Study of <i>θ<sup>ϕ</sup></i> Networks via Zagreb Connection Indices
Graph theory can be used to optimize interconnection network systems. The compatibility of such networks mainly depends on their topology. Topological indices may characterize the topology of such networks. In this work, we studied a symmetric network <inline-formula><math xmlns="http:...
| Main Authors: | , , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-10-01
|
| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/13/11/1991 |
| _version_ | 1797508401132994560 |
|---|---|
| author | Muhammad Asif Bartłomiej Kizielewicz Atiq ur Rehman Muhammad Hussain Wojciech Sałabun |
| author_facet | Muhammad Asif Bartłomiej Kizielewicz Atiq ur Rehman Muhammad Hussain Wojciech Sałabun |
| author_sort | Muhammad Asif |
| collection | DOAJ |
| description | Graph theory can be used to optimize interconnection network systems. The compatibility of such networks mainly depends on their topology. Topological indices may characterize the topology of such networks. In this work, we studied a symmetric network <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>θ</mi><mi>ϕ</mi></msup></semantics></math></inline-formula> formed by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula> time repetition of the process of joining <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula> copies of a selected graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> in such a way that corresponding vertices of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> in all the copies are joined with each other by a new edge. The symmetry of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>θ</mi><mi>ϕ</mi></msup></semantics></math></inline-formula> is ensured by the involvement of complete graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mi>θ</mi></msub></semantics></math></inline-formula> in the construction process. The free hand to choose an initial graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> and formation of chemical graphs using <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>θ</mi><mi>ϕ</mi></msup><mo>Ω</mo></mrow></semantics></math></inline-formula> enhance its importance as a family of graphs which covers all the pre-defined graphs, along with space for new graphs, possibly formed in this way. We used Zagreb connection indices for the characterization of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>θ</mi><mi>ϕ</mi></msup><mo>Ω</mo></mrow></semantics></math></inline-formula>. These indices have gained worth in the field of chemical graph theory in very small duration due to their predictive power for enthalpy, entropy, and acentric factor. These computations are mathematically novel and assist in topological characterization of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>θ</mi><mi>ϕ</mi></msup><mo>Ω</mo></mrow></semantics></math></inline-formula> to enable its emerging use. |
| first_indexed | 2024-03-10T05:02:31Z |
| format | Article |
| id | doaj.art-d919e146a84b4c949bc47ff07ace5cbf |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T05:02:31Z |
| publishDate | 2021-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-d919e146a84b4c949bc47ff07ace5cbf2023-11-23T01:43:00ZengMDPI AGSymmetry2073-89942021-10-011311199110.3390/sym13111991Study of <i>θ<sup>ϕ</sup></i> Networks via Zagreb Connection IndicesMuhammad Asif0Bartłomiej Kizielewicz1Atiq ur Rehman2Muhammad Hussain3Wojciech Sałabun4Department of Mathematics, COMSATS University Islamabad (Lahore Campus), Lahore 53713, PakistanResearch Team on Intelligent Decision Support Systems, Department of Artificial Intelligence and Applied Mathematics, Faculty of Computer Science and Information Technology, West Pomeranian University of Technology in Szczecin, ul. Żołnierska 49, 71-210 Szczecin, PolandDepartment of Mathematics, COMSATS University Islamabad (Lahore Campus), Lahore 53713, PakistanDepartment of Mathematics, COMSATS University Islamabad (Lahore Campus), Lahore 53713, PakistanResearch Team on Intelligent Decision Support Systems, Department of Artificial Intelligence and Applied Mathematics, Faculty of Computer Science and Information Technology, West Pomeranian University of Technology in Szczecin, ul. Żołnierska 49, 71-210 Szczecin, PolandGraph theory can be used to optimize interconnection network systems. The compatibility of such networks mainly depends on their topology. Topological indices may characterize the topology of such networks. In this work, we studied a symmetric network <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>θ</mi><mi>ϕ</mi></msup></semantics></math></inline-formula> formed by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula> time repetition of the process of joining <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula> copies of a selected graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> in such a way that corresponding vertices of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> in all the copies are joined with each other by a new edge. The symmetry of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>θ</mi><mi>ϕ</mi></msup></semantics></math></inline-formula> is ensured by the involvement of complete graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mi>θ</mi></msub></semantics></math></inline-formula> in the construction process. The free hand to choose an initial graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> and formation of chemical graphs using <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>θ</mi><mi>ϕ</mi></msup><mo>Ω</mo></mrow></semantics></math></inline-formula> enhance its importance as a family of graphs which covers all the pre-defined graphs, along with space for new graphs, possibly formed in this way. We used Zagreb connection indices for the characterization of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>θ</mi><mi>ϕ</mi></msup><mo>Ω</mo></mrow></semantics></math></inline-formula>. These indices have gained worth in the field of chemical graph theory in very small duration due to their predictive power for enthalpy, entropy, and acentric factor. These computations are mathematically novel and assist in topological characterization of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>θ</mi><mi>ϕ</mi></msup><mo>Ω</mo></mrow></semantics></math></inline-formula> to enable its emerging use.https://www.mdpi.com/2073-8994/13/11/1991Zagreb connection indicesgraph invariantsinterconnection networksmk graphstopological index |
| spellingShingle | Muhammad Asif Bartłomiej Kizielewicz Atiq ur Rehman Muhammad Hussain Wojciech Sałabun Study of <i>θ<sup>ϕ</sup></i> Networks via Zagreb Connection Indices Symmetry Zagreb connection indices graph invariants interconnection networks mk graphs topological index |
| title | Study of <i>θ<sup>ϕ</sup></i> Networks via Zagreb Connection Indices |
| title_full | Study of <i>θ<sup>ϕ</sup></i> Networks via Zagreb Connection Indices |
| title_fullStr | Study of <i>θ<sup>ϕ</sup></i> Networks via Zagreb Connection Indices |
| title_full_unstemmed | Study of <i>θ<sup>ϕ</sup></i> Networks via Zagreb Connection Indices |
| title_short | Study of <i>θ<sup>ϕ</sup></i> Networks via Zagreb Connection Indices |
| title_sort | study of i θ sup ϕ sup i networks via zagreb connection indices |
| topic | Zagreb connection indices graph invariants interconnection networks mk graphs topological index |
| url | https://www.mdpi.com/2073-8994/13/11/1991 |
| work_keys_str_mv | AT muhammadasif studyofithsupphsupinetworksviazagrebconnectionindices AT bartłomiejkizielewicz studyofithsupphsupinetworksviazagrebconnectionindices AT atiqurrehman studyofithsupphsupinetworksviazagrebconnectionindices AT muhammadhussain studyofithsupphsupinetworksviazagrebconnectionindices AT wojciechsałabun studyofithsupphsupinetworksviazagrebconnectionindices |