Ordering graphs by their largest (least) Aα-eigenvalues.

Let G be a simple undirected graph. For real number α ∈ [0, 1], Nikiforov defined the Aα-matrix of G as Aα(G) = αD(G) + (1 − α)A(G), where A(G) and D(G) are the adjacency matrix and the degree diagonal matrix of G respectively. In this paper, we obtain a sharp upper bound on the largest eigenvalue ρα(G) of Aα(G) for α ∈ [1/2, 1). Employing this upper bound, we prove that 'For connected G1 and G2 with n vertices and m edges, if the maximum degree Δ(G1) ≥ 2α(1 − α)(2m − n + 1) + 2α and Δ(G1) > Δ(G2), then ρ α (G 1) > ρ α (G 2) '. Let λα(G) denote the least eigenvalue of Aα(G). For α ∈ (1/2, 1), we prove that 'For two connected G1 and G2, if the minimum degree δ (G 1) ≤ 1 1 − α − 2 and δ(G1) < δ(G2), then λα(G1) < λα(G2)'. [ABSTRACT FROM AUTHOR]