Power Graphs of Finite Groups Determined by Hosoya Properties
Suppose <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula> is a finite group. The power graph represented by <...
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MDPI AG
2022-01-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/24/2/213 |
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| author | Fawad Ali Bilal Ahmad Rather Anwarud Din Tareq Saeed Asad Ullah |
| author_facet | Fawad Ali Bilal Ahmad Rather Anwarud Din Tareq Saeed Asad Ullah |
| author_sort | Fawad Ali |
| collection | DOAJ |
| description | Suppose <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula> is a finite group. The power graph represented by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">P</mi><mo>(</mo><mi mathvariant="script">G</mi><mo>)</mo></mrow></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula> is a graph, whose node set is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>, and two different elements are adjacent if and only if one is an integral power of the other. The Hosoya polynomial contains much information regarding graph invariants depending on the distance. In this article, we discuss the Hosoya characteristics (the Hosoya polynomial and its reciprocal) of the power graph related to an algebraic structure formed by the symmetries of regular molecular gones. As a consequence, we determined the Hosoya index of the power graphs of the dihedral and the generalized groups. This information is useful in determining the renowned chemical descriptors depending on the distance. The total number of matchings in a graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Γ</mi></semantics></math></inline-formula> is known as the <i>Z</i>-index or Hosoya index. The <i>Z</i>-index is a well-known type of topological index, which is popular in combinatorial chemistry and can be used to deal with a variety of chemical characteristics in molecular structures. |
| first_indexed | 2024-03-09T22:01:17Z |
| format | Article |
| id | doaj.art-2187eb18fba443f19ec0a998661cf5d6 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-09T22:01:17Z |
| publishDate | 2022-01-01 |
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| record_format | Article |
| series | Entropy |
| spelling | doaj.art-2187eb18fba443f19ec0a998661cf5d62023-11-23T19:47:48ZengMDPI AGEntropy1099-43002022-01-0124221310.3390/e24020213Power Graphs of Finite Groups Determined by Hosoya PropertiesFawad Ali0Bilal Ahmad Rather1Anwarud Din2Tareq Saeed3Asad Ullah4Institute of Numerical Sciences, Kohat University of Science & Technology, Kohat 26000, PakistanMathematical Sciences Department, College of Science, United Arab Emirates University, Al Ain P.O. Box 15551, United Arab EmiratesDepartment of Mathematics, Sun Yat-sen University, Guangzhou 510275, ChinaNonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi ArabiaDepartment of Mathematical Sciences, University of Lakki Marwat, Lakki Marwat 28420, PakistanSuppose <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula> is a finite group. The power graph represented by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">P</mi><mo>(</mo><mi mathvariant="script">G</mi><mo>)</mo></mrow></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula> is a graph, whose node set is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>, and two different elements are adjacent if and only if one is an integral power of the other. The Hosoya polynomial contains much information regarding graph invariants depending on the distance. In this article, we discuss the Hosoya characteristics (the Hosoya polynomial and its reciprocal) of the power graph related to an algebraic structure formed by the symmetries of regular molecular gones. As a consequence, we determined the Hosoya index of the power graphs of the dihedral and the generalized groups. This information is useful in determining the renowned chemical descriptors depending on the distance. The total number of matchings in a graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Γ</mi></semantics></math></inline-formula> is known as the <i>Z</i>-index or Hosoya index. The <i>Z</i>-index is a well-known type of topological index, which is popular in combinatorial chemistry and can be used to deal with a variety of chemical characteristics in molecular structures.https://www.mdpi.com/1099-4300/24/2/213molecular structurechemical graphspower graphsfinite groupsHosoya indexHosoya polynomial |
| spellingShingle | Fawad Ali Bilal Ahmad Rather Anwarud Din Tareq Saeed Asad Ullah Power Graphs of Finite Groups Determined by Hosoya Properties Entropy molecular structure chemical graphs power graphs finite groups Hosoya index Hosoya polynomial |
| title | Power Graphs of Finite Groups Determined by Hosoya Properties |
| title_full | Power Graphs of Finite Groups Determined by Hosoya Properties |
| title_fullStr | Power Graphs of Finite Groups Determined by Hosoya Properties |
| title_full_unstemmed | Power Graphs of Finite Groups Determined by Hosoya Properties |
| title_short | Power Graphs of Finite Groups Determined by Hosoya Properties |
| title_sort | power graphs of finite groups determined by hosoya properties |
| topic | molecular structure chemical graphs power graphs finite groups Hosoya index Hosoya polynomial |
| url | https://www.mdpi.com/1099-4300/24/2/213 |
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