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

Slim Belhaiza, Nair Maria Maia Abreu, Pierre Hansen and Carla Silva Oliveira

Chapter Chapter 1 in Graph Theory and Combinatorial Optimization, 2005, pp 1-16 from Springer

Abstract: 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.

Keywords: Graph Theory; Connected Graph; Variable Neighborhood Search; Laplacian Matrix; Extremal Graph (search for similar items in EconPapers)
Date: 2005
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DOI: 10.1007/0-387-25592-3_1

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