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

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.