Distribution of deformed Laplacian limit points

2025/12/04 by Elismar R. Oliveira, Oliveira, Elismar R., Jonas Szutkoski +3
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2512.04836

openalex publication_date 2025/12/04 · openalex created_date 2025/12/06 · openalex updated_date 2026/07/28

Abstract

This paper investigates limit points of the deformed Laplacian matrix, which merges the Laplacian and signless Laplacian matrices of a graph through a quadractic one-parameter family of matrices. First, we show that any value greater or equal to 1 is a deformed Laplacian limit point (for different values of the parameter s) using a simple family of trees. Second, we define (T_k)_{k ∈ ℕ} the Shearer's sequence of caterpillars for λ>1 and we present a convergence criterion based on Shearer's approach. Our main result is that for any fixed value λ₀>1 there exists a unique value 0

Citations

• Spectral properties of the deformed Laplacian matrix of trees and H-join graphs
• Limit points of (signless) Laplacian spectral radii of linear trees
• Applications of rational difference equations to spectral graph theory: expanded version
• Merging the A- and Q-spectral theories
• The Deformed Consensus Protocol: Extended Version

Related

• Limit points of (signless) Laplacian spectral radii of linear trees
• Limit points of A_α-matrices of graphs
• Spectral properties of the deformed Laplacian matrix of trees and H-join graphs
• Fiedler Vectors and Elongation of Graphs: A Threshold Phenomenon on a Particular Class of Trees
• Perturbation of Fiedler vector: interest for graph measures and shape analysis