Lower bounds for Laplacian spread and relations with invariant parameters revisited

Let $G=(V\left(G\right),E\left(G\right))$ be an $\left( n,m\right) $-graphand$X$anonemptypropersubsetof$V\left( G\right) $\mathrm{.}Let$X^{c}=V\left( G\right) \backslash X$\mathrm{.}Theedgedensityof$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 $$ 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.