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Some Applications of Eigenvalues of Graphs

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Structural Analysis of Complex Networks

Abstract

The main goal of spectral graph theory is to relate important structural properties of a graph to its eigenvalues. In this chapter, we survey some old and new applications of spectral methods in graph partitioning, ranking, and epidemic spreading in networks and clustering.

MSC2000: Primary 15A18; Secondary 68R10, 05C99

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References

  1. Alon N (1986) Eigenvalues and expanders. Combinatorica 6:83–96

    Article  MATH  MathSciNet  Google Scholar 

  2. Alon N (1996) Bipartite subgraphs. Combinatorica 16:301–311

    Article  MATH  MathSciNet  Google Scholar 

  3. Alon N, Bollobás B, Krivelevich M, Sudakov B (2003) Maximum cuts and judicious partitions of graphs without short cycles. J Combin Theory B 88:329–346

    Article  MATH  Google Scholar 

  4. Alon N, Capalbo M (2007) Finding disjoint paths in expanders deterministically and online. In: FOCS, pp 518–524

    Google Scholar 

  5. Alon N, Krivelevich M, Sudakov B (1999) List coloring of random and pseudo-random graphs. Combinatorica 19:453–472

    Article  MATH  MathSciNet  Google Scholar 

  6. Alon N, Milman V (1985) λ1, Isoperimetric inequalities for graphs and superconcentrators. J Combin Theory B 38:73–88

    Article  MATH  MathSciNet  Google Scholar 

  7. Andersen R, Chung F, Lang K (2006) Local graph partitioning using pagerank vectors. In: Proceedings of the 47th annual IEEE symposium on foundation of computer science, pp 475–486

    Google Scholar 

  8. Bezrukov S, Elsässer R, Monien B, Preis R, Tillich J-P (2004) New spectral lower bounds on the bisection width of graphs. Theor Comput Sci 320:155–174

    Article  MATH  Google Scholar 

  9. Bollobás B (1998) Modern graph theory. Graduate texts in mathematics, vol 184. Springer, New York

    Google Scholar 

  10. Bollobás B, Nikiforov V (2004) Graphs and Hermitian matrices: eigenvalue interlacing. Discrete Math 289:119–127

    Article  MATH  MathSciNet  Google Scholar 

  11. Brin S, Page L (1998) The anatomy of a large-scale hypertextual Web search engine. Comput Netw ISDN Syst 30:107–117

    Article  Google Scholar 

  12. Brouwer A, Haemers W (2005) Eigenvalues and perfect matchings. Lin Algebra Appl 395:155–162

    Article  MATH  MathSciNet  Google Scholar 

  13. Brouwer A, Haemers W (2010) Spectra of graphs. Monograph in preparation +200pp, available at http://homepages.cwi.nl/~aeb/math/ipm.pdf

  14. Brouwer A, Koolen J (2009) The vertex connectivity of a distance regular graph. Eur J Combinator 30:668–673

    Article  MATH  MathSciNet  Google Scholar 

  15. Brouwer A, Mesner DM (1985) The connectivity of strongly regular graphs. Eur J Combinator 6:215–216

    MATH  MathSciNet  Google Scholar 

  16. Bui TN, Chaudhuri S, Leighton FT, Sipser M (1987) Graph bisection algorithms with good average case behaviour. Combinatorica 2:171–191

    Article  MathSciNet  Google Scholar 

  17. Butler S, Chung F (2010) Small spectral gap in the combinatorial Laplacian implies Hamiltonian. Ann Combinator (in press)

    Google Scholar 

  18. Chung FRK (1997) Spectral graph theory. American Mathematical Society, Providence, RI

    MATH  Google Scholar 

  19. Chung FRK (2004) Discrete isoperimetric inequalities. Surveys in differential geometry IX. International Press, Somerville, MA, pp 53–82

    Google Scholar 

  20. Chung FRK (2007) Four proofs of the Cheeger inequality and graph paritition algorithms. In: Proceedings of ICCM, vol II

    Google Scholar 

  21. Chung FRK (2007) Random walks and local cuts in graphs. Lin Algebra Appl 423:22–32

    Article  MATH  Google Scholar 

  22. Cioabă SM (2010) Eigenvalues and edge-connectivity of regular graphs. Lin Algebra Appl 432:458–470

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  24. Cvetković D, Doob M, Sachs H (1980) Spectra of graphs – theory and application. Academic, New York, 3rd edn, Johann Ambrosius Barth (1995)

