Metric Dimension and Exchange Property for Resolving Sets in Rotationally-Symmetric Graphs

A family 𝒢 of connected graphs is a family with constant metric dimension if dim(G) is finite and does not depend upon the choice of G in 𝒢. In this paper, we show that the graph An∗ and the graph Anp obtained from the antiprism graph have constant metric dimension.

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On 2-metric resolvability in rotationally-symmetric graphs

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-210040 ◽

2021 ◽

pp. 1-9

Author(s):

Humera Bashir ◽

Zohaib Zahid ◽

Agha Kashif ◽

Sohail Zafar ◽

Jia-Bao Liu

Keyword(s):

Network Optimization ◽

Intelligent Systems ◽

Robot Navigation ◽

Data Transformation ◽

Metric Dimension ◽

Intelligent Networks ◽

Symmetric Graphs ◽

Symmetric Plane ◽

Rotationally Symmetric ◽

Sensor Networking

The 2-metric resolvability is an extension of metric resolvability in graphs having several applications in intelligent systems for example network optimization, robot navigation and sensor networking. Rotationally symmetric graphs are important in intelligent networks due to uniform rate of data transformation to all nodes. In this article, 2-metric dimension of rotationally symmetric plane graphs Rn, Sn and Tn is computed and found to be independent of the number of vertices.

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Power graphs and exchange property for resolving sets

Open Mathematics ◽

10.1515/math-2019-0093 ◽

2019 ◽

Vol 17 (1) ◽

pp. 1303-1309 ◽

Cited By ~ 1

Author(s):

Ghulam Abbas ◽

Usman Ali ◽

Mobeen Munir ◽

Syed Ahtsham Ul Haq Bokhary ◽

Shin Min Kang

Keyword(s):

Present Article ◽

Linear Algebra ◽

Finite Groups ◽

Robot Navigation ◽

Simple Graph ◽

Exchange Property ◽

Metric Dimension ◽

Sufficient Condition ◽

Resolving Set ◽

Dihedral Groups

Abstract Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups.

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Existence and extendibility of rotationally symmetric graphs with a prescribed higher mean curvature function in Euclidean and Minkowski spaces

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2016.09.022 ◽

2017 ◽

Vol 446 (1) ◽

pp. 1046-1059 ◽

Cited By ~ 3

Author(s):

Daniel de la Fuente ◽

Alfonso Romero ◽

Pedro J. Torres

Keyword(s):

Mean Curvature ◽

Curvature Function ◽

Symmetric Graphs ◽

Minkowski Spaces ◽

Rotationally Symmetric

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On Metric Dimensions of Symmetric Graphs Obtained by Rooted Product

Mathematics ◽

10.3390/math6100191 ◽

2018 ◽

Vol 6 (10) ◽

pp. 191 ◽

Cited By ~ 4

Author(s):

Shahid Imran ◽

Muhammad Siddiqui ◽

Muhammad Imran ◽

Muhammad Hussain

Keyword(s):

Connected Graph ◽

The Other ◽

Metric Dimension ◽

Resolving Set ◽

Minimum Cardinality ◽

Four Dimensions ◽

Symmetric Graphs ◽

Metric Dimensions ◽

Cycle Path

Let G = (V, E) be a connected graph and d(x, y) be the distance between the vertices x and y in G. A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by dim(G). In this paper, Cycle, Path, Harary graphs and their rooted product as well as their connectivity are studied and their metric dimension is calculated. It is proven that metric dimension of some graphs is unbounded while the other graphs are constant, having three or four dimensions in certain cases.

