Acta Mathematicae Applicatae Sinica

, Volume 13, Issue 4, pp 410–413 | Cite as

The diameters of almost all Cayley digraphs

  • Meng Jixiang 
  • Liu Xin 
Article

Abstract

LetG be a finite group of ordern andS be a subset ofG not containing the identity element ofG. Letp (0<p<1) be a fixed number. We define the set of all labelled Cayley digraphsX(G,S) (S<-G\{1}) ofG as a sample space and assign a probability measure by requiringP(aεS)=p for anyaG\{1}. Here it is shown that the probability of the set of Cayley digraphs ofG with diameter 2 approaches 1 as the ordern ofG approaches infinity.

Key words

Random Cayley digraph 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    W. Imrich. On the Connectivity of Cayley Graphs.J. Combin. Theory. (Series B), 1979, 26: 323–326.Google Scholar
  2. [2]
    Y.O. Hamidoune. On the Connectivity of Cayley Digraphs.Europ. J. Combinatorics, 1984,5: 309–312.Google Scholar
  3. [3]
    D. Witte and J.A. Gallian. A Survey: Hamiltonian Cycles in Cayley Graphs.Discrete Math., 1984, 51: 293–304.Google Scholar
  4. [4]
    C.D. Gosil. On Cayley Graph Isomorphisms.Ars. Combin., 1983, 15: 231–246.Google Scholar
  5. [5]
    Y.O. Hamidoune. The Minimum Order of a Cayley Graph with Given Degree and Diameter.Networks, 1993, 23: 283–287.Google Scholar
  6. [6]
    D.Z. Du, D.F. Hsu and F.K. Hwang. Doubly Linked Ring Networks.IEEE Trans. Comput., 1985, 34: 853–855.Google Scholar
  7. [7]
    S.B. Akers and B. Krishnamurthy. A Graph Theoretic Model for Symmetric Interconnection Networks.IEEE. Trans. Comput., 1989, 38(4): 555–566.Google Scholar
  8. [8]
    J.C. Bermond and D. Tzvieli. Minimal Diameter Double Loop Networks, Dense Optimal Familes.Networks, 1991, 21: 1–9.Google Scholar
  9. [9]
    J. Bond and C. Delorme. A Note on Partial Cayley Graphs.Discrete Math., 1993, 114: 63–74.Google Scholar
  10. [10]
    B. Bollobas. Random Graphs. Academic Press, New York, 1985.Google Scholar
  11. [11]
    L. Babai and C.D. Godsil. On the Automorphism Groups of Almost all Cayley Graphs.J. Europ. Combinatorics, 1982, 3: 9–15.Google Scholar
  12. [12]
    C.D. Godsil. On the Full Automorphism Group of a Graph.Combinatorica, 1981, 1: 243–262.Google Scholar
  13. [13]
    J.A. Bondy and U.S.R. Murty. Graph Theory with Applications. American Elsevier, New York, 1976.Google Scholar
  14. [14]
    N.L. Biggs. Algebraic Graph Theory. Cambridge, MA: Cambridge University Press, 1974.Google Scholar
  15. [15]
    P. Billingsley. Probability and Measure. John Wiley & Sons, New York, 1979.Google Scholar
  16. [16]
    Y.O. Hamidoune. Sur les Atoms Dun Graphe Orinte.Acad. Sci. Paris, 1977, 284: 1253–1256.Google Scholar
  17. [17]
    N. Brand. Almost all Steinhaus Graphs Have Diameter 2.J. Graph Theory, 1992, 16(3): 213–219.Google Scholar
  18. [18]
    N. Brand, S. Curran, S. Das and T. Jacob. Probability of Diameter Two for Steinhaus Graphs.Discrete Applied Math., 1993, 41: 165–171.Google Scholar

Copyright information

© Science Press, Beijing, China and Allerton Press, Inc., New York, U.S.A. 1997

Authors and Affiliations

  • Meng Jixiang 
    • 1
  • Liu Xin 
    • 2
  1. 1.Department of MathematicsXinjiang UniversityUrumqiChina
  2. 2.Institute of Applied Mathematicsthe Chinese Academy of SciencesBeijingChina

Personalised recommendations