  For a connected graph $G$ of order $n$, let $Diag(Tr)$ be the diagonal matrix
of vertex transmissions and $D(G)$ be the distance matrix of $G$. The distance
Laplacian matrix of $G$ is defined as $D^L(G)=Diag(Tr)-D(G)$ and the
eigenvalues of $D^{L}(G)$ are called the distance Laplacian eigenvalues of $G$.
Let $\partial_{1}^{L}(G)\geq \partial_{2}^{L}(G)\geq \dots \geq
\partial_{n}^{L}(G)$ be the distance Laplacian eigenvalues of $G$. Given an
interval $I$, let $m_{D^{L} (G)} I$ (or simply $m_{D^{L} } I$) be the number of
distance Laplacian eigenvalues of $G$ which lie in the interval $I$. For a
prescribed interval $I$, we determine $m_{D^{L} }I$ in terms of independence
number $\alpha(G)$, chromatic number $\chi$, number of pendant vertices and
diameter $d$ of the graph $G$. In particular, we prove that $m_{D^{L}(G)
}[n,n+2)\leq \chi-1$, ~$m_{D^{L}(G) }[n,n+\alpha(G))\leq n-\alpha(G)$ and we
show that the inequalities are sharp. We also show that $m_{D^{L} (G )}\bigg(
n,n+\left\lceil\frac{n}{\chi}\right\rceil\bigg)\leq n-
\left\lceil\frac{n}{\chi}\right\rceil-C_{\overline{G}}+1 $, where
$C_{\overline{G}}$ is the number of components in $\overline{G}$, and discuss
some cases where the bound is best possible. In addition, we prove that
$m_{D^{L} (G )}[n,n+p)\leq n-p$, where $p\geq 1$ is the number of pendant
vertices. Also, we characterize graphs of diameter $d\leq 2$ which satisfy
$m_{D^{L}(G) } (2n-1,2n )= \alpha(G)-1=\frac{n}{2}-1$. At the end, we propose
some problems of interest.
