Laplacian eigenvalue distribution and diameter of graphs

Let G be a connected graph on n vertices with diameter d. It is known that if 2≤d≤n−2, there are at most n−d Laplacian eigenvalues in the interval [n−d+2,n]. In this paper, we show that if 1≤d≤n−3, there are at most n−d+1 Laplacian eigenvalues in the interval [n−d+1,n]. Moreover, we try to identify the connected graphs on n vertices with diameter d, where 2≤d≤n−3, such that there are at most n−d Laplacian eigenvalues in the interval [n−d+1,n].