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Polynomials related to chromatic polynomials

Profile image of Fengming DongFengming Dong

2020, arXiv (Cornell University)

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On the chromatic polynomial of a graph
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Let P(G, λ) be the chromatic polynomial of a graph G with n vertices, independence number α and clique number ω. We show that for every λ ≥ n,

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A NOTE ON THE CONDITIONAL CHROMATIC POLYNOMIAL
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In this note we consider a finite graph without loops and multiple edges. The colouring of a graph G in A colours is the colouring of its vertices in such a way that no two of adjacent vertices have the same colours and the number of used colours does not exceed A [1,4]. Two colourings of graph G are called different if there exists at least one vertex which changes colour when passing from one colouring to another. If F n is a full (also known as 'complete') graph on n vertices, & < n , then P(F k c F H , pi, A) = M (*>(A-*)<""*>, where M <*> = M (M-l)(/i-2). .. (/ *-* + 1). In particular P(F t <= F,,, A, A) = A W (A-*)<"-*> = A (n). If £ " is empty (also known as 'null' or 'totally disconnected' graph on n vertices) then P(E k a E, n fi, A) = fi k \"-k , in particular P(E k c £ " , A, A) = A". Let G = (X,V), \X\ = n be a graph and G o = (* " , K o), X 0 ^X 0 \X 0 \ = m be an induced subgraph of G. Let also x,y <= Xbe two non-adjacent vertices in G. We construct the graph G, from G by joining x and _y by an edge and the graph G 2 , obtained from G by contracting x and y into single vertex. Then we can observe that the following equality is true: P(G 0 ^G, ti, A) = P(G l 0 ^G u fi,\) + P(Gl<=G 2 , /i,A), (1) where Gi = Go = G o if x,_y ^ A'o, and G o , Go are the subgraphs induced respectively by X ih X 0 U{x,y} otherwise. It is so because P (G 0 c G 2 , M , A) equals the number of colourings for which x and y have the same colour. If we perform this operation further as much as possible we obtain P(G 0 = G, n, A) = X P{F kl c Fn,, M, A). /=i Since P(F k <= F n ,fj.,A) = P(F' k <^F n ,p,A) for any two fc-vertex complete subgraphs F A ., F' k of F,,, we can write P(G 0 cz G, n, A) = 5 t o, y P(F y ^ F h M , A). In order to clear up the meaning of coefficients a, y we consider one more class of colourings. A colouring of a graph is called an /-colouring, if exactly / colours are used, but Glasgow Math. J. 36 (1994) 265-267.

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New expressions for order polynomials and chromatic polynomials
Fengming Dong

Journal of Graph Theory

Let G = (V, E) be a simple graph with V = {1, 2, • • • , n} and χ(G, x) be its chromatic polynomial. For an ordering π = (v 1 , v 2 , • • • , v n) of elements of V , let δ G (π) be the number of i's, where 1 ≤ i ≤ n − 1, with either v i < v i+1 or v i v i+1 ∈ E. Let W(G) be the set of subsets {a, b, c} of V , where a < b < c, which induces a subgraph with ac as its only edge. We show that W(G) = ∅ if and only if (−1) n χ(G, −x) = π x+δG(π) n , where the sum runs over all n! orderings π of V. To prove this result, we establish an analogous result on order polynomials of posets and apply Stanley's work on the relation between chromatic polynomials and order polynomials.

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Connection between Graphs' Chromatic and Ehrhart Polynomials
Dr. Shatha A.Salman

Journal of applied sciences and nanotechnology, 2023

Graph Theory is a discipline of mathematics with numerous outstanding issues and applications in various sectors of mathematics and science. The chromatic polynomial is a type of polynomial that has valuable and attractive qualities. Ehrhart's polynomials and chromatic analysis are two essential techniques for graph analysis. They both provide insight into the graph's structure but in different ways. The relationship between chromatic and Ehrhart polynomials is an area of active research that has implications for graph theory, combinatorial, and other fields. By understanding the relationship between these two polynomials, one can better understand the structure of graphs and how they interact. This can help us to solve complex problems in our lives more efficiently and effectively. This work gives the relationship between these two essential polynomials and the proof of theorems, and an application related to these works, the model Physical Cell ID (PCID), was discussed.

