On comparison between the distance energies of a connected graph

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 asDE(G)=∑i=1n|υiD|,DLE(G)=∑i=1n|υiL−Tr‾|andDSLE(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‾=2W(G)n 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.

Keywords

• Distance matrix
• Distance Laplacian matrix
• Transmission regular graph
• Distance (signless) Laplacian energy