Limit points of (signless) Laplacian spectral radii of linear trees

TL;DR

This paper investigates the limit points of Laplacian spectral radii of linear trees, adapting and extending existing methods to identify potential density regions of these spectral limits.

Contribution

It adapts Shearer's method to Laplacian matrices, reveals its limitations, and generalizes it to linear trees to better understand the distribution of spectral limit points.

Findings

Identifies an interval where Shearer's method fails to produce limit points.

Extends the method to linear trees, generating more limit points.

Provides tools for proving the density of Laplacian limit points in a specified range.

Abstract

We study limit points of the spectral radii of Laplacian matrices of graphs. We adapted the method used by J. B. Shearer in 1989, devised to prove the density of adjacency limit points of caterpillars, to Laplacian limit points. We show that this fails, in the sense that there is an interval for which the method produces no limit points. Then we generalize the method to Laplacian limit points of linear trees and prove that it generates a larger set of limit points. The results of this manuscript may provide important tools for proving the density of Laplacian limit points in $[4.38+, \infty)$.