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:...

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Autores principales:	Muhammad Asif, Bartłomiej Kizielewicz, Atiq ur Rehman, Muhammad Hussain, Wojciech Sałabun
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	Zagreb connection indices graph invariants interconnection networks mk graphs topological index Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/d919e146a84b4c949bc47ff07ace5cbf
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spelling	oai:doaj.org-article:d919e146a84b4c949bc47ff07ace5cbf2021-11-25T19:05:50ZStudy of <i>θ<sup>ϕ</sup></i> Networks via Zagreb Connection Indices10.3390/sym131119912073-8994https://doaj.org/article/d919e146a84b4c949bc47ff07ace5cbf2021-10-01T00:00:00Zhttps://www.mdpi.com/2073-8994/13/11/1991https://doaj.org/toc/2073-8994Graph 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.Muhammad AsifBartłomiej KizielewiczAtiq ur RehmanMuhammad HussainWojciech SałabunMDPI AGarticleZagreb connection indicesgraph invariantsinterconnection networksmk graphstopological indexMathematicsQA1-939ENSymmetry, Vol 13, Iss 1991, p 1991 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Zagreb connection indices graph invariants interconnection networks mk graphs topological index Mathematics QA1-939
spellingShingle	Zagreb connection indices graph invariants interconnection networks mk graphs topological index Mathematics QA1-939 Muhammad Asif Bartłomiej Kizielewicz Atiq ur Rehman Muhammad Hussain Wojciech Sałabun Study of <i>θ<sup>ϕ</sup></i> Networks via Zagreb Connection Indices
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.
format	article
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
title	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_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_sort	study of <i>θ<sup>ϕ</sup></i> networks via zagreb connection indices
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/d919e146a84b4c949bc47ff07ace5cbf
work_keys_str_mv	AT muhammadasif studyofithsupphsupinetworksviazagrebconnectionindices AT bartłomiejkizielewicz studyofithsupphsupinetworksviazagrebconnectionindices AT atiqurrehman studyofithsupphsupinetworksviazagrebconnectionindices AT muhammadhussain studyofithsupphsupinetworksviazagrebconnectionindices AT wojciechsałabun studyofithsupphsupinetworksviazagrebconnectionindices
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Study of <i>θ<sup>ϕ</sup></i> Networks via Zagreb Connection Indices

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