scholarly journals Computing LF-Metric Dimension of Generalized Gear Networks

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Hassan Zafar ◽  
Muhammad Javaid ◽  
Ebenezer Bonyah
Keyword(s):  
Structural Properties ◽  
Metric Dimension ◽  
Sharp Bounds ◽  
Shortest Distance ◽  
Metric Dimensions

The parameter of distance in the theory of networks plays a key role to study the different structural properties of the understudy networks or graphs such as symmetry, assortative, connectivity, and clustering. For the purpose, with the help of the parameter of distance, various types of metric dimensions have been defined to find the locations of machines (or robots) with respect to the minimum consumption of time, the shortest distance among the destinations, and the lesser number of utilized nodes as places of the objects. In this article, the latest derived form of metric dimension called as LF-metric dimension is studied, and various results for the generalized gear networks are obtained in the form of exact values and sharp bounds under certain conditions. The LF-metric dimension of some particular cases of generalized gear networks (called as generalized wheel networks) is also illustrated. Moreover, the bounded and unboundedness of the LF-metric dimension for all obtained results is also presented.

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

Mathematics ◽  
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.

scholarly journals Lower and Upper Bounds of Fractional Metric Dimension of Connected Networks

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.

scholarly journals Boundedness of Convex Polytopes Networks via Local Fractional Metric Dimension

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Muhammad Javaid ◽  
Hassan Zafar ◽  
Amer Aljaedi ◽  
Abdulaziz Mohammad Alanazi
Keyword(s):  
Convex Polytopes ◽  
Chemical Properties ◽  
Chemical Compounds ◽  
Molecular Networks ◽  
Metric Dimension ◽  
Optimal Solutions ◽  
Weighted Version ◽  
Shortest Distance ◽  
Metric Dimensions ◽  
And Chemical Properties

Metric dimension is one of the distance-based parameter which is frequently used to study the structural and chemical properties of the different networks in the various fields of computer science and chemistry such as image processing, pattern recognition, navigation, 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, and lesser number of the utilized nodes and to characterize the chemical compounds having unique presentation in molecular networks. The fractional metric dimension being a latest developed weighted version of the metric dimension is used in the distance-related problems of the aforementioned fields to find their nonintegral optimal solutions. In this paper, we have formulated the local resolving neighborhoods with their cardinalities for all the edges of the convex polytopes networks to compute their local fractional metric dimensions in the form of exact values and sharp bounds. Moreover, the boundedness of all the obtained results is also proved.

scholarly journals Dimensi Metrik Kuat Lokal Graf Hasil Operasi Kali Kartesian

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.

scholarly journals Sharp Bounds of Local Fractional Metric Dimensions of Connected Networks

IEEE Access ◽  
2020 ◽  
Vol 8 ◽  
pp. 172329-172342
Author(s):  
Muhammad Javaid ◽  
Mohsin Raza ◽  
Poom Kumam ◽  
Jia-Bao Liu
Keyword(s):  
Sharp Bounds ◽  
Metric Dimensions

scholarly journals On Metric Dimensions of Symmetric Graphs Obtained by Rooted Product

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 191 ◽  
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.

scholarly journals The Metric Dimension and Local Metric Dimension of Relative Prime Graph

CAUCHY ◽  
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.

scholarly journals Fractional Metric Dimension of Generalized Sunlet Networks

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Muhammad Javaid ◽  
Hassan Zafar ◽  
Ebenezer Bonyah
Keyword(s):  
Shortest Path ◽  
Metric Dimension ◽  
Sharp Bounds ◽  
Connected Network

Let N = V N , E N be a connected network with vertex V N and edge set E N ⊆ V N , E N . For any two vertices a and b , the distance d a , b is the length of the shortest path between them. The local resolving neighbourhood (LRN) set for any edge e = ab of N is a set of all those vertices whose distance varies from the end vertices a and b of the edge e . A real-valued function Φ from V N to 0,1 is called a local resolving function (LRF) if the sum of all the labels of the elements of each LRN set remains greater or equal to 1. Thus, the local fractional metric dimension (LFMD) of a connected network N is dim lf N = min Φ : Φ   is minimal LRF of   N . In this study, LFMD of various types of sunlet-related networks such as sunlet network ( S m ), middle sunlet network ( MS m ), and total sunlet network ( TS m ) are studied in the form of exact values and sharp bounds under certain conditions. Furthermore, the unboundedness and boundedness of all the obtained results of LFMD of the sunlet networks are also checked.

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

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].

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

CAUCHY ◽  
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 &gt; 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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