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More connections between the matching polynomial and the chromatic polynomial
Marcelino Ramírez

AKCE International Journal of Graphs and Combinatorics, 2019

The connection between the matching polynomial and the chromatic polynomial for triangle-free graphs was revealed in the work of Farrell and Whitehead. We extend this result to all graph by mirroring the corresponding result of Godsil and Gutman for the acyclic polynomial and the characteristic polynomial. We also reintroduce the clique polynomial of Farrell as an evaluation of the U-polynomial of Noble and Welsh, which also happens to contain the matching and chromatic polynomials. c

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Chromatic polynomials of some families of graphs I: theorems and conjectures
Norman Biggs

2005

The chromatic polynomials of 'bracelets' can be studied by means of a theory based on representations of the symmetric group. This paper contains a detailed study of the theory as it relates to one type of bracelet. The underlying theory is presented rather more clearly than hitherto, and some surprising features are exhibited. The methods involve an extension of the standard theory of distance-regular graphs, and they lead to several plausible conjectures.

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Chromatic polynomials and chromaticity of graphs
Fengming Dong

2005

Preface For a century, one of the most famous problems in mathematics was to prove the four-colour theorem. This has spawned the development of many use-ful tools for solving graph colouring problems. In a paper in 1912, Birkhoff proposed a way of tackling the ...

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Some Polynomials of Flower Graphs
eunice Mphako-banda

2007

We define a class of graphs called flower and give some properties of these graphs. Then the explicit expressions of the chromatic polynomial and the flow polynomial is given. Further, we give classes of graphs with the same chromatic and flow polynomials.

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Edge colouring models for the Tutte polynomial and related graph invariants
Andrew Goodall

For integer q&gt;1, we derive edge q-colouring models for (i) the Tutte polynomial of a graph G on the hyperbola H_q, (ii) the symmetric weight enumerator of the set of group-valued q-flows of G, and (iii) a more general vertex colouring model partition function that includes these polynomials and the principal specialization order q of Stanley&#39;s symmetric monochrome polynomial. In the second half of the paper we exhibit a family of non-symmetric edge q-colouring models defined on k-regular graphs, whose partition functions for q &gt;= k each evaluate the number of proper edge k-colourings of G when G is Pfaffian.

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Note on chromatic polynomials of the threshold graphs
Miloud Mihoubi

Electronic Journal of Graph Theory and Applications, 2019

Let G be a threshold graph. In this paper, we give, in first hand, a formula relating the chromatic polynomial of G (the complement of G) to the chromatic polynomial of G. In second hand, we express the chromatic polynomials of G and G in terms of the generalized Bell polynomials.

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References (63)

