Bounds for the largest and the smallest Aα eigenvalues of a graph in terms of vertex degrees

Let G be a graph with adjacency matrix A(G) and with D(G) the diagonal matrix of its vertex degrees. Nikiforov defined the matrix A alpha(G), with alpha is an element of [0,1], as A alpha(G) = alpha D(G) + (1 - alpha)A(G). The largest and the smallest eigenvalues of A(alpha)(G) are respectively denoted by rho(G) and lambda(n)(G). In this paper, we present a tight upper bound for rho(G) in terms of the vertex degrees of G for alpha not equal 1. For any graph G, rho(G) <= max(u similar to w){alpha(d(u)+d(w))+root alpha(2)(d(u)+d(w))(2)+4(1-2 alpha)d(u)d(w)/2}. If G is connected, for a alpha is an element of [0,1), the equality holds if and only if G is regular or bipartite semi-regular. As an application, we solve the problem raised by Nikiforov in [15]. For the smallest eigenvalue lambda(n)(G) of a bipartite graph G of order n with no isolated vertices, for a alpha is an element of [0,1), then lambda(G) >= min(u similar to w){alpha(d(u)+d(w))-root alpha(2)(d(u)+d(w))(2)+4(1-2 alpha)d(u)d(w)/2}. If G is connected and alpha not equal 1/2, the equality holds if and only if G is regular or semi-regular. (C) 2020 Elsevier Inc. All rights reserved.