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.