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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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  • 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. Mauro Napoletano & Stefano Battiston & Michael D König & Frank Schweitzer, 2008. "The efficiency and evolution of R&D Networks," Sciences Po publications 2008-31, Sciences Po.
    2. repec:wsi:acsxxx:v:13:y:2010:i:04:n:s0219525910002700 is not listed on IDEAS
    3. 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.
    4. Zenou, Yves & König, Michael D. & Tessone, Claudio J., 2014. "Nestedness in networks: A theoretical model and some applications," Theoretical Economics, Econometric Society, vol. 9(3), September.
    5. 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.
    6. 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.
    7. repec:spo:wpecon:info:hdl:2441/9935 is not listed on IDEAS
    8. repec:spo:wpecon:info:hdl:2441/7346 is not listed on IDEAS

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