On comparison between the distance energies of a connected graph

Abstract

Let G be a simple connected graph of order n having Wiener index $W\left(G\right)$ . The distance, distance Laplacian and the distance signless Laplacian energies of G are respectively defined as $DE\left(G\right)=\sum _{i=1}^n\left|{\upsilon }_i^D\right|,DLE\left(G\right)=\sum _{i=1}^n|{\upsilon }_i^L-\overline{Tr}|\text{and}DSLE\left(G\right)=\sum _{i=1}^n|{\upsilon }_i^Q-\overline{Tr}|,$ where ${\upsilon }_i^D,{\upsilon }_i^L$ and ${\upsilon }_i^Q,1\le i\le n$ are respectively the distance, distance Laplacian and the distance signless Laplacian eigenvalues of G and $\overline{Tr}=\frac{2W\left(G\right)}{n}$ is the average transmission degree. In this paper, we will study the relation between $DE\left(G\right)$ , $DLE\left(G\right)$ and $DSLE\left(G\right)$ . We obtain some necessary conditions for the inequalities $DLE\left(G\right)\ge DSLE\left(G\right),DLE\left(G\right)\le DSLE\left(G\right),DLE\left(G\right)\ge DE\left(G\right)$ and $DSLE\left(G\right)\ge DE\left(G\right)$ to hold. We will show for graphs with one positive distance eigenvalue the inequality $DSLE\left(G\right)\ge DE\left(G\right)$ always holds. Further, we will show for the complete bipartite graphs the inequality $DLE\left(G\right)\ge DSLE\left(G\right)\ge DE\left(G\right)$ holds. We end this paper by computational results on graphs of order at most 6.

Keywords: 05C12; 05C50; 15A18; Distance (signless) Laplacian energy; Distance Laplacian matrix; Distance matrix; Transmission regular graph.