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The spectrum of a graph

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Combinatorial Mathematics V

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 622))

Abstract

We survey the results obtained by a large number of authors concerning the spectrum of a graph. The questions of characterisation by spectrum, cospectral graphs and information derived from the spectrum are discussed.

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References

  1. M. Aigner, The uniqueness of the cubic lattice graph, J. Comb. Theory, 6 (1969), 282–297.

    Article  MathSciNet  MATH  Google Scholar 

  2. J.S.O. Aliev and E. Seiden, Steiner triple systems and strongly regular graphs, J. Comb. Theory, 6 (1969), 33–39.

    Article  MathSciNet  MATH  Google Scholar 

  3. G.A. Baker, Drum shapes and isospectral graphs, J. Math. Phys., 7 (1966), 2238–2242.

    Article  MATH  Google Scholar 

  4. I.S. Berezin and N.P. Zhidkov, Computing Methods II, Pergamon Press, Oxford, 1965.

    MATH  Google Scholar 

  5. N.Biggs, Perfect codes and distance-transitive graphs, in Combinatorics, London Mathematical Society Lecture Notes, 13, C.U.P., (1974), 1–8.

    Google Scholar 

  6. R.C. Bose and R. Laskar, Eigenvalues of the adjacency matrix of tetrahedral graphs, Aequat. Math., 4 (1970), 37–43.

    Article  MathSciNet  MATH  Google Scholar 

  7. L.C. Chang, The uniqueness and non-uniqueness of the triangular association schemes, Science Record 3 (1959), 604–613.

    MATH  Google Scholar 

  8. L.C. Chang, Association schemes of partially balanced designs with parameters v=28, n1=12, n2=15 and p11 2=4, Science Record 4 (1960), 12–18.

    Google Scholar 

  9. F.H. Clarke, A graph polynomial and its applications, Discrete Math., 3 (1972), 305–313.

    Article  MathSciNet  MATH  Google Scholar 

  10. L. Collatz and U. Sinogowitz, Spektren der Graphen, Fiat Reviews of German Science, Naturforschung U. Medizin in Deutschland, (1939–1946), Bd. 2, (Reine Math.), 251.

    Google Scholar 

  11. L. Collatz and U. Sinogowitz, Spektren endlicher Grafen, Abh. Math. Sem. Univ. Hamburg, 21 (1957), 63–77.

    Article  MathSciNet  MATH  Google Scholar 

  12. W.S. Connor, The uniqueness of the triangular association scheme, Ann. Math. Stat., 29 (1958), 262–266.

    Article  MathSciNet  MATH  Google Scholar 

  13. C.A. Coulson and A. Streitweiser, Dictionary of π-electron calculations, Pergamon Press, Oxford, 1965.

    Google Scholar 

  14. D.M. Cvetković, Spectrum of the graph of n-tuples, Univ. Beograd. Publ. Elektrotehn Fak., Ser. Mat. Fiz., Nos. 274–301 (1969), 91–95.

    Google Scholar 

  15. D.M. Cvetković, New characterisations of the cubic lattice graph, Publ. Inst. Math., 10 (1970), 195–198.

    MATH  Google Scholar 

  16. D.M. Cvetković, Graphs and their spectra, Univ. Beograd. Publ. Elektrotehn Fak., Ser. Mat. Fiz., Nos. 354–356 (1971), 1–50.

    MathSciNet  MATH  Google Scholar 

  17. D.M. Cvetković, Cubic integral graphs, Univ. Beograd. Publ. Elektrotehn Fak., Ser. Mat. Fiz., Nos. 498–541 (1975), 107–113.

    MathSciNet  MATH  Google Scholar 

  18. D.M. Cvetković and I.M. Gutman, The algebraic multiplicity of the number zero in the spectrum of a bipartite graph, Matem. Vesnik, 9 (1972), 141–150.

    MathSciNet  MATH  Google Scholar 

  19. D.M. Cvetković, I.M. Gutman and N. Trinajstić, Conjugated molecules having integral graph spectra, Chem. Phys. Letters, 29 (1974), 65–68.

    Article  Google Scholar 

  20. M. Doob, Graphs with a small number of distinct eigenvalues, Ann. N.Y. Acad. Sci., 175 (Art. 1), (1970), 104–110.

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Doob, On characterising certain graphs with four eigenvalues by their spectra, Lin. Alg. Appl., 3 (1970), 461–482.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Doob, An interrelation between line graphs, eigenvalues and matroids, J. Comb. Theory, 15B (1973), 40–50.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Doob, Eigenvalues of a graph and its imbeddings, J. Comb. Theory, 17B (1974), 244–248.

    Article  MathSciNet  MATH  Google Scholar 

  24. M. Doob, A spectral characterization of the line graph of a b.i.b.d. with λ=1, Lin. Alg. Appl., 12 (1975), 11–20.

    Article  MathSciNet  Google Scholar 

  25. M. Fiedler, Algebraic connectivity of graphs, Czech. Math. J., 23 (1973), 298–305.

    MathSciNet  MATH  Google Scholar 

  26. H.J. Finck and G. Grohmann, Vollständiges Produkt, chromatische Zahl und characteristisches Polynom regularer Graphen I, Wiss. Z. Techn. Hochsch. Ilmenau, 11 (1965), 1–3.

