The comparative study of resolving parameters for a family of ladder networks

Mohra Zayed; Ali Ahmad; Muhammad Faisal Nadeem; Muhammad Azeem; Mohra Zayed; Ali Ahmad; Muhammad Faisal Nadeem; Muhammad Azeem

doi:10.3934/math.2022908

AIMS Mathematics

2022, Volume 7, Issue 9: 16569-16589. doi: 10.3934/math.2022908

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Research article

The comparative study of resolving parameters for a family of ladder networks

1.
Mathematics Department, College of Science, King Khalid University, Abha, Saudi Arabia
2.
College of Computer Science & Information Technology Jazan University, Jazan, Saudi Arabia
3.
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan
4.
Department of Mathematics, Riphah Institute of Computing and Applied Sciences, Riphah International University Lahore, Pakistan

Received: 11 April 2022 Revised: 28 June 2022 Accepted: 04 July 2022 Published: 11 July 2022
MSC : 05C09, 05C12, 05C92

For a simple connected graph $ G = (V, E) $, a vertex $ x\in V $ distinguishes two elements (vertices or edges) $ x_1\in V, y_1 \in E $ if $ d(x, x_1)\neq d(x, y_1). $ A subset $ Q_m\subset V $ is a mixed metric generator for $ G, $ if every two distinct elements (vertices or edges) of $ G $ are distinguished by some vertex of $ Q_m. $ The minimum cardinality of a mixed metric generator for $ G $ is called the mixed metric dimension and denoted by $ dim_m(G). $ In this paper, we investigate the mixed metric dimension for different families of ladder networks. Among these families, we consider Möbius ladder, hexagonal Möbius ladder, triangular Möbius ladder network and conclude that all these families have constant-metric, edge metric and mixed metric dimension.
- Möbius ladder network,
- hexagonal Möbius ladder network,
- triangular Möbius ladder network,
- mixed metric dimension,
- mixed metric generator
Citation: Mohra Zayed, Ali Ahmad, Muhammad Faisal Nadeem, Muhammad Azeem. The comparative study of resolving parameters for a family of ladder networks[J]. AIMS Mathematics, 2022, 7(9): 16569-16589. doi: 10.3934/math.2022908

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Abstract

For a simple connected graph $ G = (V, E) $, a vertex $ x\in V $ distinguishes two elements (vertices or edges) $ x_1\in V, y_1 \in E $ if $ d(x, x_1)\neq d(x, y_1). $ A subset $ Q_m\subset V $ is a mixed metric generator for $ G, $ if every two distinct elements (vertices or edges) of $ G $ are distinguished by some vertex of $ Q_m. $ The minimum cardinality of a mixed metric generator for $ G $ is called the mixed metric dimension and denoted by $ dim_m(G). $ In this paper, we investigate the mixed metric dimension for different families of ladder networks. Among these families, we consider Möbius ladder, hexagonal Möbius ladder, triangular Möbius ladder network and conclude that all these families have constant-metric, edge metric and mixed metric dimension.

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