On comparison between the distance energies of a connected graph

Let G be a simple connected graph of order n having Wiener index $$ . The distance, distance Laplacian and the distance signless Laplacian energies of G are respectively defined as where $$ and $$ are respectively the distance, distance Laplacian and the distance signless Laplacian eigenvalues of G and $$ is the average transmission degree. In this paper, we will study the relation between $$ , $$ and $$ . We obtain some necessary conditions for the inequalities $$ and $$ to hold. We will show for graphs with one positive distance eigenvalue the inequality $$ always holds. Further, we will show for the complete bipartite graphs the inequality $$ holds. We end this paper by computational results on graphs of order at most 6.