. Tutte WT (1967) On dichromatic polynomials. J Combin Theory 2:301-320
105. Tutte WT (1984) Graph theory. Cambridge University Press, Cambridge
106. Tutte WT (2004) Graph-polynomials. Special issue on the Tutte polynomial. Adv Appl Math 32:5-9
107. Vertigan D (1998) Bicycle dimension and special points of the Tutte polynomial. J Combin Theory B 74:378-396
108. Vertigan DL, Welsh D J A (1992) The computational complexity of the Tutte plane: the bipartite case. Combinator Probab Comput 1:181-187
109. Welsh D J A (1993) Complexity: knots, colorings and counting. Cambridge University Press, Cambridge
110. Welsh D J A (1999) The Tutte polynomial, in statistical physics methods in discrete probability, combinatorics, and theoretical computer science. Random Struct Algorithm 15:210-228
111. Welsh D J A, Merino C (2000) The Potts model and the Tutte polynomial. J Math Phys 41:1127-1152
112. Whitney H (1932) A logical expansion in mathematics. Bull Am Math Soc 38:572-579
113. Woodall D (1992) A zero-free interval for chromatic polynomials. Discrete Math 101:333-341
114. Yetter D (1990) On graph invariants given by linear recurrence relations. J Combin Theory B 48:6-18
115. Zaslavsky T (1987) The Möbius function and the characteristic polynomial. In: White N (ed) Combinatorial geometries, encyclopedia of mathematics and its applications. Cambridge University Press, Cambridge
116. Zhang CQ (1997) Integer flows and cycle covers of graphs. Marcel Dekker Inc., New York.