On the Constant Partition Dimension of Some Generalized Families of Toeplitz Graph
The use of graph theory is prevalent in the field of network design, whereby it finds utility in several domains such as the development of integrated circuits, communication networks, and transportation systems. The comprehension of partition dimensions may facilitate the enhancement of network des...
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| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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Wiley
2024-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2024/4721104 |
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| author | Ali N. A. Koam Ali Ahmad Maryam Salem Alatawi Adnan Khalil Muhammad Azeem |
| author_facet | Ali N. A. Koam Ali Ahmad Maryam Salem Alatawi Adnan Khalil Muhammad Azeem |
| author_sort | Ali N. A. Koam |
| collection | DOAJ |
| description | The use of graph theory is prevalent in the field of network design, whereby it finds utility in several domains such as the development of integrated circuits, communication networks, and transportation systems. The comprehension of partition dimensions may facilitate the enhancement of network designs in terms of efficiency and reliability. Let VG be a vertex set of a connected graph and S⊂VG, the distance between a vertex v and subset S is defined as dv,S=mindv,xx∈S. An k-ordered partition of VG is Rp=Rp1,Rp2,…,Rpk and the identification code of vertex v with respect to Rp is the k-tuple rvRp=dv,Rp1,dv,Rp2,…,dv,Rpk. The k-partition Rp is said to be a partition resolving if rvRp, ∀v∈VG are distinct. Partition dimension is the minimum number k in the partition resolving set, symbolized by pdG. In this paper, we considered the families of graph named as Toeplitz network, and proved that the partition dimension of Tnt1,t2, where t1=2,3, and gcdt1,t2=1 is constant. |
| format | Article |
| id | doaj-art-e6c13b2eb1274fcd9a214dcd745dabf2 |
| institution | Kabale University |
| issn | 2314-4785 |
| language | English |
| publishDate | 2024-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-e6c13b2eb1274fcd9a214dcd745dabf22025-02-03T10:09:59ZengWileyJournal of Mathematics2314-47852024-01-01202410.1155/2024/4721104On the Constant Partition Dimension of Some Generalized Families of Toeplitz GraphAli N. A. Koam0Ali Ahmad1Maryam Salem Alatawi2Adnan Khalil3Muhammad Azeem4Department of MathematicsDepartment of Computer ScienceDepartment of MathematicsDepartment Computer SciencesDepartment of MathematicsThe use of graph theory is prevalent in the field of network design, whereby it finds utility in several domains such as the development of integrated circuits, communication networks, and transportation systems. The comprehension of partition dimensions may facilitate the enhancement of network designs in terms of efficiency and reliability. Let VG be a vertex set of a connected graph and S⊂VG, the distance between a vertex v and subset S is defined as dv,S=mindv,xx∈S. An k-ordered partition of VG is Rp=Rp1,Rp2,…,Rpk and the identification code of vertex v with respect to Rp is the k-tuple rvRp=dv,Rp1,dv,Rp2,…,dv,Rpk. The k-partition Rp is said to be a partition resolving if rvRp, ∀v∈VG are distinct. Partition dimension is the minimum number k in the partition resolving set, symbolized by pdG. In this paper, we considered the families of graph named as Toeplitz network, and proved that the partition dimension of Tnt1,t2, where t1=2,3, and gcdt1,t2=1 is constant.http://dx.doi.org/10.1155/2024/4721104 |
| spellingShingle | Ali N. A. Koam Ali Ahmad Maryam Salem Alatawi Adnan Khalil Muhammad Azeem On the Constant Partition Dimension of Some Generalized Families of Toeplitz Graph Journal of Mathematics |
| title | On the Constant Partition Dimension of Some Generalized Families of Toeplitz Graph |
| title_full | On the Constant Partition Dimension of Some Generalized Families of Toeplitz Graph |
| title_fullStr | On the Constant Partition Dimension of Some Generalized Families of Toeplitz Graph |
| title_full_unstemmed | On the Constant Partition Dimension of Some Generalized Families of Toeplitz Graph |
| title_short | On the Constant Partition Dimension of Some Generalized Families of Toeplitz Graph |
| title_sort | on the constant partition dimension of some generalized families of toeplitz graph |
| url | http://dx.doi.org/10.1155/2024/4721104 |
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