Improved McClelland and Koolen-Moulton bounds for distance energy of a graph

Let G be a graph with n vertices and m edges. The term energy of a graph G was introduced by I. Gutman in chemistry due to its relevance to the total π-electron energy of a carbon compound. An analogous energy $$ , called the distance energy, was defined by Indulal et al. (MATCH Commun. Math. Comput. Chem. 60:461-472, 2008) in 2008. McClelland and Koolen-Moulton bounds for distance energy were established subsequently by Ramane et al. (Kragujev. J. Math. 31:59-68, 2008). The lower and upper bounds for $$ obtained in this paper are better than the McClelland and Koolen-Moulton bounds.