On Generalized Moore Digraphs

  • Michael Sampels
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3019)

Abstract

The transmission of a strongly connected digraph D is defined as the sum of all distances in D. A lower bound for the transmission in terms of the order n and the maximal outdegree Δ +  of D can be regarded as a generalization of the Moore bound for digraphs. Bridges and Toueg showed that Moore digraphs in the strong sense exist only for the trivial cases Δ + =1 or Δ + =n-1. Using techniques founded on Cayley digraphs, we constructed vertex-symmetric generalized Moore digraphs. Such graphs are applicable to interconnection networks of parallel computers, routers, switches, backbones, etc.

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References

  1. 1.
    Besche, H.U., Eick, B.: The groups of order at most 1000 except 512 and 768. Journal of Symbolic Computation 27, 405–413 (1999)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Biggs, N.: Algebraic Graph Theory, 2nd edn. Cambridge University Press, Cambridge (1993)Google Scholar
  3. 3.
    Bollobás, B.: Extremal Graph Theory. Academic Press, London (1978)MATHGoogle Scholar
  4. 4.
    Bridges, W.G., Toueg, S.: On the impossibility of directed Moore graphs. Journal of Combinatorial Theory, Series B 29, 339–341 (1980)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Buset, D.: Maximal cubic graphs with diameter 4. Discrete Applied Mathematics 101, 53–61 (2000)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Buskens, R.W., Rogers, M.J., Stanton, R.G.: A census of tetravalent generalized Moore networks. Congressus Numerantium 52, 255–296 (1986)MathSciNetGoogle Scholar
  7. 7.
    Buskens, R.W., Stanton, R.G.: The generalized Moore graphs on eleven vertices. Journal of Combinatorial Mathematics and Combinatorial Computing 1, 23–66 (1987)MATHMathSciNetGoogle Scholar
  8. 8.
    Buskens, R.W., Stanton, R.G.: Generalized Moore graphs on twelve and thirteen vertices. Ars Combinatoria 23-B, 87–132 (1987)MathSciNetGoogle Scholar
  9. 9.
    Cerf, V.G., Cowan, D.D., Mullin, R.C., Stanton, R.G.: Computer networks and generalized Moore graphs. Congressus Numerantium 9, 379–398 (1973)Google Scholar
  10. 10.
    Cerf, V.G., Cowan, D.D., Mullin, R.C., Stanton, R.G.: Trivalent generalized Moore networks on sixteen nodes. Utilitas Mathematica 6, 259–283 (1974)MATHMathSciNetGoogle Scholar
  11. 11.
    Cerf, V.G., Cowan, D.D., Mullin, R.C., Stanton, R.G.: A partial census of trivalent generalized Moore networks. In: Street, A.P., Wallis, W.D. (eds.) Proceedings of the 3rd Australian Conference on Combinatorial Mathematics. Lecture Notes in Mathematics, vol. 452, pp. 1–27. Springer, Berlin (1975)CrossRefGoogle Scholar
  12. 12.
    Comellas, F., Fiol, M.A.: Vertex symmetric digraphs with small diameter. Discrete Applied Mathematics 58, 1–11 (1995)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Comellas, F.: The (degree, diameter) problem for graphs, http://wwwmat.upc.es/grup_de_grafs/table_g.html
  14. 14.
    Hoffman, J., Singleton, R.R.: On Moore graphs with diameters 2 and 3. IBM Journal of Research and Development 4, 497–504 (1960)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lakshmivarahan, S., Jwo, J.-S., Dhall, S.K.: Symmetry in interconnection networks based on Cayley graphs of permutation groups: A survey. Parallel Computing 19, 361–407 (1993)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Miller, M., Fris, I.: Maximum order digraphs for diameter 2 or degree 2. In: Rees, R.S. (ed.) Graphs, matrices, and designs. Lecture Notes in Pure and Applied Mathematics, vol. 139, pp. 269–278. Dekker, New York (1993)Google Scholar
  17. 17.
    Sampels, M.: Large networks with small diameter. In: Möhring, R.H. (ed.) WG 1997. LNCS, vol. 1335, pp. 288–302. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  18. 18.
    Sampels, M.: Algebraic Construction of Efficient Interconnection Networks (in German: Algebraische Konstruktion effizienter Verbindungsnetzwerke). Dissertation, University of Oldenburg, Germany (1998)Google Scholar
  19. 19.
    Sampels, M., Vilents, M.: Symmetric interconnection networks in the design of switches for WANs and LANs. In: Baum, D., Müller, N., Rödler, R. (eds.) Proceedings of the 10th GI/ITG Special Interest Conference on Measurement, Modelling and Evaluation of Computer and Communication Systems (MMB 1999), pp. 43–48. University of Trier (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Michael Sampels
    • 1
  1. 1.Institut de Recherches Interdisciplinaires et de, Développements en Intelligence ArtificielleUniversité Libre de BruxellesBruxellesBelgium

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