On comparison between the distance energies of a connected graph.
| Autor: | Ganie HA; Department of School Education JK Govt. Kashmir, India., Rather BA; Department of Mathematical Sciences, Samarkand International University of Technology, Samarkand 140100, Uzbekistan., Shang Y; Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK. |
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| Jazyk: | angličtina |
| Zdroj: | Heliyon [Heliyon] 2024 Nov 13; Vol. 10 (22), pp. e40316. Date of Electronic Publication: 2024 Nov 13 (Print Publication: 2024). |
| DOI: | 10.1016/j.heliyon.2024.e40316 |
| Abstrakt: | 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. Competing Interests: The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: The authors declare that Y. Shang is a Section editor for Heliyon. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. (© 2024 Published by Elsevier Ltd.) |
| Databáze: | MEDLINE |
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