  1. F. Brenti, Expansions of chromatic polynomials and log-concavity, Trans. Amer. Math. Soc. 332 (1992), 729-756.
  2. F. Brenti, G Royle and D. Wagner, location of zeros of chromatic and related polynomials of graphs, Canad. J. Math. 46 (1994), 55-80.
  3. T. Brylawski and J. Oxley, The Tutte Polynomial and its Applications. In: White, N. (ed) Matroid Applications, Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1992.
  4. R. D. Cameron, C. J. Colbourn, R. C. Read and N. C. Wormald, Cat- aloguing the graphs on 10 vertices, J. Graph Theory 9 (1985), 551-562.
  5. L. E. Chávez-Lomelí, C. Merino, S. D. Noble and M. Ramírez-Ibáñez, Some inequalities for the Tutte polynomial, European J. of Combin. 32 (2011), 422-433.
  6. Fengming Dong, On graphs having no flow zeros in (1,2), Electron. J. Combin. 22 (2015), Paper #P1.82.
  7. Fengming Dong, On zero-free intervals of flow polynomials, J. Combin. Theory Ser. B 111 (2015) 181-200.
  8. Fengming Dong, On graphs whose flow polynomials have real roots only, Electron. J. Combin. 25(3) (2018), #P3.26.
  9. Fengming Dong, A survey on real zeros of flow polynomials, J. Graph Theory 92 (2019), 361-376.
  10. Fengming Dong, New expressions for order polynomials and chromatic polynomials, J. Graph Theory 94 (2020), 30-58.
  11. F.M. Dong, K.L. Teo, C.H.C. Little and M.D. Hendy, Zeros of adjoint polynomials of paths and cycles, Australasian J. Combin. 25 (2002) 167-174.
  12. G. Farr, A correlation inequality involving stable set and chromatic poly- nomials, J. Combin. Theory Ser. B 58 (1993), 14-21.
  13. R. Fernández and A. Procacci, Cluster expansion for abstract polymer models: New bounds from an old approach, Comm. Math. Phys. 274 (2007), 123-140.
  14. A. J. Goodall1, A. Mier and S. D. Noble, The Tutte polynomial char- acterizes simple outerplanar graphs, Electronic Note on Discrete Math. 38 (2011), 639-644.
  15. A. J. Goodall, C. Merino, A. Mier, and M. Noy, On the evaluation of the Tutte polynomial at the points (1, -1) and (2, -1), Ann. Combin. 12 (2009), 479-492.
  16. I. Gutman and F. Harary, Generalizations of the matching polynomial, Utilitas Mathematica 24 (1983), 97-106.
  17. G. Haggard, D.J. Pearce and G.F. Royle, Computing Tutte polynomials, ACM Trans. Math. Software 37 (2010), article 24.
  18. T. Helgason. Aspects of the theory of hypermatroids, pages 191-213. Lec- ture Notes in Mathematics 411. Springer, Berlin, 1974.
  19. B. Jackson, An inequality for Tutte polynomials, Combinatorica 30 (2010), 69-81. http://dx.doi.org/10.1007/s00493-0102484-4.
  20. B. Jackson, A zero-free interval for flow polynomials of near-cubic graphs, Combin. Probab. Comput. 16 (2007) 85-108.
  21. B. Jackson, Zeros of chromatic and flow polynomials of graphs, J. Geom. 76 (2003), 95-109.
  22. B. Jackson and A. Sokal, Zero-free regions for multivariate Tutte poly- nomials (alias Potts-model partition functions) of graphs and matroids, J. Combin. Theory Ser. B 99 (2009), 869-903.
  23. J. L. Jacobsen and J. Salas, Is the five-flow conjecture almost false? J. Combin. Theory Ser. B 103 (2013), 532-565.
  24. F. Jaeger, Nowhere-zero flow problems, in: L.W. Beineke, R.J. Wilson (Eds.), Selected Topics in Graph Theory, vol. 3, Academic Press, 1988, 71-95.
  25. W. Kook, V. Reiner, D. Stanton, A Convolution Formula for the Tutte Polynomial, J. Combin. Theory Ser. B 76 (1999), 297-300.
  26. J.P.S. Kung, a multiplicity identity for characteristic polynomial of a matroid, Advance in Applied Math. 32 (2004), 319-326.
  27. J.P.S. Kung and G. Royle, Graphs whose flow polynomials have only integral roots, European J. of Combin. 32 (2011), 831-840.
  28. R.Y. Liu, A new method to find chromatic polynomial of graphs and its applications, Kexue Tongbao 32 (1987), 1508-1509 (In Chinese, English summary).
  29. P. Martin, Remarkable valuation of the dichromatic polynomial of planar multigraphs, J. Combin. Theory Ser. B 24 (1978), 318-324.
  30. P. Martin, Enumérations eulériennes dans le multigraphs et invariants de Tutte-Gröthendieck. PhD Thesis, Grenoble, 1977.
