On the sum of powers of the $ A_{\alpha} $-eigenvalues of graphs

Abstract

<abstract><p>Let $ A(G) $ and $ D(G) $ be the adjacency matrix and the degree diagonal matrix of a graph $ G $, respectively. For any real number $ \alpha \in[0, 1] $, Nikiforov recently defined the $ A_{\alpha} $-matrix of $ G $ as $ A_{\alpha}(G) = \alpha D(G)+(1-\alpha)A(G) $. The graph invariant $ S_{\alpha}^{p}(G) $ is the sum of the $ p $-th power of the $ A_{\alpha} $-eigenvalues of $ G $ for $ \frac{1}{2} < \alpha < 1 $, which has a close relation to the $ \alpha $-Estrada index. In this paper, we establish some bounds on $ S_{\alpha}^{p}(G) $ and characterize the extremal graphs. In particular, we present some bounds on $ S_{\alpha}^{p}(G) $ in terms of the degree sequences, order and size of $ G $ by using majorization techniques. Moreover, we give lower and upper bounds for $ S_{\alpha}^{p}(G) $ of a bipartite graph and characterize the extremal graphs.</p></abstract>