Some results involving the A_α-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_α-matrix of G, A_α(G), as a linear convex combination of $A(G)$ and $D(G)$, the following way, A_α(G):=α A(G)+(1-α)D(G), where α∈[0,1]. In this paper, we present some bounds for the eigenvalues of A_α(G) and for the largest and smallest eigenvalues of A_α(l(G)). Extremal graphs attaining some of these bounds are characterized.