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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.
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    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.

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    Bibliographic Info

    Article provided by Elsevier in its journal European Journal of Operational Research.

    Volume (Year): 191 (2008)
    Issue (Month): 3 (December)
    Pages: 661-676

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    Handle: RePEc:eee:ejores:v:191:y:2008:i:3:p:661-676

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    Web page: http://www.elsevier.com/locate/eor

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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. repec:spo:wpecon:info:hdl:2441/7346 is not listed on IDEAS
    2. König, Michael & Tessone, Claudio J. & Zenou, Yves, 2009. "A Dynamic Model of Network Formation with Strategic Interactions," CEPR Discussion Papers 7521, C.E.P.R. Discussion Papers.
    3. Michael Koenig & Claudio Tessone & Yves Zenou, 2012. "Nestedness in Networks: A Theoretical Model and Some Applications," Discussion Papers 11-003, Stanford Institute for Economic Policy Research.
    4. 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.
    5. 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.
    6. repec:spo:wpecon:info:hdl:2441/9935 is not listed on IDEAS
    7. 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.

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