Bounds on Aα-eigenvalues using graph invariants

In 2017, Nikiforov introduced the Aα-matrix of a graph G, defined by the convex linear combination Aα(G) = αD(G) + (1 − α)A(G), forα ∈ [0, 1], where A(G) is the adjacency matrix and D(G) is the diagonal matrix of vertex degrees of G. This family provides a unified framework that includes the adjacency matrix, the degree matrix and, up to a scalar factor, the signless Laplacian matrix. In this paper, we establish upper bounds for the largest Aα-eigenvalue in terms of several graph invariants, such as the order, the size, the diameter, the maximum degree, the second largest degree, the clique number and the first Zagreb index, as well as upper and lower bounds for the smallest Aα-eigenvalue. For one of these bounds, we characterize all graphs attaining equality. Our results extend known bounds for adjacency and signless Laplacian eigenvalues to the Aα-setting.