Abstract
In this survey of graph polynomials, we emphasize the Tutte polynomial and a selection of closely related graph polynomials such as the chromatic, flow, reliability, and shelling polynomials. We explore some of the Tutte polynomial’s many properties and applications and we use the Tutte polynomial to showcase a variety of principles and techniques for graph polynomials in general. These include several ways in which a graph polynomial may be defined and methods for extracting combinatorial information and algebraic properties from a graph polynomial. We also use the Tutte polynomial to demonstrate how graph polynomials may be both specialized and generalized, and how they can encode information relevant to physical applications. We conclude with a brief discussion of computational complexity considerations.
MSC2000: Primary 05-02; Secondary 05C15, 05A15, 05C99
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References
Aigner M, Ziegler GM (2001) Proofs from the book. Springer, Berlin, Heidelberg
Alon N, Frieze AM, Welsh DJA (1995) Polynomial time randomized approximation schemes for Tutte–Gröthendieck invariants: the dense case. Random Struct Algorithm 6:459–478
Andrzejak A (1998) An algorithm for the Tutte polynomials of graphs of bounded treewidth. Discrete Math 190:39–54
Bak P, Tang C, Wiesenfeld K (1988) Self-organized criticality. Phys Rev A 38:364–374
Benashski J, Martin R, Moore J, Traldi L (1995) On the β-invariant for graphs. Congr Numer 109:211–221
Biggs N (1996) Algebraic graph theory, 2nd edn. Cambridge University Press, Cambridge
Biggs N (1996) Chip firing and the critical group of a graph. Research report, London school of economics, London
Birkhoff GD (1912) A determinant formula for the number of ways of coloring a map. Ann Math 14:42–46
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
Bodlaender HL (1993) A tourist guide through treewidth. Acta Cybernet 11:1–21
Bollobás B (1998) Modern graph theory. Graduate texts in mathematics. Springer, New York
Borgs C (2006) Absence of zeros for the chromatic polynomial on bounded degree graphs. Combinator Probab Comput 15:63–74
Brown JI, Hickman CA, Sokal AD, Wagner DG (2001) On the chromatic roots of generalized theta graphs. J Combin Theory B 83:272–297
Brylawski T (1971) A combinatorial model for series–parallel networks. Trans Am Math Soc 154:1–22
Brylawski T (1972) A decomposition for combinatorial geometries. Trans Am Math Soc 171:235–282
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)
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
Chang SC, Shrock R (2001) Exact Potts model partition functions on wider arbitrary-length strips of the square lattice. Physica A 296:234–288
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
Chari MK, Colbourn CJ (1997) Reliability polynomials: a survey. J Combin Inf System Sci 22:177–193
Chia GL (1997) A bibliography on chromatic polynomials. Discrete Math 172:175–191
Choe YB, Oxley JG, Sokal AD, Wagner DG (2004) Homogeneous multivariate polynomials with the half-plane property. Adv Appl Math 32:88–187
Crapo HH (1967) A higher invariant for matroids. J Combin Theory 2:406–417
Crapo HH (1969) The Tutte polynomial. Aeq Math 3:211–229
Dhar D (1990) Self-organized critical state of sandpile automaton models. Phys Rev Lett 64:1613–1616
Diestel R (2000) Graph theory. Graduate texts in mathematics. Springer, New York
Dong FM (2004) The largest non-integer zero of chromatic polynomials of graphs with fixed order. Discrete Math 282:103–112
Dong FM, Koh KM (2004) On upper bounds for real roots of chromatic polynomials. Discrete Math 282:95–101
Dong FM, Koh KM (2007) Bounds for the coefficients of flow polynomials. J Combin Theory B 97:413–420
Dong FM, Koh KM, Teo KL (2005) Chromatic polynomials and chromaticity of graphs. World Scientific, Hackensack, NJ
Duffin RJ (1965) Topology of series–parallel networks. J Math Anal Appl 10:303–318
Ellis-Monaghan J (2004) Exploring the Tutte–Martin connection. Discrete Math 281:173–187
Ellis-Monaghan J (2004) Identities for the circuit partition polynomials, with applications to the diagonal Tutte polynomial. Adv Appl Math 32:188–197
Etienne G, Las Vergnas M (1998) External and internal elements of a matroid basis. Discrete Math 179:111–119
Farr GE (2006) The complexity of counting colourings of subgraphs of the grid. Combinator Probab Comput 15:377–383
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
Fernandez R, Procacci A (2008) Regions without complex zeros for chromatic polynomials on graphs with bounded degree. Combinator Probab Comput 17:225–238
