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

Abstract

Let $G=(V,E)$ be a simple connected graph of order $n\ge 2$, size $m$ with
normalized Laplacian eigenvalues $\gamma_1\ge \gamma_2 \ge \cdots \ge
\gamma_{n-1}>\gamma_n =0$. Denote with $s_{\alpha}(G)=\sum_{i=1}^{n-1}\gamma_i^{\alpha}$, where $\alpha$ 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 $s_{\alpha}(G)$, for various real $\alpha$ are proved.
Our results not only generalize and improve some previous results on $s_{\alpha}(G)$, Kemeny constant and Laplacian incidence energy, but also present new bounds for these graph invariants.