On comparison between the distance energies of a connected graph
Аннотация
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 D E ( G ) = ∑ i = 1 n | υ i D | , D L E ( G ) = ∑ i = 1 n | υ i L − T r ‾ | and D S L E ( G ) = ∑ i = 1 n | υ i Q − T r ‾ | , where υ i D , υ i L and υ i Q , 1 ≤ i ≤ n are respectively the distance, distance Laplacian and the distance signless Laplacian eigenvalues of G and T r ‾ = 2 W ( G ) n is the average transmission degree. In this paper, we will study the relation between D E ( G ) , D L E ( G ) and D S L E ( G ) . We obtain some necessary conditions for the inequalities D L E ( G ) ≥ D S L E ( G ) , D L E ( G ) ≤ D S L E ( G ) , D L E ( G ) ≥ D E ( G ) and D S L E ( G ) ≥ D E ( G ) to hold. We will show for graphs with one positive distance eigenvalue the inequality D S L E ( G ) ≥ D E ( G ) always holds. Further, we will show for the complete bipartite graphs the inequality D L E ( G ) ≥ D S L E ( G ) ≥ D E ( G ) holds. We end this paper by computational results on graphs of order at most 6.