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

TL;DR

This work introduces the deformed Laplacian $M_G(s)=I-sA+s^2(D-I)$, situating it as a bridge between the Laplacian and signless Laplacian. It develops general spectral bounds and subgraph monotonicity results, and provides detailed eigenvalue localization for trees using the Diagonalize algorithm. The authors derive a complete spectrum description for graphs formed via $H$-join operations, with explicit formulas in key cases (e.g., $H=P_r$ and $H=C_r$), enabling closed-form eigenvalue expressions. The findings illuminate how global spectral structure arises from local regular blocks and inter-block connections, with potential implications for network dynamics and graph-based computations. Overall, the paper advances understanding of parametrized graph matrices and their spectra, especially in tree and join-graph constructions, through rigorous algebraic characterizations and constructive methods.

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.

Figures (3)

• Figure 1: Initial values for the graph $K_{1,n}$.
• Figure 2: Graphs $G$ (left) and $G'$(right).
• Figure 3: Graph $P_{n}$.

Theorems & Definitions (24)

• Definition 1
• Theorem 1
• Example 1
• Proposition 2
• Example 2
• Remark 1
• Lemma 1
• Theorem 3
• Corollary 1
• Theorem 4