Some upper bounds for the energy of graphs

Let G = (V,E) be a graph with n vertices and e edges. Denote V(G) = {v1,v2,...,vn}. The 2-degree of vi, denoted by ti, is the sum of degrees of the vertices adjacent to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_i, 1\leqslant i\leqslant n$$\end{document}. Let σi be the sum of the 2-degree of vertices adjacent to vi. In this paper, we present two sharp upper bounds for the energy of G in terms of n, e, ti, and σi, from which we can get some known results. Also we give a sharp bound for the energy of a forest, from which we can improve some known results for trees.