Bounding the sum of powers of normalized Laplacian eigenvalues of a graph

Let G be a simple connected graph of order n. Its normalized Laplacian eigenvalues are λ1≥λ2≥⋯≥λn−1≥λn=0. In this paper, new bounds on Sβ*(G)=∑i=1n−1λiβ (β ≠ 0, 1) are derived.