An Upper Bound on the Diameter of a Graph from Eigenvalues Associated with Its Laplacian

The authors give a new upper bound for the diameter $D( G )$ of a graph G in terms of the eigenvalues of the Laplacian of G. The bound is \[ D ( G ) ≤ {cosh- 1 ( n - 1 )}{cosh- 1 ( λ _n + λ _2 /λ _n - λ _2 )} + 1, \] where 0 ≤ λ ₂ ≤ ⋯ ≤ λ ₙ are the eigenvalues of the Laplacian of G and where is the floor function.