Gabrielov A (1993) Abelian avalanches and the Tutte polynomials. Physica A 195:253–274
Garey MR, Johnson DS (1979) Computers and intractability – a guide to the theory of NP-completeness. W.H. Freeman, San Francisco
Gioan E (2007) Enumerating degree sequences in digraphs and a cycle–cocycle reversing system. Eur J Combinator 28:1351–1366
Gioan E, Las Vergnas M (2005) Activity preserving bijections between spanning trees and orientations in graphs. Discrete Math 298:169–188
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
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
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
Jackson B (1993) A zero-free interval for chromatic polynomials of graphs. Combinator Probab Comput 2:325–336
Jackson B (2003) Zeros of chromatic and flow polynomials of graphs. J Geom. 76:95–109
Jackson B (2007) Zero-free intervals for flow polynomials of near-cubic graphs. Combinator Probab Comput 16:85–108
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
Jaeger F (1988) Nowhere-zero flow problems. In: Beineke LW, Wilson RJ (eds) Selected topics in graph theory, vol 3. Academic, New York
Jaeger F, Vertigan DL, Welsh DJA (1990) On the computational complexity of the Jones and Tutte polynomials. Math Proc Cambridge Philos Soc 108:35–53
Jerrum MR, Sinclair A (1993) Polynomial time approximation algorithms for the Ising model. SIAM J Comput 22:1087–1116
Kasteleyn PW (1961) The statistics of dimers on a lattice. Physica 27:1209–1225
Kleitman DJ, Winston KJ (1981) Forests and score vectors. Combinatorica 1:49–54
Kook W, Reiner V, Stanton D (1999) A convolution formula for the Tutte polynomial. J Combin Theory B 76:297–300
Las Vergnas M (1977) Acyclic and totally cyclic orientations of combinatorial geometries. Discrete Math 20:51–61
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
Las Vergnas M (1988) On the evaluation at (3,3) of the Tutte polynomial of a graph. J Combin Theory B 44:367–372
Las Vergnas M. The Tutte polynomial of a morphism of matroids V. Derivatives as generating functions, Preprint
Lieb EH (1967) Residual entropy of square ice. Phys Rev 162:162–172
Makowsky JA (2005) Colored Tutte polynomials and Kauffman brackets for graphs of bounded tree width. Discrete Appl Math 145:276–290
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
Martin P (1977) Enumérations eulériennes dans le multigraphs et invariants de Tutte–Gröthendieck. PhD thesis, Grenoble
Martin P (1978) Remarkable valuation of the dichromatic polynomial of planar multigraphs. J Combin Theory B 24:318–324
McKee TA (2001) Recognizing dual-chordal graphs. Congr Numer 150:97–103
Merino C (1997) Chip firing and the Tutte polynomial. Ann Combin 1:253–259
Merino C (2001) The chip firing game and matroid complex. Discrete Mathematics and Theoretical Computer Science, Proceedings, vol AA, pp 245–256
Merino C, de Mier A, Noy M (2001) Irreducibility of the Tutte polynomial of a connected matroid. J Combin Theory B 83:298–304
Noble SD (1998) Evaluating the Tutte polynomial for graphs of bounded tree-width. Combinator Probab Comput 7:307–321
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
Oum S, Seymour PD (2006) Approximating clique-width and branch-width. J Combin Theory B 96:514–528
Oxley J (1982) On Crapo’s beta invariant for matroids. Stud Appl Math 66:267–277
Oxley J, Welsh DJA (1979) The Tutte polynomial and percolation. In: Bondy JA, Murty USR (eds) Graph theory and related topics. Academic, London
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
Procacci A, Scoppola B, Gerasimov V (2003) Potts model on infinite graphs and the limit of chromatic polynomials. Commun Math Phys 235:215–231
Provan JS, Billera LJ (1980) Decompositions of simplicial complexes related to diameters of convex polyhedra. Math Oper Res 5:576–594
Read RC (1968) An introduction to chromatic polynomials. J Combin Theory B 4:52–71
Read RC, Rosenstiehl P (1978) On the principal edge tripartition of a graph. Ann Discrete Math 3:195–226
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
Robertson N, Seymour PD (1984) Graph minors. I. Excluding a forest. J Combin Theory B 35:39–61
Robertson N, Seymour PD (1984) Graph minors. III. Planar tree-width. J Combin Theory B 36:49–64
Robertson N, Seymour PD (1986) Graph minors. II. Algorithmic aspects of tree-width. J Algorithm 7:309–322
Royle G (2007) Graphs with chromatic roots in the interval (1,2). Electron J Combinator 14(1):N18
