An Approach to the Geometric-Arithmetic Index for Graphs under Transformations’ Fact over Pendent Paths
Graph theory is a dynamic tool for designing and modeling of an interconnection system by a graph. The vertices of such graph are processor nodes and edges are the connections between these processors nodes. The topology of a system decides its best use. Geometric-arithmetic index is one of the most...
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| Main Authors: | , , , , , |
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | Complexity |
| Online Access: | http://dx.doi.org/10.1155/2021/3745862 |
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| _version_ | 1832561437833166848 |
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| author | Muhammad Asif Hamad Almohamedh Muhammad Hussain Khalid M Alhamed Abdulrazaq A. Almutairi Sultan Almotairi |
| author_facet | Muhammad Asif Hamad Almohamedh Muhammad Hussain Khalid M Alhamed Abdulrazaq A. Almutairi Sultan Almotairi |
| author_sort | Muhammad Asif |
| collection | DOAJ |
| description | Graph theory is a dynamic tool for designing and modeling of an interconnection system by a graph. The vertices of such graph are processor nodes and edges are the connections between these processors nodes. The topology of a system decides its best use. Geometric-arithmetic index is one of the most studied graph invariant to characterize the topological aspects of underlying interconnection networks or graphs. Transformation over graph is also an important tool to define new network of their own choice in computer science. In this work, we discuss transformed family of graphs. Let Γnk,l be the connected graph comprises by k number of pendent path attached with fully connected vertices of the n-vertex connected graph Γ. Let AαΓnk,l and AαβΓnk,l be the transformed graphs under the fact of transformations Aα and Aαβ, 0≤α≤l, 0≤β≤k−1, respectively. In this work, we obtained new inequalities for the graph family Γnk,l and transformed graphs AαΓnk,l and AαβΓnk,l which involve GAΓ. The presence of GAΓ makes the inequalities more general than all those which were previously defined for the GA index. Furthermore, we characterize extremal graphs which make the inequalities tight. |
| format | Article |
| id | doaj-art-81a5abb1ae924e9b9693bd372ea8296c |
| institution | Kabale University |
| issn | 1076-2787 1099-0526 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Complexity |
| spelling | doaj-art-81a5abb1ae924e9b9693bd372ea8296c2025-02-03T01:25:01ZengWileyComplexity1076-27871099-05262021-01-01202110.1155/2021/37458623745862An Approach to the Geometric-Arithmetic Index for Graphs under Transformations’ Fact over Pendent PathsMuhammad Asif0Hamad Almohamedh1Muhammad Hussain2Khalid M Alhamed3Abdulrazaq A. Almutairi4Sultan Almotairi5Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, PakistanKing Abdulaziz City for Science and Technology (KACST) Riyadh, Riyadh, Saudi ArabiaDepartment of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, PakistanIT Programs Center, Faculty of IT Department, Institute of Public Administration, Riyadh 11141, Saudi ArabiaInformation and Computer Center, The Public Authority for Applied Education and Training, The Ministry of Education, Kuwait, KuwaitDepartment of Natural and Applied Sciences, Faculty of Community College, Majmaah University, Majmaah 11952, Saudi ArabiaGraph theory is a dynamic tool for designing and modeling of an interconnection system by a graph. The vertices of such graph are processor nodes and edges are the connections between these processors nodes. The topology of a system decides its best use. Geometric-arithmetic index is one of the most studied graph invariant to characterize the topological aspects of underlying interconnection networks or graphs. Transformation over graph is also an important tool to define new network of their own choice in computer science. In this work, we discuss transformed family of graphs. Let Γnk,l be the connected graph comprises by k number of pendent path attached with fully connected vertices of the n-vertex connected graph Γ. Let AαΓnk,l and AαβΓnk,l be the transformed graphs under the fact of transformations Aα and Aαβ, 0≤α≤l, 0≤β≤k−1, respectively. In this work, we obtained new inequalities for the graph family Γnk,l and transformed graphs AαΓnk,l and AαβΓnk,l which involve GAΓ. The presence of GAΓ makes the inequalities more general than all those which were previously defined for the GA index. Furthermore, we characterize extremal graphs which make the inequalities tight.http://dx.doi.org/10.1155/2021/3745862 |
| spellingShingle | Muhammad Asif Hamad Almohamedh Muhammad Hussain Khalid M Alhamed Abdulrazaq A. Almutairi Sultan Almotairi An Approach to the Geometric-Arithmetic Index for Graphs under Transformations’ Fact over Pendent Paths Complexity |
| title | An Approach to the Geometric-Arithmetic Index for Graphs under Transformations’ Fact over Pendent Paths |
| title_full | An Approach to the Geometric-Arithmetic Index for Graphs under Transformations’ Fact over Pendent Paths |
| title_fullStr | An Approach to the Geometric-Arithmetic Index for Graphs under Transformations’ Fact over Pendent Paths |
| title_full_unstemmed | An Approach to the Geometric-Arithmetic Index for Graphs under Transformations’ Fact over Pendent Paths |
| title_short | An Approach to the Geometric-Arithmetic Index for Graphs under Transformations’ Fact over Pendent Paths |
| title_sort | approach to the geometric arithmetic index for graphs under transformations fact over pendent paths |
| url | http://dx.doi.org/10.1155/2021/3745862 |
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