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