Royle G (2008) Planar triangulations with real chromatic roots arbitrarily close to four. Ann Combinator 12:195–210
Schwärzler W (1993) The coefficients of the Tutte polynomial are not unimodal. J Combin Theory B 58:240–242
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
Seymour PD (1981) Nowhere-zero 6-flows. J Combin Theory B, 30, 130–135
Seymour PD, Welsh DJA (1975) Combinatorial applications of an inequality of statistical mechanics. Math Proc Cambridge Philos Soc 77:485–495
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
Shrock R, Tsai SH (1997) Families of graphs with W r (G, q) functions that are nonanalytic at \(1/q = 0\). Phys Rev E 56:3935–3943
Shrock R, Tsai SH (1998) Ground state entropy of Potts antiferromagnets: cases with noncompact W boundaries having multiple points at \(1/q = 0\). Physica A 259:315–348
Shrock R (2001) Chromatic polynomials and their zeros and asymptotic limits for families of graphs. Discrete Math 231:421–446
Sokal AD (2001) A personal list of unsolved problems concerning lattice gases and antiferromagnetic Potts models. Markov Process Relat Fields 7:21–38
Sokal AD (2001) Bounds on the complex zeros of (di)chromatic polynomials and Potts-model partition functions. Combinator Probab Comput 10:41–77
Sokal AD (2004) Chromatic roots are dense in the whole complex plane. Combinator Probab Comput 13:221–261
Stanley R (1973) Acyclic orientations of graphs. Discrete Math 5:171–178
Stanley R (1980) Decomposition of rational polytopes. Ann Discrete Math 6:333–342
Stanley R (1996) Combinatorics and commutative algebra, 2nd edn. Progress in Mathematics, vol 41. Birkhäuser, Boston Besel Stuttgart
Stanley R (1996) Enumerative combinatorics, vol 1. Cambridge University Press, Cambridge
Stanley R (1999) Enumerative combinatorics, vol 2. Cambridge University Press, Cambridge
Thomassen C (1997) The zero-free intervals for chromatic polynomials of graphs. Combinator Probab Comput 6:497–506
Traldi L (2006) On the colored Tutte polynomial of a graph of bounded treewidth. Discrete Appl Math 154:1032–1036
Tutte WT (1947) A ring in graph theory. Proc Cambridge Philos Soc 43:26–40
Tutte WT (1948) An algebraic theory of graphs, PhD thesis, University of Cambridge
Tutte WT (1954) A contribution to the theory of chromatic polynomials. Can J Math 6:80–91
Tutte WT (1967) On dichromatic polynomials. J Combin Theory 2:301–320
Tutte WT (1984) Graph theory. Cambridge University Press, Cambridge
Tutte WT (2004) Graph-polynomials. Special issue on the Tutte polynomial. Adv Appl Math 32:5–9
Vertigan D (1998) Bicycle dimension and special points of the Tutte polynomial. J Combin Theory B 74:378–396
Vertigan DL, Welsh DJA (1992) The computational complexity of the Tutte plane: the bipartite case. Combinator Probab Comput 1:181–187
Welsh DJA (1993) Complexity: knots, colorings and counting. Cambridge University Press, Cambridge
Welsh DJA (1999) The Tutte polynomial, in statistical physics methods in discrete probability, combinatorics, and theoretical computer science. Random Struct Algorithm 15:210–228
Welsh DJA, Merino C (2000) The Potts model and the Tutte polynomial. J Math Phys 41:1127–1152
Whitney H (1932) A logical expansion in mathematics. Bull Am Math Soc 38:572–579
Woodall D (1992) A zero-free interval for chromatic polynomials. Discrete Math 101: 333–341
Yetter D (1990) On graph invariants given by linear recurrence relations. J Combin Theory B 48:6–18
Zaslavsky T (1975) Facing up to arrangements: face-count formulas for partitions of space by hyperplanes. Mem Am Math Soc 154. American Mathematical Society, Providence, RI
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
Zhang CQ (1997) Integer flows and cycle covers of graphs. Marcel Dekker Inc., New York
Acknowledgments
We thank all the friends and colleagues who offered many helpful comments and suggestions during the writing of this chapter.
The first author was supported by the National Security Agency and by the Vermont Genetics Network through Grant Number P20 RR16462 from the INBRE Program of the National Center for Research Resources (NCRR), a component of the National Institutes of Health (NIH).
The second author was supported by CONACYT of Mexico, Grant 83977.
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Ellis-Monaghan, J.A., Merino, C. (2011). Graph Polynomials and Their Applications I: The Tutte Polynomial. In: Dehmer, M. (eds) Structural Analysis of Complex Networks. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4789-6_9
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