A New Property of Hamming Graphs and Mesh of d-ary Trees

  • Alain Bretto
  • Cerasela Jaulin
  • Luc Gillibert
  • Bernard Laget
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5081)


In this article we characterize two well-known graphs used in many applications, particularly in network applications: Hamming graphs and meshes of d-ary trees MT(d,1). More precisely, we show that they are so-called \(\mathbb{G}\)-graphs. \(\mathbb{G}\)-graphs are a new type of graphs constructed from a group. They have nice algebraic proprieties and can be regular or semi-regular.


Automorphism Group Cayley Graph Left Action Graph Automorphism London Mathematical Society Student Text 
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  1. 1.
    Babai, L.: Handbook of combinatorics. In: Automorphism groups, isomorphism, reconstruction, ch. 27 (1994)Google Scholar
  2. 2.
    Bretto, A., Faisant, A.: Another way for associating a graph to a group. Math.Slovaca 55(1), 1–8 (2005)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Bretto, A., Gilibert, L., Laget, B.: Symmetric and Semisymmetric Graphs Construction Using G-graphs. In: Kauers, M. (ed.) International Symposium on Symbolic and Algebraic Computation (ISSAC 2005), Beijing, China, July 24-27, 2005, pp. 61–67. ACM press, New York (2005) ISBN:1-59593-095-7CrossRefGoogle Scholar
  4. 4.
    Bretto, A., Gilibert, L.: G-graphs for the cage problem: a new upper bound. In: International Symposium on Symbolic and Algebraic Computation (ISSAC 2007), Waterloo, Ontario, Canada, July-29 August-1st 2007. ACM press, New York (2007) ISBN:978-1-59593-743-8 Google Scholar
  5. 5.
    Bretto, A., Faisant, A., Gillibert, L.: G-graphs: A new representation of groups. Journal of Symbolic Computation 42(5), 549–560 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cannon, J.J., Holt, D.F.: Automorphism group computation and isomorphism testing in finite groups. Journal of Symbolic Computation 35(3), 241–267 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cannon, J.J., Holt, D.F.: Computing conjugacy class representatives in permutation groups. Journal of Algebra 300(1), 213–222 (2006) MR2228644zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cooperman, G., Finkelstein, L., Sarawagi, N.: Applications of Cayley Graphs. In: Sakata, S. (ed.) AAECC 1990. LNCS, vol. 508, pp. 367–378. Springer, Heidelberg (1991)Google Scholar
  9. 9.
    Cooperman, G., Finkelstein, L.: New Methods for Using Cayley Graphs in Interconnection Networks. Discrete Applied Mathematics 37/38, 95–118 (1992)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Fraigniaud, P., Konig, J.-C., Lazard, E.: Oriented hypercubes. Networks 39(2), 98–106 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Zhang, G.-F., Gao, X.-S.: Spatial geometric constraint solving based on k-connected graph decomposition. In: Proceedings of the ACM symposium on Applied computing Dijon, France 2006 SESSION: Geometric computing and reasoning (GCR), pp. 979–983. ACM Press, New York (2006)Google Scholar
  12. 12.
    The GAP Team, (06 May 2002), GAP - Reference Manual, Release 4.3,
  13. 13.
    Lauri, J., Scapellato, R.: Topics in Graphs Automorphisms and Reconstruction, London Mathematical Society Student Texts (2003)Google Scholar
  14. 14.
    Lauri, J.: Constructing graphs with several pseudosimilar vertices or edges. Discrete Mathematics 267(1-3), 197–211 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Rockmore, D., Hordijk, W., Kostelec, P., Stadler, P.F.: Fast Fourier Transform for Fitness Landscapes. Applied and Computational Harmonic Analysis 12(1), 57–76 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Sunil Chandran, L., Kavitha, T.: The carvingwidth of hypercubes. Discrete Mathematics 306(18), 2270–2274 (2006)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Alain Bretto
    • 1
  • Cerasela Jaulin
    • 1
  • Luc Gillibert
    • 1
  • Bernard Laget
    • 2
  1. 1.Université de Caen, GREYC CNRS UMR-6072Caen cedexFrance
  2. 2.ENISESaint Etienne Cedex 02France

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