    MathSciNet  MATH  Google Scholar 

  27. M.E. Fisher, On hearing the shape of a drum, J. Comb. Theory, 1 (1966), 105–125.

    Article  MathSciNet  MATH  Google Scholar 

  28. F. Gantmacher, Applications of the theory of matrices, Vol. 2, Chelsea, New York, 1959.

    Google Scholar 

  29. C. Godsil and B. McKay, Products of graphs and their spectra, Proceedings of Fourth Australian Comb. Conf., Adelaide, 1975, to appear.

    Google Scholar 

  30. C. Godsil and B. McKay, Some computational results on the spectra of graphs, Proceedings of Fourth Australian Comb. Conf., Adelaide, 1975, to appear.

    Google Scholar 

  31. J.M. Goethals and J.J. Seidel, Orthogonal matrices with zero diagonal, Can. J. Math., 19 (1967), 1001–1010.

    Article  MathSciNet  MATH  Google Scholar 

  32. F. Harary, C. King, A. Mowshowitz and R.C. Read, Cospectral graphs and digraphs, Bull. Lond. Math. Soc., 3 (1971), 321–328.

    Article  MathSciNet  MATH  Google Scholar 

  33. F. Harary and A.J. Schwenk, Which graphs have integral spectra? in Graphs and Combinatorics, Springer Lecture Notes in Mathematics, 406, Springer-Verlag, Berlin, (1974), 45–75.

    Chapter  Google Scholar 

  34. E.V. Haynsworth, Applications of a theorem on partitioned matrices, J. Res. Nat. Bur. Stand., (B), 63 (1959), 73–78.

    Article  MathSciNet  MATH  Google Scholar 

  35. A.J. Hoffman, On the exceptional case in a characterization of the arcs of a complete graph, IBM J. Res. Dev., 4 (1960), 487–496.

    Article  MathSciNet  MATH  Google Scholar 

  36. A.J. Hoffman, On the uniqueness of the triangular association scheme, Ann. Math. Stat., 31 (1960), 492–497.

    Article  MathSciNet  MATH  Google Scholar 

  37. A.J. Hoffman, On the line graph of a projective plane, Proc. Amer. Math. Soc., 16 (1965), 297–302.

    Article  MathSciNet  MATH  Google Scholar 

  38. A.J.Hoffman, The eigenvalues of the adjacency matrix of a graph, Combinatorial Mathematics and its Applications, Proc. Chapel Hill Conf., (1967), 578–584.

    Google Scholar 

  39. A.J. Hoffman, Some recent results on spectral properties of graphs, Beiträge zur Graphentheorie, Leipzig, (1968), 75–80.

    Google Scholar 

  40. A.J. Hoffman, Eigenvalues and partitioning of the edges of a graph, Lin. Alg. Appl., 5 (1972), 137–146.

    Article  MATH  Google Scholar 

  41. A.J. Hoffman and D.K. Ray-Chaudhuri, On the line graph of a symmetric b.i.b.d., Trans. Amer. Math. Soc., 116 (1965), 238–252.

    Article  MathSciNet  MATH  Google Scholar 

  42. A.J. Hoffman and D.K. Ray-Chaudhuri, On the line graph of a finite affine plane, Can. J. Math., 17 (1965), 687–694.

    Article  MathSciNet  MATH  Google Scholar 

  43. H. Hosoya, M. Murakami and M. Gotoh, Distance polynomial and characterization of a graph, Nat. Sci. Report, Ochanomizu University, 24 (1973), 27–34.

    MathSciNet  MATH  Google Scholar 

  44. M. Kac, Can one hear the shape of a drum? Amer. Math. Monthly, 74 (4, Part II), (1966), 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  45. C. King, Characteristic polynomials of 7-node graphs, Scientific Report, Univ. of West Indies/CC 6, (AFOSR project 1026-66).

    Google Scholar 

  46. V. Krishnamoorthy and K.R. Parthasarathy, A note on non-isomorphic cospectral digraphs, J. Comb. Theory, 17B (1974), 39–40.

    Article  MathSciNet  MATH  Google Scholar 

  47. W.W. Kuhn, An algorithm for graph isomorphism using adjacency matrices, Ph.D. Dissertation, Univ. of Pennsylvania, 1971.

    Google Scholar 

  48. R. Laskar, Eigenvalues of the adjacency matrix of cubic lattice graphs, Pac. J. Math., 29 (1969), 623–629.

    Article  MathSciNet  MATH  Google Scholar 

  49. H.C. Longuet-Higgins, Resonance structures and MO in unsaturated hydrocarbons, J. Chem. Phys., 18 (1950), 265–274.

    Article  Google Scholar 

  50. B.D. McKay, On the spectral characterisation of trees, in preparation.

    Google Scholar 

  51. A. Mowshowitz, The group of a graph whose adjacency matrix has all distinct eigenvalues, Proof Techniques in Graph Theory, Academic Press, New York, (1969), 109–110.