    Google Scholar 

  25. Cvetković D, Rowlinson P, Simic S (1997) Eigenspaces of graphs. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  26. de Klerk E, Pasechnik DV (2007) A note on the stability of an orthogonality graph. Eur J Combinator 28:1971–1979

    Article  MATH  Google Scholar 

  27. Delsarte P (1973) An algebraic approach to the association schemes of coding theory. Philips Res Rep Suppl 10:1–97

    MathSciNet  Google Scholar 

  28. Diáz J, Do N, Serna MJ, Wormald NC (2003) Bounds on the max and min bisection of random cubic and random 4-regular graphs. Theor Comput Sci 307:531–548

    Article  MATH  Google Scholar 

  29. Donath WE, Hoffman AJ (1973) Lower bounds for the partitioning of the graphs. IBM J Res Dev 17:420–425

    Article  MATH  MathSciNet  Google Scholar 

  30. Draief M, Ganesh A, Massoulié L. Thresholds for virus spread on networks, available at http://arxiv.org/abs/math/0606514

  31. Elsässer R, Lücking T, Monien B (2003) On spectral bounds for the k-partitioning of graphs. Theor Comput Syst 36:461–478

    Article  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  33. Fiedler M (1975) A property of eigenvectors of non-negative symmetric matrices and its applications to graph theory. Czech Math J 85:619–633

    MathSciNet  Google Scholar 

  34. Garey MR, Johnson DS, Stockmeyer L (1976) Some simplified NP-complete graph problems. Theor Comput Sci 1:237–267

    Article  MATH  MathSciNet  Google Scholar 

  35. Gilbert J, Hutchinson J, Tarjan R (1984) A separation theorem for graphs of bounded genus. J Algorithm 5:391–407

    Article  MATH  MathSciNet  Google Scholar 

  36. Godsil CD, Newman MW (2008) Coloring an orthogonality graph. SIAM J Discrete Math 22:683–692

    Article  MATH  MathSciNet  Google Scholar 

  37. Godsil CD, Newman MW (2008) Eigenvalue bounds for independent sets. J Combin Theory B 98:721–734

    Article  MATH  MathSciNet  Google Scholar 

  38. Godsil CD, Royle G (2001) Algebraic graph theory. Springer, Berlin

    MATH  Google Scholar 

  39. Guattery S, Miller GL (1998) On the quality of spectral separators. SIAM J Matrix Anal Appl 19:701–719

    Article  MATH  MathSciNet  Google Scholar 

  40. Haemers W (1995) Interlacing eigenvalues and graphs. Lin Algebra Appl 226/228:593–616

    Article  MathSciNet  Google Scholar 

  41. Haveliwala TH (2003) Topic-sensitive PageRank: a context-sensitive ranking algorithm for Web search. IEEE Trans Knowl Data Eng 15:784–796

    Article  Google Scholar 

  42. Hoffman AJ (1970) On eigenvalues and colorings of graphs. In: Graph theory and its applications. (Proceedings of an advanced seminar, mathematical research center, University of Wisconsin, Madison, WI, 1969). Academic, New York, pp 79–91

    Google Scholar 

  43. Hoffman AJ, Wielandt H (1953) The variation of the spectrum of a normal matrix. Duke Math J 29:37–39

    Article  MathSciNet  Google Scholar 

  44. Hoory S, Linial N, Wigderson A (2006) Expander graphs and their applications. Bull AMS 43:439–561

    Article  MATH  MathSciNet  Google Scholar 

  45. Horn RA, Johnson CR (1985) Matrix analysis. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  46. Kannan R, Vempala S, Vetta A (2004) On clusterigs: good, bad and spectral. J ACM 51:497–515

    Article  MATH  MathSciNet  Google Scholar 

  47. Kelner JA (2004) Spectral partitioning, eigenvalue bounds, and circle packings for graphs of bounded genus. In: Proceedings of the symposium on the theory of Computing (STOC)

    Google Scholar 

  48. Kendall MG (1955) Further contributions to the theory of paired comparisons. Biometrics 11:43–62

    Article  MathSciNet  Google Scholar 

  49. Kirchhoff G (1847) Über die Auflösung der Gleichungen auf welche man bei der Untersuchung der Linearen Vertheilung galvanischer Ströme geführt wird. Ann Phys Chem 72:497–508

    Article  Google Scholar 

  50. Kirkland SJ, Molitierno JJ, Neumann M, Shader BL (2002) On graphs with equal algebraic and vertex connectivity. Lin Algebra Appl 341:45–56

