  Let $G=\left( V\left( G\right) ,E\left( G\right) \right) $ be an $\left(
n,m\right) $-graph and $X$ a nonempty proper subset of $V\left( G\right) $. Let
$X^{c}=V\left( G\right) \backslash X$.\ The edge density of $X$ in $G$ is given
by \begin{equation*} \rho _{G}\left( X\right) =\frac{n\left\vert E_{X}\left(
G\right) \right\vert }{\left\vert X\right\vert \left\vert X^{c}\right\vert },
\end{equation*} where $E_{X}\left( G\right) \ $ is the set of edges in $G$ with
one end in $% X $ and the other in $X^{c}$. The Laplacian spread of a graph is
the difference between the greatest Laplacian eigenvalue and the algebraic
connectivity. In this paper, we use the edge density of some nonempty proper
subsets of vertices in $G$ to establish new lower bounds for the Laplacian
spread. Also, using some known numerical inequalities some lower bounds for the
Laplacian spread of a graph with a prescribed degree sequence are presented.
