Spectral properties of the deformed Laplacian matrix of trees and H-join graphs

TL;DR

This paper explores the spectral characteristics of a family of deformed Laplacian matrices, providing new insights into eigenvalue localization for trees and explicit spectra for H-join graphs.

Contribution

It introduces a unified framework for analyzing the deformed Laplacian, with novel eigenvalue localization results for trees and explicit spectral formulas for H-join graphs.

Findings

Eigenvalue localization for trees

Explicit spectrum of H-join graphs

Unified analysis framework for deformed Laplacian

Abstract

This paper investigates spectral properties of the deformed Laplacian matrix, which merges the Laplacian and signless Laplacian matrices of a graph through a one-parameter family of matrices. We present general results on the eigenvalues of these matrices for simple undirected graphs. Additionally, we analyze the spectrum of the deformed Laplacian in the specific cases of trees and H-join graphs. For trees, we derive strong results on the localization of eigenvalues, while for H-join graphs, we explicitly compute the spectrum of the deformed Laplacian.