Some results involving the $A_\alpha$-eigenvalues for graphs and line graphs
<|reference_start|>Some results involving the $A_\alpha$-eigenvalues for graphs and line graphs: Let $G$ be a simple graph with adjacency matrix $A(G)$, signless Laplacian matrix $Q(G)$, degree diagonal matrix $D(G)$ and let $l(G)$ be the line graph of $G$. In 2017, Nikiforov defined the $A_\alpha$-matrix of $G$, $A_\alpha(G)$, as a linear convex combination of $A(G)$ and $D(G)$, the following way, $A_\alpha(G):=\alpha A(G)+(1-\alpha)D(G),$ where $\alpha\in[0,1]$. In this paper, we present some bounds for the eigenvalues of $A_\alpha(G)$ and for the largest and smallest eigenvalues of $A_\alpha(l(G))$. Extremal graphs attaining some of these bounds are characterized.<|reference_end|>
@article{junior2024some,
    title={Some results involving the $A_\alpha$-eigenvalues for graphs and line
  graphs},
    author={Joao Domingos Gomes da Silva Junior, Carla Silva Oliveira, Liliana
  Manuela Gaspar C. da Costa},
    journal={arXiv preprint arXiv:2402.15470},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.15470},
    primaryClass={cs.DM math.CO}
}