Bounds for Aα-eigenvalues

Abstract

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 Aα(G), as a convex combination of A(G) and D(G), the following way, Aα(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 Aα-matrix. Moreover, extremal graphs attaining some of these bounds are characterized.