Eigenvalues of 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.