Eigenvalues of ${\mathbf{L}}_{\hat{\mathbf{I}}}\pm -$matrices under graph operations

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 $L_{\hat{I}}\pm$ as the convex linear combination: \[ L_α(G) = αD(G) + (α- 1)A(G), \] where $\hat{I}\pm \in [0,1]$. 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 $L_{\hat{I}}\pm$ under graph operations and within specific families of graphs.

13 pages, 2 figures