  For a simple connected graph $G$ of order $n$ having distance Laplacian
eigenvalues $ \rho^{L}_{1}\geq \rho^{L}_{2}\geq \cdots \geq \rho^{L}_{n}$, the
distance Laplacian energy $DLE(G)$ is defined as
$DLE(G)=\sum_{i=1}^{n}\left|\rho^{L}_i-\frac{2 W(G)}{n}\right|$, where $W(G)$
is the Wiener index of $G$. We obtain a relationship between the Laplacian
energy and distance Laplacian energy for graphs with diameter 2. We obtain
lower bounds for the distance Laplacian energy $DLE(G)$ in terms of the order
$n$, the Wiener index $W(G)$, independence number, vertex connectivity number
and other given parameters. We characterize the extremal graphs attaining these
bounds. We show that the complete bipartite graph has the minimum distance
Laplacian energy among all connected bipartite graphs and complete split graph
has the minimum distance Laplacian energy among all connected graphs with given
independence number. Further, we obtain the distance Laplacian spectrum of the
join of a graph with the union of two other graphs. We show that the graph
$K_{k}\bigtriangledown(K_{t}\cup K_{n-k-t}), 1\leq t \leq
\lfloor\frac{n-k}{2}\rfloor $, has the minimum distance Laplacian energy among
all connected graphs with vertex connectivity $k$. We conclude this paper with
a discussion on trace norm of a matrix and the importance of our results in the
theory of trace norm of the matrix $D^L(G)-\frac{2W(G)}{n}I_n$.
