On Sharp Bounds on Partition Dimension of Convex Polytopes

Chu, Yu‐Ming; Nadeem, Muhammad Faisal; Azeem, Muhammad; Siddiqui, Muhammad Kamran

doi:10.1109/access.2020.3044498

Cited by 33 publications

(25 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [38] bounds of uni-cyclic graphs are presented. For more recent literature and results, we refer to [13], [17], [20], [26], [28], [30]- [32], [34], [37], [38].…”

Section: Introduction and Some Basic Definitionsmentioning

confidence: 99%

On the Partition Dimension of Tri-Hexagonal α-Boron Nanotube

Shabbir

Azeem

2021

IEEE Access

Self Cite

View full text Add to dashboard Cite

The production of low-cost, small in size, and high in efficiency objects is the topic of research in almost all scientific fields, especially of engineering. In this scenario, nanotechnology becomes of great importance. To achieve these tasks, one needs to study the different physical and chemical aspects of a chemical compound. In mathematical graph theory, chemical compounds are transformed in a unique mathematical representation and then examined under various parameters for these purposes. The Partition Dimension is also one of these parametric tools, which are used to identify each atom or vertex of a chemical structure under some chosen conditions. In the present paper, we use this tool to show the unique identification of each atom of a tri-hexagonal lattice of the α-boron nanotube.INDEX TERMS α-boron nanotube, tri-hexagonal boron nanotube, molecular graph, resolving partition set, partition dimension.

show abstract

“…In [38] bounds of uni-cyclic graphs are presented. For more recent literature and results, we refer to [13], [17], [20], [26], [28], [30]- [32], [34], [37], [38].…”

Section: Introduction and Some Basic Definitionsmentioning

confidence: 99%

On the Partition Dimension of Tri-Hexagonal α-Boron Nanotube

Shabbir

Azeem

2021

IEEE Access

Self Cite

View full text Add to dashboard Cite

show abstract

“…Fehar et al studied the metric dimension of Cayley digraphs [7]. For further studies of metric dimension of convex polytopes, Caylay and Toeplitz networks, see [8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Improved Lower Bound of LFMD with Applications of Prism-Related Networks

Javaid

Zafar

Zhu

et al. 2021

Mathematical Problems in Engineering

View full text Add to dashboard Cite

The different distance-based parameters are used to study the problems in various fields of computer science and chemistry such as pattern recognition, image processing, integer programming, navigation, drug discovery, and formation of different chemical compounds. In particular, distance among the nodes (vertices) of the networks plays a supreme role to study structural properties of networks such as connectivity, robustness, completeness, complexity, and clustering. Metric dimension is used to find the locations of machines with respect to minimum utilization of time, lesser number of the utilized nodes as places of the objects, and shortest distance among destinations. In this paper, lower bound of local fractional metric dimension for the connected networks is improved from unity and expressed in terms of ratio obtained by the cardinalities of the under-study network and the local resolving neighbourhood with maximum order for some edges of network. In the same context, the LFMDs of prism-related networks such as circular diagonal ladder, antiprism, triangular winged prism, and sun flower networks are computed with the help of obtained criteria. At the end, the bounded- and unboundedness of the obtained results is also shown numerically.

show abstract

“…As we discussed above that this has numerous use in the chemical field, much work has done with graph prospectives and metric dimension also consider important to study different structures with it, like the structure of H-Naphtalenic and V C 5 C 7 nano-tubes discussed with metric concept [20], some upper bounds of cellulose network considering metric dimension as a point of discussion [43], metric of silicate star are computed in [44], a two-dimensional lattice of αboron nanotubes discussed with specific applications in terms of metric dimension in [19]. For the partition dimension, a graph with n − 3, partition dimension discussed in [3], (4, 6) is a special type of fullerene structure and it is also studied by [29] with the concept of partition dimension, There are few very recent research on the bounded partition dimension, we encourage to have a look the articles [10], [27], [31]. The bounds of partitioning on the specific type of nanotube are studied in [39].…”

Section: Introductionmentioning

confidence: 99%

Metric and Fault-Tolerant Metric Dimension of Hollow Coronoid

et al. 2021

Self Cite

View full text Add to dashboard Cite

Coronoid systems actually arrangements of hexagons into six sides of benzenoids. By nature, it is an organic chemical structure. Hollow coronoids are primitive and catacondensed coronoids. It is also known as polycyclic conjugated hydrocarbons. The mathematical study of chemicals is of great interest to different specialties researchers. While graph theory always played a significant role to make chemical structures understandable and blessed with applications also. After transforming the chemical structure into a graph, one can implement different theoretical and implicative studies on structures. Metric dimension is considered as one of the most studied and implicative parameters of graph theory. In this concept, few suggested vertices are chosen such as the remaining vertices have unique locations or identifications. In this study, we discussed different metric-based parameters for the hollow coronoid structure. INDEX TERMS Hollow coronoid; metric dimension; resolving set; fault-tolerant metric dimension.

show abstract

On Sharp Bounds on Partition Dimension of Convex Polytopes

Cited by 33 publications

References 30 publications

On the Partition Dimension of Tri-Hexagonal α-Boron Nanotube

On the Partition Dimension of Tri-Hexagonal α-Boron Nanotube

Improved Lower Bound of LFMD with Applications of Prism-Related Networks

Metric and Fault-Tolerant Metric Dimension of Hollow Coronoid

Contact Info

Product

Resources

About