On comparison between the distance energies of a connected graph
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 dista...
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Elsevier
2024-11-01
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| Series: | Heliyon |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2405844024163475 |
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| author | Hilal A. Ganie Bilal Ahmad Rather Yilun Shang |
| author_facet | Hilal A. Ganie Bilal Ahmad Rather Yilun Shang |
| author_sort | Hilal A. Ganie |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-6f7ffc8d0677436f9c99d91a81ffe5c2 |
| institution | Kabale University |
| issn | 2405-8440 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Heliyon |
| spelling | doaj-art-6f7ffc8d0677436f9c99d91a81ffe5c22024-11-30T07:12:30ZengElsevierHeliyon2405-84402024-11-011022e40316On comparison between the distance energies of a connected graphHilal A. Ganie0Bilal Ahmad Rather1Yilun Shang2Department of School Education JK Govt. Kashmir, India; Corresponding authors.Department of Mathematical Sciences, Samarkand International University of Technology, Samarkand 140100, UzbekistanDepartment of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK; Corresponding authors.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.http://www.sciencedirect.com/science/article/pii/S2405844024163475Distance matrixDistance Laplacian matrixTransmission regular graphDistance (signless) Laplacian energy |
| spellingShingle | Hilal A. Ganie Bilal Ahmad Rather Yilun Shang On comparison between the distance energies of a connected graph Heliyon Distance matrix Distance Laplacian matrix Transmission regular graph Distance (signless) Laplacian energy |
| title | On comparison between the distance energies of a connected graph |
| title_full | On comparison between the distance energies of a connected graph |
| title_fullStr | On comparison between the distance energies of a connected graph |
| title_full_unstemmed | On comparison between the distance energies of a connected graph |
| title_short | On comparison between the distance energies of a connected graph |
| title_sort | on comparison between the distance energies of a connected graph |
| topic | Distance matrix Distance Laplacian matrix Transmission regular graph Distance (signless) Laplacian energy |
| url | http://www.sciencedirect.com/science/article/pii/S2405844024163475 |
| work_keys_str_mv | AT hilalaganie oncomparisonbetweenthedistanceenergiesofaconnectedgraph AT bilalahmadrather oncomparisonbetweenthedistanceenergiesofaconnectedgraph AT yilunshang oncomparisonbetweenthedistanceenergiesofaconnectedgraph |