Abstract
Let G be a simple connected graph of order n having Wiener index W(G) . The distance, distance Laplacian and the distance signless Laplacian energies of G are respectively defined as DE(G) = ∑ i = 1n|υiD|, DLE(G) = ∑ i = 1n|υiL - Tr‾| and DSLE(G) = ∑ i = 1n|υiQ - Tr‾|, where υiD, υiL and υiQ, 1 ≤ i ≤ n are respectively the distance, distance Laplacian and the distance signless Laplacian eigenvalues of G and Tr‾ = 2/ is the average transmission degree. In this paper, we will study the relation between DE(G) , DLE(G) and DSLE(G) . We obtain some necessary conditions for the inequalities DLE(G) ≥ DSLE(G), DLE(G) ≤ DSLE(G), DLE(G) ≥ DE(G) and DSLE(G) ≥ DE(G) to hold. We will show for graphs with one positive distance eigenvalue the inequality DSLE(G) ≥ DE(G) always holds. Further, we will show for the complete bipartite graphs the inequality DLE(G) ≥ DSLE(G) ≥ DE(G) holds. We end this paper by computational results on graphs of order at most 6.