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.

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 On the Geodesic Identification of Vertices in Convex Plane Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Fawaz E. Alsaadi ◽  
Muhammad Salman ◽  
Masood Ur Rehman ◽  
Abdul Rauf Khan ◽  
Jinde Cao ◽  
...  
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Metric Dimension ◽  
Plane Graphs ◽  
Convex Plane ◽  
Strong Metric Dimension ◽  
Minimum Number

A shortest path between two vertices u and v in a connected graph G is a u − v geodesic. A vertex w of G performs the geodesic identification for the vertices in a pair u , v if either v belongs to a u − w geodesic or u belongs to a v − w geodesic. The minimum number of vertices performing the geodesic identification for each pair of vertices in G is called the strong metric dimension of G . In this paper, we solve the strong metric dimension problem for three convex plane graphs by performing the geodesic identification of their vertices.

scholarly journals BI-DIMENSI METRIK DARI GRAF ANTIPRISMA

2020 ◽  
Vol 20 (2) ◽  
pp. 53
Author(s):  
Hendy Hendy ◽  
M. Ismail Marzuki
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Minimum Cardinality

Let G = (V, E) be a simple and connected graph. For each x ∈ V(G), it is associated with a vector pair (a, b), denoted by S x , corresponding to subset S = {s1 , s2 , ... , s k } ⊆ V(G), with a = (d(x, s1 ), d(x, s2 ), ... , d(x, s k )) and b = (δ(x, s1 ), δ(x, s2 ), ... , δ(x, s k )). d(v, s) is the length of shortest path from vertex v to s, and δ(v, s) is the length of the furthest path from vertex v to s. The set S is called the bi-resolving set in G if S x ≠ S y for any two distinct vertices x, y ∈ V(G). The bi- metric dimension of graph G, denoted by β b (G), is the minimum cardinality of the bi-resolving set in graph G. In this study we analyze bi-metric dimension in the antiprism graph (A n ). From the analysis that has been done, it is obtained the result that bi-metric dimension of graph A n , β b (A n ) is 3. Keywords: Antiprism graph, bi-metric dimension, bi-resolving set. .

scholarly journals On the strong metric dimension of the strong products of graphs

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Dorota Kuziak ◽  
Ismael G. Yero ◽  
Juan A. Rodríguez-Velázquez
Keyword(s):  
Connected Graph ◽  
Metric Dimension ◽  
Np Hard ◽  
Factor Graphs ◽  
Resolving Set ◽  
Sharp Bounds ◽  
Strong Product ◽  
Product Graphs ◽  
Strong Metric Dimension

AbstractLet G be a connected graph. A vertex w ∈ V.G/ strongly resolves two vertices u,v ∈ V.G/ if there exists some shortest u-w path containing v or some shortest v-w path containing u. A set S of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of S. The smallest cardinality of a strong resolving set for G is called the strong metric dimension of G. It is well known that the problem of computing this invariant is NP-hard. In this paper we study the problem of finding exact values or sharp bounds for the strong metric dimension of strong product graphs and express these in terms of invariants of the factor graphs.

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.

The Map-Based Shortest Route Selection by Using Ant Colony Optimization Algorithm

2018 ◽  
Vol 1 (1) ◽  
pp. 15
Author(s):  
Achmad Fanany Onnilita Gaffar ◽  
Agusma Wajiansyah ◽  
Supriadi Supriadi
Keyword(s):  
Ant Colony Optimization ◽  
Shortest Path ◽  
Optimization Algorithm ◽  
Optimization Problems ◽  
Google Earth ◽  
Ant Colony ◽  
Food Sources ◽  
Shortest Route ◽  
Study Results ◽  
Destination Point

The shortest path problem is one of the optimization problems where the optimization value is a distance. In general, solving the problem of the shortest route search can be done using two methods, namely conventional methods and heuristic methods. The Ant Colony Optimization (ACO) is the one of the optimization algorithm based on heuristic method. ACO is adopted from the behavior of ant colonies which naturally able to find the shortest route on the way from the nest to the food sources. In this study, ACO is used to determine the shortest route from Bumi Senyiur Hotel (origin point) to East Kalimantan Governor's Office (destination point). The selection of the origin and destination points is based on a large number of possible major roads connecting the two points. The data source used is the base map of Samarinda City which is cropped on certain coordinates by using Google Earth app which covers the origin and destination points selected. The data pre-processing is performed on the base map image of the acquisition results to obtain its numerical data. ACO is implemented on the data to obtain the shortest path from the origin and destination point that has been determined. From the study results obtained that the number of ants that have been used has an effect on the increase of possible solutions to optimal. The number of tours effect on the number of pheromones that are left on each edge passed ant. With the global pheromone update on each tour then there is a possibility that the path that has passed the ant will run out of pheromone at the end of the tour. This causes the possibility of inconsistent results when using the number of ants smaller than the number of tours.

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