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

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.