Bounds on $A_{\alpha }$-eigenvalues using graph invariants

In 2017, Nikiforov introduced the concept of the $A_{\alpha }$-matrix, as a linear convex combination of the adjacency matrix and the degree diagonal matrix of a graph. This matrix has attracted increasing attention in recent years, as it serves as a unifying structure that combines the adjacency matrix and the signless Laplacian matrix. In this paper, we present some bounds for the largest and smallest eigenvalue of $A_{\alpha }$-matrix involving invariants associated to graphs.