  31. C. Merino, The number of 0-1-2 increasing trees as two different evalua- tions of the Tutte polynomial of a complete graph, Electron. J. Combin. 15 (2008) # N28.
  32. C. Merino, M. Ibañez and M. G. Rodríguez, A note on some inequalities for the Tutte polynomial of a matroid, Electronic Note on Discrete Math. 34 (2009), 603-607.
  33. C. Merino and D.J.A. Welsh, Forests, colourings and acyclic orientations of the square lattice, Ann. Combin. 3 (1999), 417-429.
  34. S. D. Noble and G. F. Royle, The Merino-Welsh conjecture holds for series-parallel graphs, European J. of Combin. 38 (2014) 24-35.
  35. J.G. Oxley, Colouring, packing and the critical problem, Quart. J. Math. Oxford 29 (1978), 11-22.
  36. R.B. Potts, Some generalized order-disorder transformations, Proc. Cambridge Philos. Soc. 48 (1952), 106-109.
  37. R.C. Read and P. Rosenstiehl, On the principal edge tripartition of a graph, Ann. Discrete Math. 3 (1978), 195-226.
  38. G. C. Rota, On the foundations of combinatorial theory I. Theory of Möbius functions, Zeitschrift für Wahrscheinlichkeitstheorie und ver- wandte Gebiete 2 (1964), 340-368.
  39. A. D. Sokal, The multivariate Tutte polynomial (alias Potts model) for graphs and matroids, Surveys in combinatorics 2005 (B. Webb, ed.), London Math. Soc. Lecture Note Ser., vol. 327, Cambridge University Press, 2005.
  40. A.D. Sokal, Chromatic roots are dense in the whole complex plane, Combin. Probab. Comput. 13 (2004), 221-261.
  41. A. D. Sokal, Bounds on the Complex Zeros of (Di)Chromatic Polynomi- als and Potts-Model Partition Functions, Combin. Probab. Comput. 10 (2001), 41-77.
  42. Polynomials related to chromatic polynomials (Sect. 9)
  43. R. Stanley, Acyclic orientations of graphs, Discrete Math. 306 (2006), 905-909. (A new version of [43])
  44. R. Stanley, Acyclic orientations of graphs, Discrete Math. 5 (1973), 171- 178.
  45. R. Stanley, A Brylawski decomposition for finite ordered sets, Discrete Math. 4 (1973), 77-82.
  46. R. Stanley, A chromatic-like polynomial for ordered sets, in: Proc. sec- ond Chapel Hill conference on combinatorial mathematics and its appli- cations (1970), 421-427.
  47. C. Thomassen, The weak 3-flow conjecture and the weak circular flow conjecture, J. Combin. Theory Ser. B 102 (2012), 521-529.
  48. C. Thomassen, Spanning trees and orientations of graphs, J. Combin. 1 (2010), 101-111.
  49. W.T. Tutte, Graph Theory, Addison-Welsey, Reading, Mass., 1984.
  50. W. T. Tutte, Codichromatic Graphs, J. Combin. Theory Ser. B 16 (1974), 168-174.
  51. W.T. Tutte, On dichromatic polynomials, J. Combin. Theory 2 (1967), 301-320.
  52. W.T. Tutte, On the Algebraic Theory of Graph Colorings, J. Combin. Theory 1 (1966), 15-50.
  53. W.T. Tutte, A contribution to the theory of chromatic polynomials, Canad. J. Math. 6 (1954), 80-91.
  54. W.T. Tutte, On the imbedding of linear graphs in surfaces, Proc. Lond. Math. Soc. 51 (1950), 474-483.
  55. W.T. Tutte, A ring in graph theory, Proc. Cambridge Phil. Soc. 43 (1947), 26-40.
  56. M. L. Vergnas, On the evaluation at (3,3) of the Tutte polynomial of a graph, J. Combin. Theory Ser. B 44 (1988), 367-372.
  57. A. Vinué, Graphs and matroids determined by their Tutte polynomi- als, Ph.D. Thesis, Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, 2003.
  58. D. Wagner, The partition polynomial of a finite set system, J. Combin. Theory Ser. A 56 (1991), 138-159.
  59. C. D. Wakelin, Chromatic Polynomials, Ph.D. Thesis, University of Not- tingham, 1994.
  60. D.J.A. Welsh, http://garden.irmacs.sfu.ca/?q=category/welsh, also cited in Ref. ([17], Conjecture 1)and in Ref. ([21], Conjecture 33).
  61. H. Whitney, 2-isomorphic graphs, American J. of Mathematics 55 (1933), 245-254.
  62. F.Y. Wu, Potts model of magnetism (invited), J. Appl. Phys. 55 (1984), 2421-2425.
  63. F.Y. Wu, The Potts model, Rev. Mod. Phys. 54 (1982), 235-268. Erra- tum 55 (1983) 315.

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