Power graphs and exchange property for resolving sets
Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchan...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2019-11-01
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| Series: | Open Mathematics |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/math-2019-0093 |
| _version_ | 1818733715476447232 |
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| author | Abbas Ghulam Ali Usman Munir Mobeen Bokhary Syed Ahtsham Ul Haq Kang Shin Min |
| author_facet | Abbas Ghulam Ali Usman Munir Mobeen Bokhary Syed Ahtsham Ul Haq Kang Shin Min |
| author_sort | Abbas Ghulam |
| collection | DOAJ |
| description | Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups. |
| first_indexed | 2024-12-17T23:53:52Z |
| format | Article |
| id | doaj.art-8aa06b43045b4538b7ef9e7164c35cd1 |
| institution | Directory Open Access Journal |
| issn | 2391-5455 |
| language | English |
| last_indexed | 2024-12-17T23:53:52Z |
| publishDate | 2019-11-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Mathematics |
| spelling | doaj.art-8aa06b43045b4538b7ef9e7164c35cd12022-12-21T21:28:07ZengDe GruyterOpen Mathematics2391-54552019-11-011711303130910.1515/math-2019-0093math-2019-0093Power graphs and exchange property for resolving setsAbbas Ghulam0Ali Usman1Munir Mobeen2Bokhary Syed Ahtsham Ul Haq3Kang Shin Min4Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University Multan, Multan, PakistanCentre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University Multan, Multan, PakistanDepartment of Mathematice, Division of Science and Technology, University of Education, Lahore, PakistanCentre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University Multan, Multan, PakistanDepartment of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju, 52828, KoreaClassical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups.https://doi.org/10.1515/math-2019-0093basisinvolutionmetric dimensionmatroidpower graphresolving set05b3505c12 |
| spellingShingle | Abbas Ghulam Ali Usman Munir Mobeen Bokhary Syed Ahtsham Ul Haq Kang Shin Min Power graphs and exchange property for resolving sets Open Mathematics basis involution metric dimension matroid power graph resolving set 05b35 05c12 |
| title | Power graphs and exchange property for resolving sets |
| title_full | Power graphs and exchange property for resolving sets |
| title_fullStr | Power graphs and exchange property for resolving sets |
| title_full_unstemmed | Power graphs and exchange property for resolving sets |
| title_short | Power graphs and exchange property for resolving sets |
| title_sort | power graphs and exchange property for resolving sets |
| topic | basis involution metric dimension matroid power graph resolving set 05b35 05c12 |
| url | https://doi.org/10.1515/math-2019-0093 |
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