On the Sum of the Powers of Distance Signless Laplacian Eigenvalues of Graphs

Let G be a connected graph with n vertices, m edges and having distance signless Laplacian eigenvalues ρ1≥ ρ2 ≥ … ≥ ρn≥ 0. For any real number α ≠ 0, let mα(G)=∑i=1nρiα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m_\alpha }\left( G \right) = \sum\nolimits_{i = 1}^n {\rho _i^\alpha } $$\end{document} be the sum of αth powers of the distance signless Laplacian eigenvalues of the graph G. In this paper, we obtain various bounds for the graph invariant mα(G), which connects it with different parameters associated to the structure of the graph G. We also obtain various bounds for the quantity DEL(G), the distance signless Laplacian-energy-like invariant of the graph G. These bounds improve some previously known bounds. We also pose some extremal problems about DEL(G).