The comparative analysis of metric and edge metric dimension of some subdivisions of the wheel graph

Asian-European Journal of Mathematics ◽

10.1142/s1793557121500625 ◽

2020 ◽

pp. 2150062

Author(s):

Zahid Raza ◽

M. S. Bataineh

Keyword(s):

Comparative Analysis ◽

Metric Dimension ◽

Wheel Graph ◽

Wheel Graphs ◽

Metric Dimensions ◽

Appl Math

The aim of this study is to compute the edge metric dimension of some subdivision of the wheel graphs. In particular, we determine and compare the metric and edge metric dimensions of the graphs obtained after the cycle, spoke and barycentric subdivisions of the wheel graph. Furthermore, some families of graphs have been constructed through subdivision process for which [Formula: see text], and also [Formula: see text] which partially answer a question in [A. Kelenc, N. Tratnik and I. G. Yero, Uniquely identifying the edges of a graph: The edge metric dimension, Discrete Appl. Math. 251 (2018) 204–220].

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The Vertex-Edge Resolvability of Some Wheel-Related Graphs

Journal of Mathematics ◽

10.1155/2021/1859714 ◽

2021 ◽

Vol 2021 ◽

pp. 1-16

Author(s):

Bao-Hua Xing ◽

Sunny Kumar Sharma ◽

Vijay Kumar Bhat ◽

Hassan Raza ◽

Jia-Bao Liu

Keyword(s):

Connected Graph ◽

Metric Dimension ◽

Resolving Set ◽

Minimum Cardinality ◽

Wheel Graph ◽

Wheel Graphs ◽

Metric Dimensions ◽

Mixed Metric

A vertex w ∈ V H distinguishes (or resolves) two elements (edges or vertices) a , z ∈ V H ∪ E H if d w , a ≠ d w , z . A set W m of vertices in a nontrivial connected graph H is said to be a mixed resolving set for H if every two different elements (edges and vertices) of H are distinguished by at least one vertex of W m . The mixed resolving set with minimum cardinality in H is called the mixed metric dimension (vertex-edge resolvability) of H and denoted by m dim H . The aim of this research is to determine the mixed metric dimension of some wheel graph subdivisions. We specifically analyze and compare the mixed metric, edge metric, and metric dimensions of the graphs obtained after the wheel graphs’ spoke, cycle, and barycentric subdivisions. We also prove that the mixed resolving sets for some of these graphs are independent.

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Bounds of Fractional Metric Dimension and Applications with Grid-Related Networks

Mathematics ◽

10.3390/math9121383 ◽

2021 ◽

Vol 9 (12) ◽

pp. 1383

Author(s):

Ali H. Alkhaldi ◽

Muhammad Kamran Aslam ◽

Muhammad Javaid ◽

Abdulaziz Mohammed Alanazi

Keyword(s):

Metric Dimension ◽

Hard Problem ◽

Np Hard ◽

Weighted Version ◽

Sharp Bounds ◽

Np Hard Problem ◽

Metric Dimensions

Metric dimension of networks is a distance based parameter that is used to rectify the distance related problems in robotics, navigation and chemical strata. The fractional metric dimension is the latest developed weighted version of metric dimension and a generalization of the concept of local fractional metric dimension. Computing the fractional metric dimension for all the connected networks is an NP-hard problem. In this note, we find the sharp bounds of the fractional metric dimensions of all the connected networks under certain conditions. Moreover, we have calculated the fractional metric dimension of grid-like networks, called triangular and polaroid grids, with the aid of the aforementioned criteria. Moreover, we analyse the bounded and unboundedness of the fractional metric dimensions of the aforesaid networks with the help of 2D as well as 3D plots.

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Dimensi Metrik Kuat Lokal Graf Hasil Operasi Kali Kartesian

Contemporary Mathematics and Applications (ConMathA) ◽

10.20473/conmatha.v1i2.17383 ◽

2020 ◽

Vol 1 (2) ◽

pp. 64

Author(s):

Nurma Ariska Sutardji ◽

Liliek Susilowati ◽

Utami Dyah Purwati

Keyword(s):

Graph Theory ◽

Cartesian Product ◽

Metric Dimension ◽

Star Graph ◽

Product Graph ◽

Path Graph ◽

Strong Metric Dimension ◽

Metric Dimensions ◽

Development Result ◽

Corona Product

The strong local metric dimension is the development result of a strong metric dimension study, one of the study topics in graph theory. Some of graphs that have been discovered about strong local metric dimension are path graph, star graph, complete graph, cycle graphs, and the result corona product graph. In the previous study have been built about strong local metric dimensions of corona product graph. The purpose of this research is to determine the strong local metric dimension of cartesian product graph between any connected graph G and H, denoted by dimsl (G x H). In this research, local metric dimension of G x H is influenced by local strong metric dimension of graph G and local strong metric dimension of graph H. Graph G and graph H has at least two order.

