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 γ_1 ≥ γ_2 ≥ ⋯ ≥ γₙ₋₁ > γ_n = 0 . Denote with s_α(G) = ∑ᵢ₌₁ⁿ⁻¹ γ_i^α , 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.