Study of modified prism networks via fractional metric dimension
For a connected network $ \Gamma $, the distance between any two vertices is the length of the shortest path between them. A vertex $ c $ in a connected network is said to resolve an edge $ e $ if the distances of $ c $ from its endpoints are unequal. The collection of all the vertices which resolve...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-03-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023551?viewType=HTML |
| _version_ | 1797866971061026816 |
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| author | Ahmed Alamer Hassan Zafar Muhammad Javaid |
| author_facet | Ahmed Alamer Hassan Zafar Muhammad Javaid |
| author_sort | Ahmed Alamer |
| collection | DOAJ |
| description | For a connected network $ \Gamma $, the distance between any two vertices is the length of the shortest path between them. A vertex $ c $ in a connected network is said to resolve an edge $ e $ if the distances of $ c $ from its endpoints are unequal. The collection of all the vertices which resolve an edge is called the local resolving neighborhood set of this edge. A local resolving function is a real-valued function is defend as $ \eta : V(\Gamma) \rightarrow [0, 1] $ such that $ \eta (R_{x}(e)) \geq 1 $ for each edge $ e \in E(\Gamma) $, where $ R_{x} (e) $ represents the local resolving neighborhood set of a connected network. Thus the local fractional metric dimension is defined as $ dim_{LF}(\Gamma) = \quad min \quad \{ |\eta|: \quad \eta \quad is \quad the \quad minimal \quad local \quad resolving \quad function \quad of \quad \Gamma\}, $ where $ |\eta| = \sum \limits _ {a \in R_{x}(e)}\eta(a) $. In this manuscript, we have established sharp bounds of the local fractional metric dimension of different types of modified prism networks and it is also proved that local fractional metric dimension remains bounded when the order of these networks approaches to infinity. |
| first_indexed | 2024-04-09T23:32:47Z |
| format | Article |
| id | doaj.art-4d1708622c204cc59cff5962218cb0e5 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-09T23:32:47Z |
| publishDate | 2023-03-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-4d1708622c204cc59cff5962218cb0e52023-03-21T01:36:54ZengAIMS PressAIMS Mathematics2473-69882023-03-0185108641088610.3934/math.2023551Study of modified prism networks via fractional metric dimensionAhmed Alamer 0Hassan Zafar 1Muhammad Javaid21. Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia2. Department of Mathematics, School of Science, University of Management and Technology, Lahore, Pakistan2. Department of Mathematics, School of Science, University of Management and Technology, Lahore, PakistanFor a connected network $ \Gamma $, the distance between any two vertices is the length of the shortest path between them. A vertex $ c $ in a connected network is said to resolve an edge $ e $ if the distances of $ c $ from its endpoints are unequal. The collection of all the vertices which resolve an edge is called the local resolving neighborhood set of this edge. A local resolving function is a real-valued function is defend as $ \eta : V(\Gamma) \rightarrow [0, 1] $ such that $ \eta (R_{x}(e)) \geq 1 $ for each edge $ e \in E(\Gamma) $, where $ R_{x} (e) $ represents the local resolving neighborhood set of a connected network. Thus the local fractional metric dimension is defined as $ dim_{LF}(\Gamma) = \quad min \quad \{ |\eta|: \quad \eta \quad is \quad the \quad minimal \quad local \quad resolving \quad function \quad of \quad \Gamma\}, $ where $ |\eta| = \sum \limits _ {a \in R_{x}(e)}\eta(a) $. In this manuscript, we have established sharp bounds of the local fractional metric dimension of different types of modified prism networks and it is also proved that local fractional metric dimension remains bounded when the order of these networks approaches to infinity.https://www.aimspress.com/article/doi/10.3934/math.2023551?viewType=HTMLmetric dimensionfractional metric dimensionmodified prism networks |
| spellingShingle | Ahmed Alamer Hassan Zafar Muhammad Javaid Study of modified prism networks via fractional metric dimension AIMS Mathematics metric dimension fractional metric dimension modified prism networks |
| title | Study of modified prism networks via fractional metric dimension |
| title_full | Study of modified prism networks via fractional metric dimension |
| title_fullStr | Study of modified prism networks via fractional metric dimension |
| title_full_unstemmed | Study of modified prism networks via fractional metric dimension |
| title_short | Study of modified prism networks via fractional metric dimension |
| title_sort | study of modified prism networks via fractional metric dimension |
| topic | metric dimension fractional metric dimension modified prism networks |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023551?viewType=HTML |
| work_keys_str_mv | AT ahmedalamer studyofmodifiedprismnetworksviafractionalmetricdimension AT hassanzafar studyofmodifiedprismnetworksviafractionalmetricdimension AT muhammadjavaid studyofmodifiedprismnetworksviafractionalmetricdimension |