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arXiv · 2402.15470

Some results involving the $A_\alpha$-eigenvalues for graphs and line graphs

Joao Domingos Gomes da Silva Junior·Carla Silva Oliveira·Liliana Manuela Gaspar C. da Costa
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Abstract

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.

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Joao Domingos Gomes da Silva Junior, Carla Silva Oliveira, Liliana Manuela Gaspar C. da Costa. 2024-02-23. Some results involving the $A_\alpha$-eigenvalues for graphs and line graphs. https://arxiv.org/abs/2402.15470

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