Bounds for Aα-eigenvalues

Let G be a graph with adjacency matrix A ( G ) and degree diagonal matrix D ( G ). In 2017, Nikiforov (V. Nikiforov, Appl. Anal. Discret. Math. 11 (2017) 81–107.) defined the matrix A α ( G ), as a convex combination of A ( G ) and D ( G ), the following way, A α ( G ) = αA ( G ) + (1 − α) D ( G ) where α ∈ [0,1]. In this paper we present some new upper and lower bounds for the largest, second largest and the smallest eigenvalue of A α -matrix. Moreover, extremal graphs attaining some of these bounds are characterized.