Proof of a conjecture on distribution of Laplacian eigenvalues and diameter, and beyond

Ahanjideh, Akbari, Fakharan and Trevisan proposed a conjecture on the distribution of the Laplacian eigenvalues of graphs: for any connected graph of order n with diameter d≥2 that is not a path, the number of Laplacian eigenvalues in the interval [n−d+2,n] is at most n−d. We show that the conjecture is true, and give a complete characterization of graphs for which the conjectured bound is attained. This establishes an interesting relation between the spectral and classical parameters.