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arXiv · 2605.10944

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

Gabriel Roberto Silva de Lima·Carla Silva Oliveira·João Domingos Gomes da Silva Junior
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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.

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Gabriel Roberto Silva de Lima, Carla Silva Oliveira, João Domingos Gomes da Silva Junior. 2026-04-28. Eigenvalues of $\boldsymbol{L_\alpha}-$matrices under graph operations. https://arxiv.org/abs/2605.10944

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