Fractional Metric Dimension of Generalized Sunlet Networks

Let N=VN,EN be a connected network with vertex VN and edge set EN⊆VN,EN. For any two vertices a and b, the distance da,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 e...

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Main Authors: Muhammad Javaid, Hassan Zafar, Ebenezer Bonyah
Format: Article
Language:English
Published: Wiley 2021-01-01
Series:Journal of Mathematics
Online Access:http://dx.doi.org/10.1155/2021/4101869
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author Muhammad Javaid
Hassan Zafar
Ebenezer Bonyah
author_facet Muhammad Javaid
Hassan Zafar
Ebenezer Bonyah
author_sort Muhammad Javaid
collection DOAJ
description Let N=VN,EN be a connected network with vertex VN and edge set EN⊆VN,EN. For any two vertices a and b, the distance da,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 VN 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 dimlfN=minΦ:Φ is minimal LRF of N. In this study, LFMD of various types of sunlet-related networks such as sunlet network (Sm), middle sunlet network (MSm), and total sunlet network (TSm) 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.
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spelling doaj-art-7265e8cc8d6044699375cec7f6ab730c2025-02-03T05:47:00ZengWileyJournal of Mathematics2314-47852021-01-01202110.1155/2021/4101869Fractional Metric Dimension of Generalized Sunlet NetworksMuhammad Javaid0Hassan Zafar1Ebenezer Bonyah2Department of MathematicsDepartment of MathematicsDepartment of Mathematics EducationLet N=VN,EN be a connected network with vertex VN and edge set EN⊆VN,EN. For any two vertices a and b, the distance da,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 VN 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 dimlfN=minΦ:Φ is minimal LRF of N. In this study, LFMD of various types of sunlet-related networks such as sunlet network (Sm), middle sunlet network (MSm), and total sunlet network (TSm) 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.http://dx.doi.org/10.1155/2021/4101869
spellingShingle Muhammad Javaid
Hassan Zafar
Ebenezer Bonyah
Fractional Metric Dimension of Generalized Sunlet Networks
Journal of Mathematics
title Fractional Metric Dimension of Generalized Sunlet Networks
title_full Fractional Metric Dimension of Generalized Sunlet Networks
title_fullStr Fractional Metric Dimension of Generalized Sunlet Networks
title_full_unstemmed Fractional Metric Dimension of Generalized Sunlet Networks
title_short Fractional Metric Dimension of Generalized Sunlet Networks
title_sort fractional metric dimension of generalized sunlet networks
url http://dx.doi.org/10.1155/2021/4101869
work_keys_str_mv AT muhammadjavaid fractionalmetricdimensionofgeneralizedsunletnetworks
AT hassanzafar fractionalmetricdimensionofgeneralizedsunletnetworks
AT ebenezerbonyah fractionalmetricdimensionofgeneralizedsunletnetworks