  Let $G$ be a connected graph on $n$ vertices with diameter $d$. It is known
that if $2\le d\le n-2$, there are at most $n-d$ Laplacian eigenvalues in the
interval $[n-d+2, n]$. In this paper, we show that if $1\le d\le n-3$, 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 $2\le d\le n-3$, such that there are at most $n-d$ Laplacian
eigenvalues in the interval $[n-d+1, n]$.
