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

Abstract

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 defined the $ A_{\alpha} $-matrix of a graph $ G $ as $ A_{\alpha}(G) = \alpha D(G)+(1-\alpha)A(G) $. Let $ S_k(A_{\alpha}(G)) $ be the sum of the $ k $ largest eigenvalues of $ A_{\alpha}(G) $. In this paper, some bounds on $ S_k(A_{\alpha}(G)) $ are obtained, which not only extends the results of the sum of the $ k $ largest eigenvalues of the adjacency matrix and signless Laplacian matrix, but it also gives new bounds on graph energy.

Keywords

• $ a_{\alpha} $-matrix
• sum of $ a_{\alpha} $-eigenvalues
• energy
• graph operation