The paper proves Kolokolnikov's conjecture: every graph on n vertices with at most 2n-4 edges has algebraic connectivity at most 2, so K_{2,n-2} is a maximizer.
Kolokolnikov, Maximizing algebraic connectivity for certain families of graphs, Linear Algebra Appl
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On a conjecture of Kolokolnikov on algebraic connectivity
The paper proves Kolokolnikov's conjecture: every graph on n vertices with at most 2n-4 edges has algebraic connectivity at most 2, so K_{2,n-2} is a maximizer.