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A Toughness Condition for Fractional (k, m)-deleted Graphs Revisited

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Abstract

In computer networks, toughness is an important parameter which is used to measure the vulnerability of the network. Zhou et al. obtains a toughness condition for a graph to be fractional (k, m)-deleted and presents an example to show the sharpness of the toughness bound. In this paper, we remark that the previous example does not work and inspired by this fact, we present a new toughness condition for fractional (k, m)-deleted graphs improving the existing one. Finally, we state an open problem.

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References

  1. Ashwin, P., Postlethwaite, C.: On designing heteroclinic networks from graphs. Physica D, 265, 26–39 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Basavanagoud, B., Desai, V. R., Patil, S.: (β, α)-Connectivity index of graphs. Applied Mathematics and Nonlinear Sciences, 2, 21–30 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bondy, J. A., Mutry, U. S. R.: Graph Theory, Springer, Berlin, 2008

    Book  Google Scholar 

  4. Chvátal, V.: Tough graphs and hamiltonian circuits. Discrete Math., 5, 215–228 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  5. Crouzeilles, R., Lorini, M. L., Grelle, C. E. D.: Applying graph theory to design networks of protected areas: using inter-patch distance for regional conservation planning. Nat. Conservacao, 9, 219–224 (2011)

    Article  Google Scholar 

  6. de Araujo, D. R. B., Martins, J. F., Bastos, C. J. A.: New graph model to design optical networks. IEEE Commun. Lett., 19, 2130–2133 (2015)

    Article  Google Scholar 

  7. Fardad, M., Lin, F., Jovanovic, M. R.: Design of optimal sparse interconnection graphs for synchronization of oscillator networks. IEEE T. Automat. Contr., 59, 2457–2462 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gao, W., Guo, Y., Wang, K. Y.: Ontology algorithm using singular value decomposition and applied in multidisciplinary. Cluster Comput., 19, 2201–2210 (2016)

    Article  Google Scholar 

  9. Gao, W., Wang, W. F.: New isolated toughness condition for fractional (g, f, n)-critical graphs. Colloq. Math., 147, 55–66 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gao, W., Wang, W. F.: The eccentric connectivity polynomial of two classes of nanotubes. Chaos Soliton & Fractals, 89, 290–294 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gao, W., Wang, W. F.: The fifth geometric arithmetic index of bridge graph and carbon nanocones. J. Differ. Equ. Appl., 23(1–2), 100–109 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gao, W., Wang, W. F.: Degree conditions for fractional (k, m)-deleted graphs. Ars Combin., 113A, 273–285 (2014)

    MathSciNet  MATH  Google Scholar 

  13. Gao, W., Wang, W. F.: A neighborhood union condition for fractional (k, m)-deleted graphs. Ars Combin., 113A, 225–233 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Guirao, J. L. G., Luo, A. C. J.: New trends in nonlinear dynamics and chaoticity. Nonlinear Dynam., 84, 1–2 (2016)

    Article  MathSciNet  Google Scholar 

  15. Haenggi, M., Andrews, J., Baccelli, F., et al.: Stochastic geometry and random graphs for the analysis and design of wireless networks. IEEE J. Sel. Area. Comm., 27, 1025–1028 (2009)

    Article  Google Scholar 

  16. Jin, J. H.: Multiple solutions of the Kirchhoff-type problem in R N. Appl. Math. Nonl. Sc., 1, 229–238 (2016)

    MATH  Google Scholar 

  17. Lanzeni, S., Messina, E., Archetti, F.: Graph models and mathematical programming in biochemical network analysis and metabolic engineering design. Comput. Math. Appl., 55, 970–983 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu, G. Z., Zhang, L. J.: Toughness and the existence of fractional k-factors of graphs. Discrete Math., 308, 1741–1748 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pishvaee, M. S., Rabbani, M.: A graph theoretic-based heuristic algorithm for responsive supply chain network design with direct and indirect shipment. Adv. Eng. Softw., 42, 57–63 (2011)

    Article  MATH  Google Scholar 

  20. Possani, V. N., Callegaro, V., Reis, A. I., et al.: Graph-based transistor network generation method for supergate design. IEEE T. VLSI Syst., 24, 692–705 (2016)

    Article  Google Scholar 

  21. Rahimi, M., Haghighi, A.: A graph portioning approach for hydraulic analysis-design of looped pipe networks. Water Resour. Manag., 29, 5339–5352 (2015)

    Article  Google Scholar 

  22. Rizzelli, G., Tornatore, M., Maier, G., et al.: Impairment-aware design of translucent DWDM networks based on the k-path connectivity graph. J. Opt. Commun. Netw., 4, 356–365 (2012)

    Article  Google Scholar 

  23. Zhou, S.: A neighborhood condition for graphs to be fractional (k, m)-deleted graphs. Glasg. Math. J., 52(1), (2010) 33–40.

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhou, S. Z., Sun, Z. R., Ye, H.: A toughness condition for fractional (k, m)-deleted graphs. Inform. Process. Lett., 113, 255–259 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhou, S. Z., Xu, L., Xu, Y.: A sufficient condition for the existence of a k-factor excluding a given r-factor. Applied Mathematics and Nonlinear Sciences, 2, 13–20 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhou, S., Yang, F., Sun, Z. R.: A neighborhood condition for fractional ID-[a, b]-factor-critical graphs. Discussiones Mathematicae Graph Theory, 36(2), 409–418 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, P. R. China

    Wei Gao & Yao Jun Chen

  2. School of Information Science and Technology, Yunnan Normal University, Kunming, 650500, P. R. China

    Wei Gao

  3. Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Hospital de Marina, 30203, Cartagena, Región de Murcia, Spain

    Juan L. G. Guirao

Authors
  1. Wei Gao
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  2. Juan L. G. Guirao
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  3. Yao Jun Chen
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Corresponding author

Correspondence to Juan L. G. Guirao.

Additional information

This work has been partially supported by MINECO (Grant No. MTM2014-51891-P and Fundación Séneca de la Región de Murcia 19219/PI/14) and National Science Foundation of China (Grant No. 11401519)

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Gao, W., Guirao, J.L.G. & Chen, Y.J. A Toughness Condition for Fractional (k, m)-deleted Graphs Revisited. Acta. Math. Sin.-English Ser. 35, 1227–1237 (2019). https://doi.org/10.1007/s10114-019-8169-z

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  • DOI: https://doi.org/10.1007/s10114-019-8169-z

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