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Metric dimension of heptagonal circular ladder

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500950 ◽

2020 ◽

pp. 2050095

Author(s):

Sunny Kumar Sharma ◽

Vijay Kumar Bhat

Keyword(s):

Open Problem ◽

Geodesic Distance ◽

Radial Symmetry ◽

Metric Dimension ◽

Plane Graphs ◽

Path Graph ◽

Symmetric Plane ◽

Rotationally Symmetric ◽

Circular Ladder ◽

Multiple Edges

Let [Formula: see text] be an undirected (i.e., all the edges are bidirectional), simple (i.e., no loops and multiple edges are allowed), and connected (i.e., between every pair of nodes, there exists a path) graph. Let [Formula: see text] denotes the number of edges in the shortest path or geodesic distance between two vertices [Formula: see text]. The metric dimension (or the location number) of some families of plane graphs have been obtained in [M. Imran, S. A. Bokhary and A. Q. Baig, Families of rotationally-symmetric plane graphs with constant metric dimension, Southeast Asian Bull. Math. 36 (2012) 663–675] and an open problem regarding these graphs was raised that: Characterize those families of plane graphs [Formula: see text] which are obtained from the graph [Formula: see text] by adding new edges in [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, by answering this problem, we characterize some families of plane graphs [Formula: see text], which possesses the radial symmetry and has a constant metric dimension. We also prove that some families of plane graphs which are obtained from the plane graphs, [Formula: see text] by the addition of new edges in [Formula: see text] have the same metric dimension and vertices set as [Formula: see text], and only 3 nodes appropriately selected are sufficient to resolve all the nodes of these families of plane graphs.

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On resolvability of a graph associated to a finite vector space

Journal of Algebra and Its Applications ◽

10.1142/s0219498819500294 ◽

2019 ◽

Vol 18 (02) ◽

pp. 1950029

Author(s):

U. Ali ◽

S. A. Bokhary ◽

K. Wahid ◽

G. Abbas

Keyword(s):

Vector Space ◽

Domination Number ◽

Exchange Property ◽

Metric Dimension ◽

Resolving Set ◽

Matroid Intersection ◽

Finite Vector ◽

Finite Vector Space ◽

Partition Dimension ◽

Matroid Intersection Problem

In this paper, the resolving parameters such as metric dimension and partition dimension for the nonzero component graph, associated to a finite vector space, are discussed. The exact values of these parameters are determined. It is derived that the notions of metric dimension and locating-domination number coincide in the graph. Independent sets, introduced by Boutin [Determining sets, resolving set, and the exchange property, Graphs Combin. 25 (2009) 789–806], are studied in the graph. It is shown that the exchange property holds in the graph for minimal resolving sets with some exceptions. Consequently, a minimal resolving set of the graph is a basis for a matroid with the set [Formula: see text] of nonzero vectors of the vector space as the ground set. The matroid intersection problem for two matroids with [Formula: see text] as the ground set is also solved.

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Local Fractional Strong Metric Dimension of Certain Rotationally Symmetric Planer Networks

IEEE Access ◽

10.1109/access.2021.3131311 ◽

2021 ◽

pp. 1-1

Author(s):

Faiza Jamil ◽

Agha Kashif ◽

Sohail Zafar ◽

Zaid Bassfar ◽

Abdulaziz Mohammed Alanazi

Keyword(s):

Metric Dimension ◽

Rotationally Symmetric ◽

Strong Metric Dimension

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On the Upper Bounds of Fractional Metric Dimension of Symmetric Networks

Journal of Mathematics ◽

10.1155/2021/8417127 ◽

2021 ◽

Vol 2021 ◽

pp. 1-20

Author(s):

Muhammad Javaid ◽

Muhammad Kamran Aslam ◽

Jia-Bao Liu

Keyword(s):

Image Processing ◽

Drug Discovery ◽

Computer Science ◽

Chemical Compounds ◽

Upper Bounds ◽

Metric Dimension ◽

Rotationally Symmetric ◽

Metric Dimensions ◽

Structural Aspects ◽

Symmetric Networks

Distance-based numeric parameters play a pivotal role in studying the structural aspects of networks which include connectivity, accessibility, centrality, clustering modularity, complexity, vulnerability, and robustness. Several tools like these also help to resolve the issues faced by the different branches of computer science and chemistry, namely, navigation, image processing, biometry, drug discovery, and similarities in chemical compounds. For this purpose, in this article, we are considering a family of networks that exhibits rotationally symmetric behaviour known as circular ladders consisting of triangular, quadrangular, and pentagonal faced ladders. We evaluate their upper bounds of fractional metric dimensions of the aforementioned networks.

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