General Randić Index of Unicyclic Graphs and Its Applications to Drugs
In this work, we determine the maximum general Randić index (a general symmetric function of vertex degrees) for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>η</mi><mn>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-01-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/16/1/113 |
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| author | Alaa Altassan Muhammad Imran |
| author_facet | Alaa Altassan Muhammad Imran |
| author_sort | Alaa Altassan |
| collection | DOAJ |
| description | In this work, we determine the maximum general Randić index (a general symmetric function of vertex degrees) for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>η</mi><mn>0</mn></msub><mo>≤</mo><mi>η</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula> among all <i>n</i>-vertex unicyclic graphs with a fixed maximum degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Δ</mi></semantics></math></inline-formula> and the maximum and the second maximum general Randić index for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>η</mi><mn>0</mn></msub><mo>≤</mo><mi>η</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula> among all <i>n</i>-vertex unicyclic graphs, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>η</mi><mn>0</mn></msub><mo>≈</mo><mo>−</mo><mn>0.21</mn></mrow></semantics></math></inline-formula>. We establish sharp inequalities and identify the graphs attaining the inequalities. Thereby, extremal graphs are obtained for the general Randić index, and certain open gaps in the theory of extremal unicyclic graphs are filled (some open problems are provided). We use computational software to calculate the Randić index for the chemical trees up to order 7 and use the statistical (linear regression) analysis to discuss the various applications of the Randić index with the physical properties of drugs on the said chemical trees. We show that the Randić index is better correlated with the heat of vaporization for these alkanes. |
| first_indexed | 2024-03-08T10:34:54Z |
| format | Article |
| id | doaj.art-df72c24cb33a4a779b9c647215852913 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-08T10:34:54Z |
| publishDate | 2024-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-df72c24cb33a4a779b9c6472158529132024-01-26T18:38:58ZengMDPI AGSymmetry2073-89942024-01-0116111310.3390/sym16010113General Randić Index of Unicyclic Graphs and Its Applications to DrugsAlaa Altassan0Muhammad Imran1Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi ArabiaDepartment of Mathematical Sciences, United Arab Emirates University, Al Ain 15551, United Arab EmiratesIn this work, we determine the maximum general Randić index (a general symmetric function of vertex degrees) for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>η</mi><mn>0</mn></msub><mo>≤</mo><mi>η</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula> among all <i>n</i>-vertex unicyclic graphs with a fixed maximum degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Δ</mi></semantics></math></inline-formula> and the maximum and the second maximum general Randić index for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>η</mi><mn>0</mn></msub><mo>≤</mo><mi>η</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula> among all <i>n</i>-vertex unicyclic graphs, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>η</mi><mn>0</mn></msub><mo>≈</mo><mo>−</mo><mn>0.21</mn></mrow></semantics></math></inline-formula>. We establish sharp inequalities and identify the graphs attaining the inequalities. Thereby, extremal graphs are obtained for the general Randić index, and certain open gaps in the theory of extremal unicyclic graphs are filled (some open problems are provided). We use computational software to calculate the Randić index for the chemical trees up to order 7 and use the statistical (linear regression) analysis to discuss the various applications of the Randić index with the physical properties of drugs on the said chemical trees. We show that the Randić index is better correlated with the heat of vaporization for these alkanes.https://www.mdpi.com/2073-8994/16/1/113Randić indexgeneral Randić indexunicyclic graphextremal graphsmaximum degree |
| spellingShingle | Alaa Altassan Muhammad Imran General Randić Index of Unicyclic Graphs and Its Applications to Drugs Symmetry Randić index general Randić index unicyclic graph extremal graphs maximum degree |
| title | General Randić Index of Unicyclic Graphs and Its Applications to Drugs |
| title_full | General Randić Index of Unicyclic Graphs and Its Applications to Drugs |
| title_fullStr | General Randić Index of Unicyclic Graphs and Its Applications to Drugs |
| title_full_unstemmed | General Randić Index of Unicyclic Graphs and Its Applications to Drugs |
| title_short | General Randić Index of Unicyclic Graphs and Its Applications to Drugs |
| title_sort | general randic index of unicyclic graphs and its applications to drugs |
| topic | Randić index general Randić index unicyclic graph extremal graphs maximum degree |
| url | https://www.mdpi.com/2073-8994/16/1/113 |
| work_keys_str_mv | AT alaaaltassan generalrandicindexofunicyclicgraphsanditsapplicationstodrugs AT muhammadimran generalrandicindexofunicyclicgraphsanditsapplicationstodrugs |