Skip to main content
SpringerLink
Log in

Large networks with small diameter

  • Conference paper
  • First Online:
Graph-Theoretic Concepts in Computer Science (WG 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1335))

Included in the following conference series:

Abstract

The construction of large networks with small diameter D for a given maximal degree Δ is a major goal in combinatorial network theory. Using genetic algorithms, together with Cayley graph techniques, new results for this degree/diameter problem can be obtained. A modification of the Todd-Coxeter algorithm yields further results and allows, with Sabidussi's representation theorem, a uniform representation of vertex-symmetric graphs. The paper contains an updated table of the best known (Δ, D)-graphs and a table with the largest known graphs for a given Δ and maximum average distance µ between the nodes.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bruce W. Arden and K. Wendy Tang. IEEE Transactions on Communications, 39(11):1533–1537, November 1991.

    Google Scholar 

  2. J. C. Bermond, C. Delorme, and G. Farhi. Large graphs with given degree and diameter III. Annals of Discrete Mathematics, 13:23–32, 1982.

    Google Scholar 

  3. J. C. Bermond, C. Delorme, and G. Farhi. Large graphs with given degree and diameter. II. iJournal of Combinatorial Theory, Series B, 36:32–48, 1984.

    Google Scholar 

  4. Jean-Claude Bermond and Béla Bollobás. The diameter of graphs–A survey. Congressus Numerantium, 3:3–27, 1981.

    Google Scholar 

  5. Jean-Claude Bermond, Charles Delorme, and J.-J. Quisquater. Strategies for interconnection networks, some results from graph theory. Journal of Parallel and Distributed Computing, 3:433–449, 1986.

    Google Scholar 

  6. Norman Biggs. Algebraic Graph Theory. Cambridge University Press, 2nd edition, 1993.

    Google Scholar 

  7. Shahid H. Bokhari and A. D. Raza. Reducing the diameters of computer networks. IEEE Transactions on Computers, C-35(8):757–761, August 1986.

    Google Scholar 

  8. Lowell Campbell et al. Small diameter symmetric networks from linear groups. IEEE Transactions on Computers, 41(2):218–220, 1992.

    Google Scholar 

  9. Alan G. Chalmers and Steve Gregory. Constructing minimum path configurations for multiprocessor systems. Parallel Computing, 19:343–355, 1993.

    Google Scholar 

  10. Fan R. K. Chung. Diameters of graphs: Old and new results. Congressus Numerantium, 60:295–317, 1987.

    Google Scholar 

  11. F. Comellas and J. Gómez. New large graphs with given degree and diameter. In Y. Alavi and A. Schwenk, editors, Graph Theory, Combinatorics and Algorithms, volume 1, pages 221–233, New York, 1995. John Wiley & Sons, Inc.

    Google Scholar 

  12. Francesc Comellas. comellas@mat.upc.es.

    Google Scholar 

  13. Robert Cypher, Friedhelm Meyer auf der Heide; Christian Scheideler, and Berthold Vöcking. Universal algorithms for store-and-forward and wormhole routing. In Proceedings of the 26th ACM-STOC, pages 356–365, 1996.

    Google Scholar 

  14. C. Delorme. Examples of products giving large graphs with given degree and diameter. Discrete Applied Mathematics, 37/38:157–167, 1992.

    Google Scholar 

  15. Charles Delorme.cd@Iri.fr.

    Google Scholar 

  16. Charles Delorme and G. Farhi. Large graphs with given degree and diameter — part I. IEEE Transactions on Computers, C-33(9):857–860, September 1984.

    Google Scholar 

  17. Michael J. Dinneen and Paul R. Hafner. New results for the degree/diameter problem. Networks, 24:359–367, 1994.

    Google Scholar 

  18. Geoffrey Exoo. Applying optimization algorithms to Ramsey problems. In Yousef Alavi, editor, Graph Theory, Combinatorics, Algorithms and Applications, pages 175–179. John Wiley & Sons, Inc, New York, 1991.

    Google Scholar 

  19. H. Felsch. Programmierung der Restklassenabzählung einer Gruppe nach Untergruppen. Numerische Mathematik, 3:250–256, 1961.

    Google Scholar 

  20. P. Hafner. Large Cayley graphs and digraphs for the degree/diameter problem: an update. (in preparation), 1997.

    Google Scholar 

  21. Berthold Hagmann. Optimierungstechniken zum Entwurf günstiger Netztopologien mittels Cayley-Graphen. Diploma thesis, Universität Oldenburg, 1997.

