Advertisement

Photonic Network Communications

, Volume 37, Issue 2, pp 187–194 | Cite as

On the information transmission delay of the lexicographic product of digraphs

Original Paper

Abstract

The maximum distance and average distance of a digraph play significant roles in analyzing efficiency of interconnection networks; it provides an efficient parameter to measure the transmission delay in the network. In this paper, we use the lexicographic product method to construct a larger digraph from several specified small digraphs. The digraph constructed by this way can contain the factor digraphs as subgraphs and preserve many desirable properties of the factor digraphs. By using the extremal values way of algebra, we investigate the distance parameters of the lexicographic product of digraphs and establish a formula for the vertex distance of the lexicographic product of digraphs.

Keywords

Digraph Network distance Time delay Lexicographic product Interconnection network 

Notes

Funding

The project was supported by the National Natural Science Foundation of China (No. 11551002).

References

  1. 1.
    Sheldon, B.A., Balakrishnan, K.: A group-theoretic model for symmetric interconnection networks. IEEE Trans. Comput. 38(4), 555–566 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Feder, T.: Stable Networks and Product Graphs. Memoirs of the American Mathematical Society, vol. 8, pp. 1–116. Stanford University (1995)Google Scholar
  3. 3.
    Fredman, M.L.: New bounds on the complexity of the shortest path problem. SIAM J. Comput. 5, 83–89 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Xu, J.M.: Topological Structure and Analysis of Interconnection Networks. Kluwer, Dordrecht (2001)CrossRefzbMATHGoogle Scholar
  5. 5.
    Kumar, M., Mao, Y.H., Wang, Y.H., Qiu, T.R., Yang, C., Zhang, W.P.: Fuzzy theoretic approach to signals and systems: static systems. Inf. Sci. 418, 668–702 (2017)CrossRefGoogle Scholar
  6. 6.
    Zhang, W.P., Yang, J.Z., Fang, Y.L., Chen, H.Y., Mao, Y.H., Kumar, M.: Analytical fuzzy approach to biological data analysis. Saudi J. Biol. Sci. 24, 563–573 (2017)CrossRefGoogle Scholar
  7. 7.
    Soares, J.: Maximum distance of regular digraphs. J. Graph Theory 16, 437–450 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Knyazey, A.V.: Diameters of pseudosymmetric graphs. Mathematics Notes. 41, 473–482 (1987)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dankelmann, P.: The diameter of directed graphs. J. Comb. Theory 94, 183–186 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Zhou, T., Xu, J.M., Liu, J.: On diameter and average distance of graphs. OR Trans. 8, 33–38 (2004)Google Scholar
  11. 11.
    Entringer, R.C., Jackson, D.E., Slater, P.J.: Geodetic connectivity of graphs. IEEE Trans. Circuits Syst. 24, 460–463 (1988)CrossRefzbMATHGoogle Scholar
  12. 12.
    Ng, C.P., Teh, H.H.: On finite graphs of diameter 2. Nanta Math. 67, 72–75 (1966)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Plesnik, J.: On the sum of all distances in a graph or digraph. J. Graph Theory 8, 1–21 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kouider, M., Winkler, P.: Mean distance and minimum degree. J. Graph Theory 25, 95–99 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chung, F.R.K.: The average distance and the independence number. J. Graph Theory 12, 229–235 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Yegnanarayanan, V., Thiripurasundari, P.R.: On some graph operations and related applications. Electron. Notes Discrete Math. 33, 123–130 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Harary, F., Hayes, J., Wu, H.J.: A survey of the theory of hypercube graphs. Comput. Math Appl. 15, 277–289 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Thenmozhi, M., Sarath Chand, G.: Forecasting stock returns based on information transmission across global markets using support vector machines. Neural Comput. Appl. 27(4), 805–824 (2016)CrossRefGoogle Scholar
  19. 19.
    Chung, M.: Effective near advertisement transmission method for smart-devices using inaudible high-frequencies. Multimed. Tools Appl. 75(10), 5871–5886 (2016)CrossRefGoogle Scholar
  20. 20.
    Wang, W., Li, F., Lu, H.L., Xu, Z.B.: Graphs determined by their generalized characteristic polynomials. Linear Algebra Appl. 434, 1378–1387 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Bondy, J.A., Murty, U.S.R.: Graph Theory with Application. Macmillan Press, London (1976)CrossRefzbMATHGoogle Scholar
  22. 22.
    Li, F., Wang, W., Xu, Z.B., Zhao, H.X.: Some results on the lexicographic product of vertex-transitive graphs. Appl. Math. Lett. 24, 1924–1926 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Chong, K., Yoo, S.: Neural network prediction model for a real-time data transmission. Neural Comput. Appl. 15(3–4), 373–382 (2006)CrossRefGoogle Scholar
  24. 24.
    Bashkow, T.R., Sullivan, H.: A large scale homogeneous full distributed parallel machine. In: Proceedings of 4th Annual Symposium on Computer Architecture, pp. 105–117 (1977)Google Scholar
  25. 25.
    Day, K., Al-Ayyoub, A.: Minimal fault diameter for highly resilient product networks. IEEE Trans. Parallel Distrib. Syst. 11, 926–930 (2000)CrossRefGoogle Scholar
  26. 26.
    Georges, P.J., Mauro, D.W., Stein, M.I.: Labeling products of complete graphs with a condition at distance two. SIAM J. Discrete Math. 14, 28–35 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Fisher, M.J., Isaak, G.: Distinguishing coloring of Cartesian products of complete graphs. Discrete Math. 308, 2240–2246 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Xu, J.M.: Connectivity of Cartesian product digraphs and fault-tolerant routing of generalized hypercube. Appl. Math. 13, 179–187 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Klavzar, S.: On the canonical metric representation, average distance, and partial Hamming graphs. Eur. J. Comb. 27, 68–73 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Balbuena, C., Garcia, P., Marcote, X.: Reliability of interconnection networks modeled by a product of graphs. Networks 48, 114–120 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Chung, F.K., Coffman, E.G., Reimon, M.I.: The forwarding index of communication networks. IEEE Trans. Inf. Theory 33, 224–232 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Xu, Z.B., Li, F., Zhao, H.X.: Vertex forwarding indices of the lexicographic product of graphs. Sci. Sin. Inf. 44, 482–497 (2014). (in Chinese) Google Scholar
  33. 33.
    Chang, C.P., Sung, T.Y., Hsu, L.H.: Edge congestion and topological properties of crossed cubes. IEEE Trans. Parallel Distrib. Syst. 11, 64–79 (2000)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Computer ScienceQinghai Normal UniversityXiningPeople’s Republic of China

Personalised recommendations