Cluster Computing

, Volume 22, Supplement 3, pp 5691–5701 | Cite as

The computation on \(\alpha \)-connectedness index of uncertain graph

  • Xiulian GaoEmail author
  • Changyou Guo
  • Xiuling Yin
  • Xuedou Yu


This paper investigates the problem about \(\alpha \)-connectedness index of uncertain graph. Since uncertainties are inherent in graph data, traditional concepts and algorithms in classic graph theory are not applicable to uncertain graph. Hence, this paper firstly proposes the concepts of \(\alpha \)-connected graph and \(\alpha \)-connectedness index of uncertain graph and shows the computing method. Based on these definitions, this paper gives and verifies the relationship between connectedness index and \(\alpha \)-connectedness index of uncertain graph.


Uncertain graph \(\alpha \)-Connectedness index Uncertainty theory 



This work was supported by National Natural Science Foundation of China Grant No. 11501082, Natural Science Foundation of Shandong Province Nos. ZR2015AL016, ZR2016FM37, ZR2015FL006 and ZR2013GL001 and Dezhou soft science research plan project NO.43 (2013).

Compliance with ethical standards

Conflict of interest

All authors have no conflict of interest in this paper.

Research involving human and animal participants

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. 1.
    Broder, A., Kumar, R., Maghoul, F., et al.: Graph structure in the web. Comput. Netw. 33(1), 309–320 (2000)Google Scholar
  2. 2.
    Ellison, N.B.: Social network sites: definition history, and scholarship. J. Comput. Mediat. Commun. 13(1), 210–230 (2007)MathSciNetGoogle Scholar
  3. 3.
    Rual, J.F., Venkatesan, K., Hao, T., et al.: Towards a proteome-scale map of the human protein- protein interaction network. Nature 437(7062), 1173–1178 (2005)Google Scholar
  4. 4.
    Erdös, P., Rényi, A.: On the strength of connectedness of a random graph. Acta Math. Hungar. 12, 261–267 (1961)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Erdös, P.: Graph theory and probability. Can. J. Math. 11, 34–38 (1959)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Erdös, P.: Graph theory and probability II. Can. J. Math. 13, 346–352 (1961)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bhattacharya, P.: Some remarks on fuzzy graphs. Pattern Recognit. Lett. 6, 197–302 (1987)zbMATHGoogle Scholar
  8. 8.
    Bhutani, K.R., Battou, A.: On M-strong fuzzy graphs. Inf. Sci. 155, 103–109 (2003)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Mathew, S., Sunitha, M.S.: Types of arcs in a fuzzy graph. Inf. Sci. 179, 1760–1768 (2009)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Mordeson, J.N., Nair, P.S.: Fuzzy Graphs and Fuzzy Hyper-Graphs. Physica-Verlag, Heidelberg (2000)zbMATHGoogle Scholar
  11. 11.
    Liu, B.: Uncertainty Theory, 2nd edn. Springer, Berlin (2007)zbMATHGoogle Scholar
  12. 12.
    Liu, B.: Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty. Springer, Berlin (2010)Google Scholar
  13. 13.
    Gao, X.L., Gao, Y.: Connectedness index of uncertainty graphs. Int. J. Uncertain. Fuzziness Knowl. 21(1), 127–137 (2013)zbMATHGoogle Scholar
  14. 14.
    Gao, X.L.: Regularity index of uncertain graph. J. Intell. Fuzzy Syst. 27(4), 1671–1678 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Gao, X.L.: Tree index of uncertain graphs. Soft Comput. 20(4), 1449–1458 (2016)zbMATHGoogle Scholar
  16. 16.
    Liu, B.: Fuzzy process, hybrid process and uncertain process. J. Uncertain Syst. 2(1), 3–16 (2008)Google Scholar
  17. 17.
    Liu, B.: Theory and Practice of Uncertain Programming, 2nd edn. Springer, Berlin (2009)zbMATHGoogle Scholar
  18. 18.
    Gao, X.: Some properties of continuous uncertain measure. Int. J. Uncertain. Fuzziness Knowl. Syst. 17(3), 419–426 (2009)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Gao, X., Gao, Y., Ralescu, D.: On Liu’s inference rule for uncertain systems. Int. J. Uncertain. Fuzziness Knowl. Syst. 18(1), 1–11 (2010)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Gao, Y.: Shortest path problem with uncertain arc lengths. Comput. Math. Appl. 62(6), 2591–2600 (2011)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Zhang, Z.Q., Ralescu, D., Liu, W.Q.: Valuation of interest rate ceiling and floor in uncertain financial market. Fuzzy Optim. Decis. Mak. 15(2), 139–154 (2016)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Sun, Y., Yao, K., Dong, J.: Asian option pricing problems of uncertain mean-reverting stock model. Soft Comput. 2, 1–10 (2017)zbMATHGoogle Scholar
  23. 23.
    Shi, G., Zhang, Z.Q., Sheng, Y.H.: Valuation of stock loan under uncertain mean-reverting stock model. J. Intell. Fuzzy Syst. 33(3), 1355–1361 (2017)zbMATHGoogle Scholar
  24. 24.
    Zhang, Z.Q., Liu, W.Q., Zhang, X.D.: Valuation of convertible bond under uncertain mean-reverting stock model. J. Ambient Intell. Humaniz. Comput. 8(5), 641–650 (2017)Google Scholar
  25. 25.
    Liu, Z.B., Zhao, R.Q., Liu, X.Y., Chen, L.: Contract designing for a supply chain with uncertain information based on confidence level. Appl. Soft Comput. 56(7), 617–631 (2017)Google Scholar
  26. 26.
    Liu, B.: Uncertainty Theory, 5th ed.
  27. 27.
    Liu, B.: Some research problems in uncertainty theory. J. Uncertain Syst. 3(1), 3–10 (2009)Google Scholar
  28. 28.
    Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. Macmillan, New York (1976)zbMATHGoogle Scholar
  29. 29.
    Guo, C.Y., Zheng, X.F., Gao, X.L.: Credible nearest neighbor query in uncertain network. J. Electron. Inf. Technol. 38(4), 811–818 (2016)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  • Xiulian Gao
    • 1
    • 4
    Email author
  • Changyou Guo
    • 2
  • Xiuling Yin
    • 1
  • Xuedou Yu
    • 3
  1. 1.College of Mathematical SciencesDezhou UniversityDezhouChina
  2. 2.School of Information ManagementDezhou UniversityDezhouChina
  3. 3.Department of Science and TechnologyDezhou UniversityDezhouChina
  4. 4.Institute of Mathematics EducationDezhou UniversityDezhouChina

Personalised recommendations