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Variable Neighborhood Search for Extremal Graphs. XI. Bounds on Algebraic Connectivity

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Graph Theory and Combinatorial Optimization

Abstract

The algebraic connectivity a(G) of a graph G = (V, E) is the second smallest eigenvalue of its Laplacian matrix. Using the AutoGraphiX (AGX) system, extremal graphs for algebraic connectivity of G in function of its order n = |V| and size m = |E| are studied. Several conjectures on the structure of those graphs, and implied bounds on the algebraic connectivity, are obtained. Some of them are proved, e.g., if G ≠ K n

$$a\left( G \right) \leqslant \left\lfloor { - 1 + \sqrt {1 + 2m} } \right\rfloor $$

which is sharp for all m ≥ 2.

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Authors
  1. Slim Belhaiza
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  2. Nair Maria Maia de Abreu
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  3. Pierre Hansen
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  4. Carla Silva Oliveira
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Editor information

Editors and Affiliations

  1. McGill University & GERAD, Montréal, Canada

    David Avis

  2. École Polytechnique de Montréal & GERAD, Montréal, Canada

    Alain Hertz

  3. Université du Québec a Montréal and GERAD, Montréal, Canada

    Odile Marcotte

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© 2005 Springer Science+Business Media, Inc.

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Belhaiza, S., de Abreu, N.M.M., Hansen, P., Oliveira, C.S. (2005). Variable Neighborhood Search for Extremal Graphs. XI. Bounds on Algebraic Connectivity. In: Avis, D., Hertz, A., Marcotte, O. (eds) Graph Theory and Combinatorial Optimization. Springer, Boston, MA. https://doi.org/10.1007/0-387-25592-3_1

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