    Google Scholar 

  52. A. Mowshowitz, Graphs, groups and matrices, Proc. 25th Summer Meeting Can. Math. Congress, Congr. Num. IV, Utilitas Math., Winnipeg, (1971), 509–522.

    Google Scholar 

  53. A. Mowshowitz, The characteristic polynomial of a graph, J. Comb. Theory, 12B (1972), 177–193.

    Article  MathSciNet  MATH  Google Scholar 

  54. D.K. Ray-Chaudhuri, Characterization of line graphs, J. Comb. Theory, 3 (1967), 201–214.

    Article  MathSciNet  MATH  Google Scholar 

  55. R. Read, Teaching graph theory to a computer, Recent Progress in Combinatorics, Academic Press, New York, (1968), 161–174.

    Google Scholar 

  56. R. Read, The coding of various kinds of unlabelled trees, Graph Theory and Computing, Academic Press, New York, (1972), 153–183.

    Chapter  Google Scholar 

  57. H. Sachs, Beziehungen zwischen den in einem Graphen enthaltenen Kreisen und seinem charakteristischen Polynom, Publ. Math. Debrecen, 11 (1963), 119–134.

    MathSciNet  MATH  Google Scholar 

  58. H. Sachs, Über Teiler, Faktoren und charakteristische Polynome von Graphen, Teil I, Wiss. Z. Techn. Hochsch. Ilmenau, 12 (1966), 7–12.

    MathSciNet  MATH  Google Scholar 

  59. H. Sachs, Über Teiler, Faktoren und characteristische Polynome von Graphen, Teil II, Wiss. Z. Techn. Hochsch, Ilmenau, 13 (1967), 405–412.

    MathSciNet  MATH  Google Scholar 

  60. H.Sachs, Struktur und Spektrum von Graphen, Mitteilungen der Math. Gesellsch. der D.D.R., (1973), 119–132.

    Google Scholar 

  61. A.J. Schwenk, Almost all trees are cospectral, New Directions in Graph Theory, Academic Press, New York, (1973), 275–307.

    Google Scholar 

  62. A.J. Schwenk, Exactly thirteen connected cubic graphs have integral spectra, to appear, Proc. International Conference on the Theory and Applications of Graphs, Kalamazoo, 1976.

    Google Scholar 

  63. A.J. Schwenk, Computing the characteristic polynomial of a graph, Lecture Notes in Mathematics No. 406, Springer, Berlin, (1974), 153–172.

    MATH  Google Scholar 

  64. J.J. Seidel, Strongly regular graphs of L2-type and triangular type, Indagationes Math., 29 (1967), 188–196.

    Article  MathSciNet  MATH  Google Scholar 

  65. J.J. Seidel, Strongly regular graphs with (−1, 1, 0)-adjacency matrix having eigenvalue 3, Lin. Alg. Appl., 1 (1968), 281–298.

    Article  MathSciNet  MATH  Google Scholar 

  66. J.J. Seidel, Strongly regular graphs, Recent Progress in Combinatorics, Academic Press, New York, (1969), 185–198.

    Google Scholar 

  67. J.J. Seidel, Graphs and two-graphs, Proc. 5th South-eastern Conf., Congr. Num. X, Utilitas Math., Winnipeg, (1974), 125–143.

    Google Scholar 

  68. S.S. Shrikhande, On a characterization of the triangular association scheme, Ann. Math. Stat., 30 (1959), 39–47.

    Article  MathSciNet  MATH  Google Scholar 

  69. S.S. Shrikhande, The uniqueness of the L2 association scheme, Ann. Math. Stat., 30 (1959), 781–798.

    Article  MathSciNet  MATH  Google Scholar 

  70. L. Spialter, The atom connectivity matrix characteristic polynomial (ACMCP) and its physico-geometric (topological) significance, J. Chem. Doc., 4 (1964), 269–274.

    Article  Google Scholar 

  71. A. Streitweiser, J.I. Brauman and C.A. Coulson, Supplement tables of molecular orbit calculations I, II, Oxford, 1965.

    Google Scholar 

  72. J. Turner, Point-symmetric graphs with a prime number of points, J. Comb. Theory, 3 (1967), 136–145.

    Article  MathSciNet  MATH  Google Scholar 

  73. D.A. Waller, Eigenvalues of graphs and operations, Combinatorics, London Math. Soc. Lecture Notes No. 13, C.U.P., (1974), 177–183.

    Google Scholar 

  74. R.J. Wilson, On the adjacency matrix of a graph, Combinatorics, I.M.A., Southend-on-Sea, (1972), 295–321.

    Google Scholar 

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Authors
  1. C. Godsil
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  2. D. A. Holton
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  3. B. McKay
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Charles H. C. Little

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© 1977 Springer-Verlag

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Godsil, C., Holton, D.A., McKay, B. (1977). The spectrum of a graph. In: Little, C.H.C. (eds) Combinatorial Mathematics V. Lecture Notes in Mathematics, vol 622. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0069184

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  • DOI: https://doi.org/10.1007/BFb0069184

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08524-9

  • Online ISBN: 978-3-540-37020-8

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