Some new bounds for the sum of powers of the normalized Laplacian eigenvalues of graphs

Let G = (V, E) be a simple connected graph of order n ≥ 2, size m with normalized Laplacian eigenvalues ₁ ₂ ₍-₁ > ₙ = 0. Denote with s_ (G) = ₈=₁^n-1 ᵢ^, where α is an arbitrary real number, the sum of powers of normalized Laplacian eigenvalues of graphs. In this paper several inequalities involving invariants of the form sα (G), for various real α are proved. Our results not only generalize and improve some previous results on sα (G), Kemeny constant and Laplacian incidence energy, but also present new bounds for these graph invariants.