    Google Scholar 

  22. J. H. Holland. Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor, 1975.

    Google Scholar 

  23. S. Lakshmivarahan, Jung-Sing Jwo, and S. K. Dhall. Symmetry in interconnection networks based on Cayley graphs of permutation groups: A survey. Parallel Computing, 19:361–407, 1993.

    Google Scholar 

  24. John Leech. Coset enumeration. In Michael D. Atkinson, editor, Computational Group Theory, pages 3–18. Academic Press, 1984.

    Google Scholar 

  25. F. Thomson Leighton. Parallel Algorithms and Architectures. Morgan Kaufmann Publishers, San Mateo, California, 1992.

    Google Scholar 

  26. Gerard Memmi and Yves Raillard. Some new results about the (d, k) graph problem. IEEE Transactions on Computers, C-31(8):784–791, August 1982.

    Google Scholar 

  27. Friedhelm Meyer auf der Heide and Berthold Vocking. A packet routing protocol for arbitrary networks. In E. W. Mayr and C. Puech, editors,Proceedings of the 12th Annual Symposium on Theoretical Aspects of Computer Science (STACS '95), LNCS 900, pages 291–302, 1995.

    Google Scholar 

  28. Gert Sabidussi. Vertex-transitive graphs. Monatshefte für Mathematik, 68:426–438, 1964.

    Google Scholar 

  29. Michael Sampels. Algebraic constructions of efficient systolic architectures. In Proceedings of the 2nd International Conference on Massively Parallel Computing Systems (MPCS '96), pages 15–22. IEEE Computer Society Press, 1996.

    Google Scholar 

  30. Michael Sampels. Cayley graphs as interconnection networks: A case study. In Proceedings of the 7th International Workshop on Parallel Processing by Cellular Automata and Arrays (PARCELLA '96), pages 67–76, Berlin, 1996. Akademie-Verlag.

    Google Scholar 

  31. Michael Sampels. Massively parallel architectures and systolic communication. In Proceedings of the 5th Euromicro Workshop on Parallel and Distributed Processing (PDP '97), pages 322–329. IEEE Computer Society Press, 1997.

    Google Scholar 

  32. Michael Sampels. Representation of vertex-symmetric interconnection networks. In Proceedings of the 2nd International Conference on Parallel Processing & Applied Mathematics. (to appear), 1997.

    Google Scholar 

  33. Michael Sampels and Stefan Schöf. Massively parallel architectures for parallel discrete event. simulation. In Proceedings of the 8th European Simulation Symposium (ESS '96), volume 2, pages 374–378. SCS, 1996.

    Google Scholar 

  34. Christian Scheideler and Berthold Vöcking. Universal continuous routing strategies. In Proceedings of the 8th ACM-SPAA, pages 142–151. ACM, 1996.

    Google Scholar 

  35. Martin Schönert. GAP — groups, algorithms and programming. Lehrstuhl D für Mathematik, RWTH Aachen, 1995.

    Google Scholar 

  36. Robert M. Storwick. Improved construction techniques for (d, k) graphs. IEEE Transactions on Computers, C-19:1214–1216, December 1970.

    Google Scholar 

  37. J. A. Todd and H. S. M. Coxeter. A practical method for enumerating cosecs of a finite abstract group. Proceeding of the Edinburgh Mathematical Society, 5(2):26–34, 1936.

    Google Scholar 

  38. Lutz Twele. Effiziente Implementierung des Todd-Coxeter Algorithmus im Hinblick auf Grad/Durchmesser-Optimierung von knotentransitiven Graphen. Diploma thesis, Universität Oldenburg, 1997.

    Google Scholar 

  39. Otto Wohlmuth. A new dense group graph discovered by an evolutionary approach. In Paralleles und Verteiltes Rechnen, Beiträge zum 4. Workshop über wissenschaftliches Rechnen. Shaker Verlag. 1996.

    Google Scholar 

  40. H. P. Yap. Some topics in graph theory. London Mathematical Society Lecture Note Series 108. Cambridge University Press, 1986.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universität Oldenburg, Fachbereich 10, 26111, Oldenburg, Germany

    Michael Sampels

Authors
  1. Michael Sampels
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Rolf H. Möhring

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Sampels, M. (1997). Large networks with small diameter. In: Möhring, R.H. (eds) Graph-Theoretic Concepts in Computer Science. WG 1997. Lecture Notes in Computer Science, vol 1335. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0024505

Download citation

  • DOI: https://doi.org/10.1007/BFb0024505

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63757-8

  • Online ISBN: 978-3-540-69643-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation