Graphs determined by polynomial invariants

Theoretical Computer Science

Volumn 307, Issue 2, 2003, Pages 365-384

  Noy, Marc ^a  

^a UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

Author keywords

Chromatic polynomial; Graph invariant; Matroid; Tutte polynomial

Indexed keywords

ENCODING (SYMBOLS); GRAPH THEORY; THEOREM PROVING;

GRAPH INVARIANTS; TUTTE POLYNOMIALS;

POLYNOMIALS;

EID: 0141525079     PISSN: 03043975     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-3975(03)00225-1     Document Type: Conference Paper

References (51)

1. Aigner M. The Penrose polynomial of graphs and matroids. Hirschfeld J.W.P. Surveys in Combinatorics, 2001. 2001;Cambridge University Press, Cambridge.
2. Asratian A.S., Denley T.M.J., Häggkvist R. Bipartite Graphs and their Applications. 1998;Cambridge University Press, Cambridge.
3. Beezer R.A., Farrell E.J. The matching polynomial of a regular graph. Discrete Math. 137:1995;7-18.
4. Biggs N.L. Algebraic Graph Theory. 2nd ed. 1993;Cambridge University Press, Cambridge.
5. Birkhoff G.D. A determinant formula for the number of ways of coloring a map. Ann. of Math. 14:1912;42-46.
6. Bollobás B. Modern Graph Theory. 1999;Springer, Berlin.
7. Bollobás B., Pebody L., Riordan O. Contraction-deletion invariants for graphs. J. Combin. Theory Ser. B. 80:2000;320-345.
8. Bollobás B., Riordan O. A Tutte polynomial for coloured graphs. Combin. Probab. Comput. 8:1999;45-93.
9. Bollobás B., Riordan O. A polynomial invariant of graphs on orientable surfaces. Proc. London Math. Soc. 83(3):2001;513-531.
10. J.E. Bonin, Strongly inequivalent representations and Tutte polynomials of matroids, Algebra Univ., to appear.
11. J.E. Bonin, A. de Mier, T -uniqueness of some families of k -chordal matroids, Adv. Appl. Math., to appear.
12. Bonin J.E., Miller W.P. Characterizing combinatorial geometries by numerical invariants. European J. Combin. 20:1999;713-724.
13. Brylawski T. Intersection theory for graphs. J. Combin. Theory Ser. B. 30:1981;233-246.
14. Brylawski T., Oxley J. The Tutte polynomial and its applications. White N. Matroid Applications. 1992;Cambridge University Press, Cambridge.
15. C.Y. Chao, E.G. Whitehead Jr., On chromatic equivalence of graphs, Springer Lecture Notes in Mathematics, Vol. 642, Springer, Berlin, 1978, pp. 121-131.
16. Chung F.R.K. Spectral Graph Theory. 1997;American Mathematical Society, Providence, RI.
17. Cvetković D.M., Doob M., Sachs H. Spectra of Graphs. 3rd ed. 1995;Johann Ambrosius Barth, Heidelberg.
18. Farrell E.J., Whitehead E.G. Jr. Connections between the matching and chromatic polynomials. Internat. J. Math. Math. Sci. 15:1992;757-766.
19. Godsil C. Algebraic Combinatorics. 1993;Chapman & Hall, London.
20. Godsil C., Royle G. Algebraic Graph Theory. 2001;Springer, Berlin.
21. W. Haemers, E. van Dam, Which graphs are determined by their spectrum?, CentER Discussion paper series Nr.: 2002-66, Tilburg University.
22. Heilmann O.J., Lieb E.H. Theory of monomer-dimer systems. Comm. Math. Phys. 25:1972;190-232.
23. F. Jaeger, On transition polynomials of 4-regular graphs, in: G. Hahn, G. Sabidussi, R.E. Woodrow (Eds.), Cycles and Rays, NATO Advances in Sci. Inst., Kluwer Academic Publishers, Dordrecht, 1990, pp. 123-150.
24. Jerrum M. Two-dimensional monomer-dimer systems are computationally intractable. J. Statist. Phys. 48:1987;121-134.
25. Kasteleyn P.W. Graph theory and crystal physics. Harary F. Graph Theory and Theoretical Physics. 1967;43-110 Academic Press, New York.
26. Koh K.M., Teo K.L. The search for chromatically unique graphs. Graphs Combin. 6:1990;259-285.
27. Koh K.M., Teo K.L. The search for chromatically unique graphs II. Discrete Math. 172:1997;59-78.
28. Kung J. Curious characterizations of projective and affine geometries. Adv. Appl. Math. 28:2002;523-543.
29. Lass B. Orientations acycliques et le polynôme chromatique. European J. Combin. 22:2001;1101-1123.
30. Li N.-Z. The list of chromatically unique graphs of order seven and eight. Discrete Math. 172:1997;193-221.
31. A. Márquez, A. de Mier, M. Noy, M.P. Revuelta, Locally grid graphs: classification and Tutte uniqueness, Discrete Math., to appear.
32. Merino C., de Mier A., Noy M. Irreducibility of the Tutte polynomial of a connected matroid. J. Combin. Theory Ser. B. 83:2001;298-304.
33. A. de Mier, M. Noy, On graphs determined by their Tutte polynomials, Graphs Combin., to appear.
34. A. de Mier, M. Noy, On matroids determined by their Tutte polynomial, submitted.
35. A. de Mier, M. Noy, Tutte uniqueness of line graphs, submitted.
36. Noble S.D., Welsh D.J.A. A weighted graph polynomial from chromatic invariants of knots. Ann. Inst. Fourier (Grenoble). 49:1999;1057-1087.
37. Oxley J.G. Matroid Theory. 1992;Oxford University Press, Oxford.
38. R. Penrose, Applications of Negative Dimensional Tensors, in: Combinatorial Mathematics and its Applications, Academic Press, New York, 1971, pp. 221-244.
39. R.C. Read, W.T. Tutte, Chromatic polynomials, in: L.W. Beineke, R.J. Wilson (Eds.), Selected Topics in Graph Theory, Vol. 3, Academic Press, New York, 1988.
40. Sarmiento I. The polychromate and a chord diagram polynomial. Ann. Combin. 4:2000;227-236.
41. A.J. Schwenk, Almost all trees are cospectral, in: New Directions in the Theory of Graphs, Academic Press, New York, 1973.
42. Seress A. Large families of cospectral graphs. Des. Codes Cryptogr. 21:2000;205-208.
43. Stanley R.P. A symmetric function generalization of the chromatic polynomial of a graph. Adv. Math. 111:1995;166-194.
44. Teo C.P., Koh K.M. The chromaticity of complete bipartite graphs with at most one edge deleted. J. Graph Theory. 14:1990;89-99.
45. Tutte W.T. A contribution to the theory of chromatic polynomials. Canad. J. Math. 6:1954;80-91.
46. Tutte W.T. Codichromatic graphs. J. Combin. Theory B. 16:1974;168-174.
47. Valiant L.G. The complexity of computing the permanent. Theoret. Comput. Sci. 8:1979;189-201.
48. Welsh D.J.A. Complexity: Knots, Colourings and Counting. 1993;Cambridge University Press, Cambridge.
49. Whitehead E.G. Jr. Chromaticity of two-trees. J. Graph Theory. 9:1985;279-284.
50. Whitney H. The coloring of graphs. Ann. of Math. 33:1932;688-718.
51. Whitney H. On the abstract properties of linear dependence. Amer. J. Math. 57:1935;509-533.