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

Ahanjideh, Akbari, Fakharan and Trevisan proposed a conjecture in [Linear Algebra Appl. 632 (2022) 1--14] on the distribution of the Laplacian eigenvalues of graphs: for any connected graph of order $n$ with diameter $d\ge 2$ 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.