Eigenvalues of $\boldsymbol{L_\alpha}-$matrices under graph operations

TL;DR

This paper studies the eigenvalues of a family of matrices called $L_eta$ matrices, which are convex combinations of degree and adjacency matrices, under various graph operations and specific graph families.

Contribution

It provides a unified spectral analysis of $L_eta$ matrices, extending understanding of their eigenvalues in different graph contexts.

Findings

Eigenvalues of $L_eta$ matrices are characterized under certain graph operations.

Spectral properties of $L_eta$ matrices are analyzed for specific graph families.

Abstract

Let $G$ be a simple graph, $A(G)$ its adjacency matrix, and $D(G)$ its diagonal degree matrix. In 2022, \citeauthor{Wang2020} (\cite{Wang2020}) defined the family of matrices $L_\alpha$ as the convex linear combination: \[ L_\alpha(G) = \alpha D(G) + (\alpha - 1)A(G), \] where $\alpha \in [0,1]$. The study of the spectrum of this family of matrices may provide a unified framework for analyzing the spectra of the adjacency, degree, and Laplacian matrices ($D(G) - A(G)$). In this work, we investigate the spectrum of $L_\alpha$ under graph operations and within specific families of graphs.