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Circulant Digraphs and Monomial Ideals

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Part of the Lecture Notes in Computer Science book series (LNTCS,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. Beker, T., Weispfenning, V.: Gröbner basis - a computational approach to commutative algebra. Graduate Texts in Mathematics. Springer, Heidelberg (1993)

    Google Scholar 

  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)

    CrossRef  Google Scholar 

  3. Boesch, F.T., Tindell, R.: Circulants and their connectivity. J. Graph Theory 8, 487–499 (1984)

    CrossRef  MATH  MathSciNet  Google Scholar 

  4. Hwang, F.K.: A complementary survey on double-loop networks. Theoretical Computer Science 263, 211–229 (2001)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. Mans, B.: Optimal Distributed algorithms in unlabeled tori and chordal rings. Journal of Parallel and Distributed Computing 46, 80–90 (1997)

    CrossRef  MATH  Google Scholar 

  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)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. Miller, E.: Resolutions and Duality for Monomial Ideals. Ph.D. thesis (2000)

    Google Scholar 

  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)

    CrossRef  Google Scholar 

  9. Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. American Mathematical Society, Providence (1996)

    MATH  Google Scholar 

  10. Wong, C.K., Coppersmith, D.: A Combinatorial Problem Related to Multimodule Memory Organizations. J. ACM 21(3), 392–402 (1974)

    CrossRef  MATH  MathSciNet  Google Scholar 

  11. Žerovnik, J., Pisanski, T.: Computing the Diameter in Multiple-Loop Networks. J. Algorithms 14(2), 226–243 (1993)

    CrossRef  MATH  MathSciNet  Google Scholar 

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© 2005 Springer-Verlag Berlin Heidelberg

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Gómez, D., Gutierrez, J., Ibeas, Á. (2005). Circulant Digraphs and Monomial Ideals. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2005. Lecture Notes in Computer Science, vol 3718. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11555964_17

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  • DOI: https://doi.org/10.1007/11555964_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-28966-1

  • Online ISBN: 978-3-540-32070-8

  • eBook Packages: Computer ScienceComputer Science (R0)