Scalable Sparse Topologies with Small Spectrum

  • Robert Elsässer
  • Rastislav Královič
  • Burkhard Monien
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2010)


One of the fundamental properties of a graph is the number of distinct eigenvalues of its adjacency or Laplacian matrix. Determining this number is of theoretical interest and also of practical impact. Graphs with small spectra exhibit many symmetry properties and are well suited as interconnection topologies. Es- pecially load balancing can be done on such interconnection topologies in a small number of steps. In this paper we are interested in graphs with maximal degree O(log n), where n is the number of vertices, and with a small number of distinct eigenvalues. Our goal is to find scalable families of such graphs with polyloga- rithmic spectrum in the number of vertices. We present also the eigenvalues of the Butterfly graph.


Load Balance Adjacency Matrix Maximal Degree Laplacian Matrix Vertex Degree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Akers, S.B., Harel, D., Krishnamurthy, B.: The Star Graph: An Attractive Alternative to the n-Cube, Proc. of the International Conference on Parallel Processing,1987, pp.393–400Google Scholar
  2. 2.
    Akers, S.B., Krishnamurthy, B.: A Group-Theoretic Model for Symmetric Interconnection Networks, IEEE Transactions on Computers 38, 1989, pp. 555–565zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Biggs, N. L.:Algebraic Graph Theory, (2nd ed.), Cambridge University Press, Cambridge, 1993Google Scholar
  4. 4.
    Brouwe, A. E., Cohen, A. M., Neumaier A.: Distance-Regular Graphs, Springer Verlag 1989Google Scholar
  5. 5.
    Chung, F.R.K.: Spectral Graph Theory, American Mathematical Society, 1994Google Scholar
  6. 6.
    Cybenko, G: Load balancing for distributed memory multiprocessors J. of Parallel and Distributed Computing 7, 1989, pp. 279–301CrossRefGoogle Scholar
  7. 7.
    Cvetković, D. M., Doob, M., Sachs, H.: Spectra of graphs, Theory and Application, Academic Press, 1980Google Scholar
  8. 8.
    Decker, T., Monien, B., Preis, R.: Towards Optimal Load Balancing Topologies, Proceedings of the 6th EuroPar Conference, LNCS, 2000, to appearGoogle Scholar
  9. 9.
    Delorme, C., Tillich, J. P.: The spectrum of DeBruijn and Kautz Graphs, European Journal of Combinatorics 19, 1998, pp. 307–319zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Diekmann R., Frommer, A., Monien, B.: Efficient schemes for nearest neighbor load balancing, Parallel Computing 25, 1999, pp. 789–812CrossRefMathSciNetGoogle Scholar
  11. 11.
    Elsässer, R., Frommer, A., Monien, B., Preis, R.: Optimal and Alternating-Direction Load Balancing, EuroPar’99, LNCS 1685, 1999, pp. 280–290Google Scholar
  12. 12.
    Flatto, L., Odlyzko, A.M., Wales, D.B.: Random Shuffles and Group Representations, The Annals of Probability 13, 1985 pp. 154–178zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Gosh, B., Muthukrishnan, S., Schultz, M.H.: First and Second order diffusive methods for rapid, coarse, distributed load balancing, SPAA, 1996, pp. 72–81Google Scholar
  14. 14.
    Hu, Y.F., Blake, R.J.: An improved diffusion algorithm for dynamic load balancing, Parallel Computing 25, 1999, pp. 417–444zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hu, Y.F., Blake, R.J., Emerson, D.R.: An optimal migration algorithm for dynamic load balancing, Concurrency: Prac. and Exp. 10, 1998, pp. 467–483zbMATHCrossRefGoogle Scholar
  16. 16.
    Hubaut, X. L.:Strongly Regular Graphs, Discrete Math. 13, 1975, pp. 357–381zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Tillich, J.-P.: The spectrum of the double-rooted tree, personal communicationGoogle Scholar
  18. 18.
    Xu, C., Lau, F.C.M.: Load Balancing in Parallel Computers, Kluwer, 1997Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Robert Elsässer
    • 1
  • Rastislav Královič
    • 2
  • Burkhard Monien
    • 1
  1. 1.University of PaderbornGermany
  2. 2.Comenius UniversityMFF-UK BratislavaSlovakia

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