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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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The Metric Dimension and Local Metric Dimension of Relative Prime Graph

CAUCHY ◽

10.18860/ca.v6i3.10103 ◽

2020 ◽

Vol 6 (3) ◽

pp. 149-161

Author(s):

Inna Kuswandari ◽

Fatmawati Fatmawati ◽

Mohammad Imam Utoyo

Keyword(s):

Prime Graph ◽

Metric Dimension ◽

Resolving Set ◽

Prime Graphs ◽

Vertex Set ◽

Integer Rings ◽

Metric Dimensions ◽

Relative Prime

This study aims to determine the value of metric dimensions and local metric dimensions of relative prime graphs formed from modulo integer rings, namely . As a vertex set is and if and are relatively prime. By finding the pattern elements of resolving set and local resolving set, it can be shown the value of the metric dimension and the local metric dimension of graphs are and respectively, where is the number of vertices groups that formed multiple 2,3, … , and is the cardinality of set . This research can be developed by determining the value of the fractional metric dimension, local fractional metric dimension and studying the advanced properties of graphs related to their forming rings.Key Words : metric dimension; modulo ; relative prime graph; resolving set; rings.

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Lower and Upper Bounds of Fractional Metric Dimension of Connected Networks

Fractal and Fractional ◽

10.3390/fractalfract5040276 ◽

2021 ◽

Vol 5 (4) ◽

pp. 276

Author(s):

Muhammad Javaid ◽

Muhammad Kamran Aslam ◽

Muhammad Imran Asjad ◽

Bander N. Almutairi ◽

Mustafa Inc ◽

...

Keyword(s):

Large Scale ◽

Chemical Compounds ◽

Vital Role ◽

Molecular Networks ◽

Metric Dimension ◽

Weighted Version ◽

Shortest Distance ◽

Metric Dimensions ◽

Large Scale Networks ◽

2D And 3D

The distance centric parameter in the theory of networks called by metric dimension plays a vital role in encountering the distance-related problems for the monitoring of the large-scale networks in the various fields of chemistry and computer science such as navigation, image processing, pattern recognition, integer programming, optimal transportation models and drugs discovery. In particular, it is used to find the locations of robots with respect to shortest distance among the destinations, minimum consumption of time, lesser number of the utilized nodes, and to characterize the chemical compounds, having unique presentations in molecular networks. After the arrival of its weighted version, known as fractional metric dimension, the rectification of distance-related problems in the aforementioned fields has revived to a great extent. In this article, we compute fractional as well as local fractional metric dimensions of web-related networks called by subdivided QCL, 2-faced web, 3-faced web, and antiprism web networks. Moreover, we analyse their final results using 2D and 3D plots.

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On the local metric dimension of t-fold wheel, Pn o Km, and generalized fan

Indonesian Journal of Combinatorics ◽

10.19184/ijc.2018.2.2.4 ◽

2018 ◽

Vol 2 (2) ◽

pp. 88

Author(s):

Rokhana Ayu Solekhah ◽

Tri Atmojo Kusmayadi

Keyword(s):

Connected Graph ◽

Ordered Set ◽

Metric Dimension ◽

Minimum Cardinality ◽

Wheel Graph ◽

Metric Set ◽

Metric Basis ◽

Fan Graph

Let G be a connected graph and let u, v ∈ V(G). For an ordered set W = {w1, w2, ..., wn} of n distinct vertices in G, the representation of a vertex v of G with respect to W is the n-vector r(v∣W) = (d(v, w1), d(v, w2), ..., d(v, wn)), where d(v, wi) is the distance between v and wi for 1 ≤ i ≤ n. The set W is a local metric set of G if r(u ∣ W) ≠ r(v ∣ W) for every pair u, v of adjacent vertices of G. The local metric set of G with minimum cardinality is called a local metric basis for G and its cardinality is called a local metric dimension, denoted by lmd(G). In this paper we determine the local metric dimension of a t-fold wheel graph, Pn ⊙ Km graph, and generalized fan graph.

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Dimensi Metrik Graf Kr+mKsr, m, r, s, En

CAUCHY ◽

10.18860/ca.v1i4.1800 ◽

2011 ◽

Vol 1 (4) ◽

pp. 165

Author(s):

Hindayani Hindayani

Keyword(s):

Metric Dimension ◽

Combinatorial Search ◽

Robotic Navigation ◽

Resolving Set ◽

Minimum Cardinality ◽

Metric Dimensions ◽

Partition Graph

<div class="standard"><a id="magicparlabel-29">The concept of minimum resolving set has proved to be useful and or related to a variety of fields such as Chemistry, Robotic Navigation, and Combinatorial Search and Optimization. So that, this thesis explains the metric dimension of graph Kr + mKsr, m, r, s E N. Resolving set of a graph G is a subset of F (G) that its distance representation is distinct to all vertices of graph G. Resolving set with minimum cardinality is called minimum resolving set, and cardinal states metric dimension of G and noted with dim (G). By drawing the graph, it will be found the resolving set, minimum resolving set and the metric dimension easily. After that, formulate those metric dimensions into a theorem. This research search for the metric dimension of Kr + mKs, m > 2, m,r,s E N and its outcome are dim (Kr + mK1)= m+ (r-2) and dim(Kr + mKs)= m(s-1)+(r-2). This research can be continued for determining the metric dimension of another graph, by changing the operation of its graph or partition graph.</a></div>

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