A Universal upper bound on Graph Diameter based on Laplacian Eigenvalues
  We prove that the diameter of any unweighted connected graph G is O(k log n/lambda_k), for any k>= 2. Here, lambda_k is the k smallest eigenvalue of the normalized laplacian of G. This solves a problem posed by Gil Kalai. 
Shayan Oveis Gharan and Luca Trevisan
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