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Variable neighborhood search for extremal graphs. 16. Some conjectures related to the largest eigenvalue of a graph

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Listed:
  • Aouchiche, M.
  • Bell, F.K.
  • Cvetkovic, D.
  • Hansen, P.
  • Rowlinson, P.
  • Simic, S.K.
  • Stevanovic, D.

Abstract

We consider four conjectures related to the largest eigenvalue of (the adjacency matrix of) a graph (i.e., to the index of the graph). Three of them have been formulated after some experiments with the programming system AutoGraphiX, designed for finding extremal graphs with respect to given properties by the use of variable neighborhood search. The conjectures are related to the maximal value of the irregularity and spectral spread in n-vertex graphs, to a Nordhaus-Gaddum type upper bound for the index, and to the maximal value of the index for graphs with given numbers of vertices and edges. None of the conjectures has been resolved so far. We present partial results and provide some indications that the conjectures are very hard.

Suggested Citation

  • Aouchiche, M. & Bell, F.K. & Cvetkovic, D. & Hansen, P. & Rowlinson, P. & Simic, S.K. & Stevanovic, D., 2008. "Variable neighborhood search for extremal graphs. 16. Some conjectures related to the largest eigenvalue of a graph," European Journal of Operational Research, Elsevier, vol. 191(3), pages 661-676, December.
    • Handle: RePEc:eee:ejores:v:191:y:2008:i:3:p:661-676
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    References listed on IDEAS

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    1. Hansen, Pierre & Mladenovic, Nenad, 2001. "Variable neighborhood search: Principles and applications," European Journal of Operational Research, Elsevier, vol. 130(3), pages 449-467, May.
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    Cited by:

    1. Michael D. König & Claudio J. Tessone & Yves Zenou, 2010. "From Assortative To Dissortative Networks: The Role Of Capacity Constraints," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 13(04), pages 483-499.
    2. , D. & Tessone, Claudio J. & ,, 2014. "Nestedness in networks: A theoretical model and some applications," Theoretical Economics, Econometric Society, vol. 9(3), September.
    3. Mauro Napoletano & Stefano Battiston & Michael D König & Frank Schweitzer, 2008. "The efficiency and evolution of R&D Networks," Working Papers hal-01066189, HAL.
    4. repec:spo:wpmain:info:hdl:2441/9935 is not listed on IDEAS
    5. repec:hal:spmain:info:hdl:2441/9935 is not listed on IDEAS
    6. Fernando, Garcia Alvarado & Antoine, Mandel, 2022. "The network structure of global tax evasion evidence from the Panama papers," Journal of Economic Behavior & Organization, Elsevier, vol. 197(C), pages 660-684.
    7. repec:hal:wpspec:info:hdl:2441/9935 is not listed on IDEAS
    8. König, Michael D. & Battiston, Stefano & Napoletano, Mauro & Schweitzer, Frank, 2012. "The efficiency and stability of R&D networks," Games and Economic Behavior, Elsevier, vol. 75(2), pages 694-713.
    9. repec:spo:wpecon:info:hdl:2441/9935 is not listed on IDEAS
    10. Réti, Tamás, 2019. "On some properties of graph irregularity indices with a particular regard to the σ-index," Applied Mathematics and Computation, Elsevier, vol. 344, pages 107-115.
    11. M. Koenig & Claudio J. Tessone & Yves Zenou, "undated". "A Dynamic Model of Network Formation with Strategic Interactions," Working Papers CCSS-09-006, ETH Zurich, Chair of Systems Design.
    12. repec:hal:wpspec:info:hdl:2441/7346 is not listed on IDEAS
    13. repec:spo:wpmain:info:hdl:2441/7346 is not listed on IDEAS
    14. repec:hal:spmain:info:hdl:2441/7346 is not listed on IDEAS
    15. repec:spo:wpecon:info:hdl:2441/7346 is not listed on IDEAS
    16. Pierre Hansen & Nenad Mladenović & José Moreno Pérez, 2010. "Variable neighbourhood search: methods and applications," Annals of Operations Research, Springer, vol. 175(1), pages 367-407, March.

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