Resolving set and exchange property in nanotube

Ali N. A. Koam; Sikander Ali; Ali Ahmad; Muhammad Azeem; Muhammad Kamran Jamil; Ali N. A. Koam; Sikander Ali; Ali Ahmad; Muhammad Azeem; Muhammad Kamran Jamil

doi:10.3934/math.20231035

AIMS Mathematics

2023, Volume 8, Issue 9: 20305-20323. doi: 10.3934/math.20231035

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Research article

Resolving set and exchange property in nanotube

1.
Department of Mathematics, College of Science, Jazan University, New Campus, Jazan 2097, Saudi Arabia
2.
Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus, Pakistan
3.
Department of Information Technology and Security, College of Computer Science and Information Technology, Jazan University, Jazan, Saudi Arabia
4.
Department of Mathematics, Riphah International University Lahore, Pakistan

Received: 20 March 2023 Revised: 06 June 2023 Accepted: 12 June 2023 Published: 20 June 2023
MSC : 05C09, 05C12, 05C92

Give us a linked graph, $ G = (V, E). $ A vertex $ w\in V $ distinguishes between two components (vertices and edges) $ x, y\in E\cup V $ if $ d_G(w, x)\neq d_G (w, y). $ Let $ W_{1} $ and $ W_{2} $ be two resolving sets and $ W_{1} $ $ \neq $ $ W_{2} $. Then, we can say that the graph $ G $ has double resolving set. A nanotube derived from an quadrilateral-octagonal grid belongs to essential and extensively studied compounds in materials science. Nano-structures are very important due to their thickness. In this article, we have discussed the metric dimension of the graphs of nanotubes derived from the quadrilateral-octagonal grid. We proved that the generalized nanotube derived from quadrilateral-octagonal grid have three metric dimension. We also check that the exchange property is also held for this structure.
- resolving set,
- metric dimension,
- nanotube,
- quadrilateral-octogonal grid,
- exchange property
Citation: Ali N. A. Koam, Sikander Ali, Ali Ahmad, Muhammad Azeem, Muhammad Kamran Jamil. Resolving set and exchange property in nanotube[J]. AIMS Mathematics, 2023, 8(9): 20305-20323. doi: 10.3934/math.20231035

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Abstract

Give us a linked graph, $ G = (V, E). $ A vertex $ w\in V $ distinguishes between two components (vertices and edges) $ x, y\in E\cup V $ if $ d_G(w, x)\neq d_G (w, y). $ Let $ W_{1} $ and $ W_{2} $ be two resolving sets and $ W_{1} $ $ \neq $ $ W_{2} $. Then, we can say that the graph $ G $ has double resolving set. A nanotube derived from an quadrilateral-octagonal grid belongs to essential and extensively studied compounds in materials science. Nano-structures are very important due to their thickness. In this article, we have discussed the metric dimension of the graphs of nanotubes derived from the quadrilateral-octagonal grid. We proved that the generalized nanotube derived from quadrilateral-octagonal grid have three metric dimension. We also check that the exchange property is also held for this structure.

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