Lower bounds for Laplacian spread and relations with invariant parameters revisited

Abstract

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*} ρ_{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.