On sum of powers of Laplacian eigenvalues and Laplacian Estrada index of graphs

Abstract

Let $G$ be a simple graph and $\alpha$ a real number. The quantity $s_{\alpha}(G)$ defined as the sum of the $\alpha$-th power of the non-zero Laplacian eigenvalues of $G$ generalizes several concepts in the literature. The Laplacian Estrada index is a newly introduced graph invariant based on Laplacian eigenvalues. We establish bounds for $s_{\alpha}$ and Laplacian Estrada index related to the degree sequences.