    Article  MATH  MathSciNet  Google Scholar 

  51. Krivelevich M, Sudakov B (2006) Pseudo-random graphs. In: More sets, graphs and numbers. Bolyai Society Mathematical Studies, vol 15, pp 199–262

    Google Scholar 

  52. Lipton RJ, Tarjan RE (1979) A separator theorem for planar graphs. SIAM J Appl Math 36:177–189

    Article  MATH  MathSciNet  Google Scholar 

  53. Mihail M (1989) Conductance and convergence of Markov chains – a combinatorial treatment of expandes. In: Proceedings of the 30th annual symposium on foundations of computer science. IEEE Computer Society, Washington, DC, pp 526–531

    Google Scholar 

  54. Mohar B (1989) Isoperimetric numbers of graphs. J Combin Theory B 47:274–291

    Article  MATH  MathSciNet  Google Scholar 

  55. Mohar B (1997) Some applications of Laplace eigenvalues of graphs. In: Hahn G, Sabidussi G (eds) Graph symmetry. Kluwer, Dordrecht, pp 225–275

    Google Scholar 

  56. Monien B, Preis R (2001) Upper bounds on the bisection width of 3 and 4-regular graphs. In: Pultr A, Sgall J, Kolman P (eds) Mathematical foundations of computer science, vol 2136. Lecture Notes in Computer Science. Springer, pp 524–536

    Google Scholar 

  57. Nikiforov V (2001) Some inequalities for the largest eigenvalue of a graph. Combinator Probab Comput 11:179–189

    MathSciNet  Google Scholar 

  58. Spielman D, Teng SH (1996) Spectral partitioning works: planar graphs and finite element meshes. In: Proceedings of the 37th annual symposium on foundations of computer science, pp 96–106

    Google Scholar 

  59. Spielman D, Teng SH (2007) Spectral partitioning works. Lin Algebra Appl 421:284–305

    Article  MATH  MathSciNet  Google Scholar 

  60. Tanner RM (1984) Explicit construction of concentrators from generalized N-gons. SIAM J Algebra Discr 5:287–294

    Article  MATH  MathSciNet  Google Scholar 

  61. van Dam E, Haemers W (2003) Which graphs are determined by their spectrum? Lin Algebra Appl 373:241–272

    Article  MATH  Google Scholar 

  62. van Dam E, Haemers W (2009) Developments on spectral characterization of graphs. Discrete Math 309:576–586

    Article  MATH  MathSciNet  Google Scholar 

  63. van Dam E, Jamakovic A, Kooij RE, Van Mieghem P (2006) Robustness of networks against viruses: the role of spectral radius. In: Proceedings of the 13th annual symposium of the IEEE/CVT Benelux, Liége, Belgium, pp 35–38

    Google Scholar 

  64. van Dam E, Kooij RE (2007) The minimal spectral radius of graphs with give diameter. Lin Algebra Appl 423:408–419

    Article  MATH  Google Scholar 

  65. Wang Y, Chakrabarti D, Wang C, Faloutsos C (2003) Epidemic spreading in real networks: an eigenvalue viewpoint. In: Proceedings of the symposium on reliable distributed computing, Florence, Italy, pp 25–34

    Google Scholar 

  66. Wei TH (1952) The algebraic foundations of ranking theory. Cambridge University Press, London

    Google Scholar 

  67. West DB (2001) Introduction to graph theory, 2nd edn. Prentice Hall, Englewood Cliffs, NJ

    Google Scholar 

  68. Whitney H (1932) Congruent graphs and the connectivity of graphs. Am J Math 54:150–168

    Article  MathSciNet  Google Scholar 

  69. Wilf H (1967) The eigenvalues of a graph and its chromatic number. J Lond Math Soc 42:330–332

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

This work is supported by a start-up grant from the Department of Mathematical Sciences at the University of Delaware. The author is grateful to the referees for their comments.

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  1. Department of Mathematical Sciences, University of Delaware, 501 Ewing Hall, Newark, DE, 19716-2553, USA

    Sebastian M. Cioabă

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  1. Sebastian M. Cioabă
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Correspondence to Sebastian M. Cioabă .

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  1. Medizinische Informatik und Technik, Institute for Bioinformatics and Transla, UMIT-Private Universität für Gesundheits, Eduard Wallnöfer-Zentrum 1, Hall in Tirol, A-6060, Austria

    Matthias Dehmer

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Cioabă, S.M. (2011). Some Applications of Eigenvalues of Graphs. In: Dehmer, M. (eds) Structural Analysis of Complex Networks. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4789-6_14

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