  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.
