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Circulant Digraphs and Monomial Ideals

  • Domingo Gómez
  • Jaime Gutierrez
  • Álvar Ibeas
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3718)

Abstract

It is known that there exists a Minimum Distance Diagram (MDD) for circulant digraphs of degree two (or double-loop computer networks) which is an L-shape. Its description provides the graph’s diameter and average distance on constant time. In this paper we clarify, justify and extend these diagrams to circulant digraphs of arbitrary degree by presenting monomial ideals as a natural tool. We obtain some properties of the ideals we are concerned. In particular, we prove that the corresponding MDD is also an L-shape in the affine r-dimensional space. We implement in PostScript language a graphic representation of MDDs for circulant digrahs with two or three jumps. Given the irredundant irreducible decomposition of the associated monomial ideal, we provide formulae to compute the diameter and the average distance. Finally, we present a new and attractive family (parametrized with the diameter d>2) of circulant digraphs of degree three associated to an irreducible monomial ideal.

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References

  1. 1.
    Beker, T., Weispfenning, V.: Gröbner basis - a computational approach to commutative algebra. Graduate Texts in Mathematics. Springer, Heidelberg (1993)Google Scholar
  2. 2.
    Bermond, J.-C., Comellas, F., Hsu, D.F.: Distributed Loop Computer Networks: A Survey. Journal of Parallel and Distributed Computing 24, 2–10 (1995)CrossRefGoogle Scholar
  3. 3.
    Boesch, F.T., Tindell, R.: Circulants and their connectivity. J. Graph Theory 8, 487–499 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Hwang, F.K.: A complementary survey on double-loop networks. Theoretical Computer Science 263, 211–229 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Mans, B.: Optimal Distributed algorithms in unlabeled tori and chordal rings. Journal of Parallel and Distributed Computing 46, 80–90 (1997)zbMATHCrossRefGoogle Scholar
  6. 6.
    Hsu, D.F., Jia, X.-D.: Extremal Problems in the Combinatorial Construction of Distributed Loop Networks. SIAM J. Discrete Math. 7(1), 57–71 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Miller, E.: Resolutions and Duality for Monomial Ideals. Ph.D. thesis (2000)Google Scholar
  8. 8.
    Miller, E., Sturmfels, B.: Monomial ideals and planar graphs. In: Fossorier, M.P.C., Imai, H., Lin, S., Poli, A. (eds.) AAECC 1999. LNCS, vol. 1719, pp. 19–28. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  9. 9.
    Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. American Mathematical Society, Providence (1996)zbMATHGoogle Scholar
  10. 10.
    Wong, C.K., Coppersmith, D.: A Combinatorial Problem Related to Multimodule Memory Organizations. J. ACM 21(3), 392–402 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Žerovnik, J., Pisanski, T.: Computing the Diameter in Multiple-Loop Networks. J. Algorithms 14(2), 226–243 (1993)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Domingo Gómez
    • 1
  • Jaime Gutierrez
    • 1
  • Álvar Ibeas
    • 1
  1. 1.Faculty of SciencesUniversity of CantabriaSantanderSpain

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