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Science in China Series A: Mathematics

, Volume 42, Issue 3, pp 264–271 | Cite as

Asymptotic enumeration theorems for the numbers of spanning trees and Eulerian trails in circulant digraphs and graphs

  • Zhang Fuji
  • Yong Xuerong
Article

Abstract

The asymptotic properties of the numbers of spanning trees and Eulerian trails in circulant digraphs and graphs are studied. Let\(C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)\) be a directed circulant graph. Let\(\left( {C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)} \right)\) and\(\left( {C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)} \right)\) be the numbers of spanning trees and of Eulerian trails, respectively. Then
$$\begin{array}{*{20}c} \begin{gathered} \lim \frac{1}{k}\sqrt[p]{{T\left( {C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)} \right)}} = 1, \hfill \\ \lim \frac{1}{{k!}}\sqrt[p]{{E\left( {C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)} \right)}} = 1, \hfill \\ \end{gathered} & {p \to \infty .} \\ \end{array} $$
Furthermore, their line digraph and iterations are dealt with and similar results are obtained for undirected circulant graphs.

Keywords

asymptotic enumeration Eulerian trails graph circulant 

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References

  1. 1.
    Wong, C. K., Coppersmith, D. A., A combinatorial problem related to multimode memory organization,J. Assoc. Comput., 1971, 21: 392.MathSciNetGoogle Scholar
  2. 2.
    Bermond, J.C., Distributed loop computer networks: A survey,J. Parallel and Distribution Computing, 1995, 24: 2.CrossRefGoogle Scholar
  3. 3.
    Bermond, J.C., Interconnected network, special issue, inDiscrete Applied Mathematics, Amsterdam: Elsevier Science, 1992, 37/38: 992.Google Scholar
  4. 4.
    Fiol, M. A., Line digraph iteration and the(d, k) digraph problem,IEEE Trans. on Computers, 1994, C33: 400.Google Scholar
  5. 5.
    Chen, W.K.,Applied Graph Theory, Amsterdam: North-Holland, 1971.MATHGoogle Scholar
  6. 6.
    Zhang, H.S. et al., On the number of spanning trees and Eulerian tours in iterated line digraphs,Discrete Appl. Mathematics, 1997, 73: 59.MATHCrossRefGoogle Scholar
  7. 7.
    Cheng, C., Maximizing the number of spanning trees in a graph: Two related problems in graph theory and optimum design theory,J. Combin. Theory., Ser B, 1981, 30: 240.CrossRefGoogle Scholar
  8. 8.
    Lovasz, L., Plummer, M.D.,Matching Theory, Amsterdam: North-Holland, 1986.MATHGoogle Scholar
  9. 9.
    Biggs, N.,Algebraic Graph Theory, Amsterdam: North-Holland, 1985.Google Scholar
  10. 10.
    Zhang, F.J., Lin G.N., The complexity of disgraphs, inGraph Theory and Its Applications (ed. Capobianco, M.F.),East and West Annal of New York Academic of Science, 1989, 171–180.Google Scholar
  11. 11.
    Jacobson, N.,Basic Algebra I, San Francisco: W H Freeman and Company, 1974.MATHGoogle Scholar
  12. 12.
    Yong, X.R., Zhang, F.J., An asymptotic property of the number of spanning trees of double fixed stop loop networks,Appl. Math. JCU, 1997, 12B: 233.MathSciNetGoogle Scholar
  13. 13.
    Ablow, C. M., Brenner, J.L., Roots and canonical forms for circulant matrices,Trans. Amer. Math. Soc., 1963, 107: 360.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Imas, M., Itoh, M., Design to minimize diameter on building networks,IEEE Trans. Computers, 1981, C30: 439.CrossRefGoogle Scholar
  15. 15.
    Imas, M., Itoh, M., A design for directed graph with minimum diameter,IEEE Trans. Computers, 1983, C32: 782.CrossRefGoogle Scholar
  16. 16.
    Li, X.L., Zhang, F.J., On the number of spanning trees and Euler tours in generalized de Brujin graphs,Discrete Mathematics, 1991, 94: 189.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Science in China Press 1999

Authors and Affiliations

  • Zhang Fuji
    • 1
  • Yong Xuerong
    • 2
    • 3
  1. 1.Department of MathematicsXiamen UniversityXiamenChina
  2. 2.Department of Computer ScienceHong Kong University of Science & TechnologyHong KongChina
  3. 3.Department of MathematicsXinjiang UniversityUrumqiChina

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