Lower bounds for Laplacian spread and relations with invariant parameters revisited

TL;DR

This paper establishes new lower bounds for the Laplacian spread of graphs by analyzing edge densities of vertex subsets and explores relations with invariant parameters, enhancing understanding of spectral graph properties.

Contribution

It introduces novel lower bounds for Laplacian spread based on subset edge densities and relates these bounds to invariant graph parameters, extending spectral graph theory insights.

Findings

New lower bounds for Laplacian spread using subset edge densities

Bounds for Laplacian spread with prescribed degree sequences

Application of numerical inequalities to spectral bounds

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*} \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.