{"title":"具有给定垂顶点数的化学树的一般Sombor指数最大化的完整解","authors":"Sultan Ahmad , Kinkar Chandra Das","doi":"10.1016/j.amc.2025.129532","DOIUrl":null,"url":null,"abstract":"<div><div>For a graph <em>G</em>, the general Sombor (<span><math><msub><mrow><mi>SO</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>) index is defined as:<span><span><span><math><msub><mrow><mi>SO</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><munder><mo>∑</mo><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></munder><msup><mrow><mo>(</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo></mrow><mrow><mi>α</mi></mrow></msup><mo>,</mo></math></span></span></span> where <span><math><mi>α</mi><mo>≠</mo><mn>0</mn></math></span> is a real number, <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the edge set and <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> denotes the degree of a vertex <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> in <em>G</em>. A chemical tree is a tree in which no vertex has a degree greater than 4, and a pendant vertex is a vertex with degree 1. This paper aims to completely characterize the <em>n</em>− vertex chemical trees with a fixed number of pendant vertices (=<em>p</em>) that maximize the <span><math><msub><mrow><mi>SO</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> index over <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>0</mn></mrow></msub><mo><</mo><mi>α</mi><mo><</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, where <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≈</mo><mn>0.144</mn></math></span> is the unique non-zero root of equation <span><math><mn>4</mn><mo>(</mo><msup><mrow><mn>32</mn></mrow><mrow><mi>α</mi></mrow></msup><mo>−</mo><msup><mrow><mn>25</mn></mrow><mrow><mi>α</mi></mrow></msup><mo>)</mo><mo>+</mo><msup><mrow><mn>8</mn></mrow><mrow><mi>α</mi></mrow></msup><mo>−</mo><msup><mrow><mn>13</mn></mrow><mrow><mi>α</mi></mrow></msup><mo>+</mo><msup><mrow><mn>5</mn></mrow><mrow><mi>α</mi></mrow></msup><mo>−</mo><msup><mrow><mn>10</mn></mrow><mrow><mi>α</mi></mrow></msup><mo>=</mo><mn>0</mn></math></span> and <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≈</mo><mn>3.335</mn></math></span> is the unique non-zero solution of equation <span><math><mn>3</mn><mspace></mspace><mo>(</mo><msup><mrow><mn>17</mn></mrow><mrow><mi>α</mi></mrow></msup><mo>−</mo><msup><mrow><mn>10</mn></mrow><mrow><mi>α</mi></mrow></msup><mo>)</mo><mo>+</mo><mn>3</mn><mspace></mspace><msup><mrow><mo>(</mo><mn>20</mn><mo>)</mo></mrow><mrow><mi>α</mi></mrow></msup><mo>−</mo><msup><mrow><mn>13</mn></mrow><mrow><mi>α</mi></mrow></msup><mo>−</mo><mn>2</mn><mspace></mspace><msup><mrow><mo>(</mo><mn>25</mn><mo>)</mo></mrow><mrow><mi>α</mi></mrow></msup><mo>=</mo><mn>0</mn></math></span>. Since <span><math><msub><mrow><mi>SO</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>SO</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msub></math></span> correspond to the classical forgotten and the Sombor indices of a graph <em>G</em>, respectively, our results apply to both indices. Moreover, Liu et al. <strong>[More on Sombor indices of chemical graphs and their applications to the boiling point of benzenoid hydrocarbons, <em>Int. J. Quantum Chem.</em> 121 (2021) #26689]</strong> addressed the problem of maximizing the Sombor index for chemical trees with even <span><math><mi>p</mi><mo>≥</mo><mn>6</mn></math></span> only, which was later extended by Du et al. <strong>[On bond incident degree index of chemical trees with a fixed order and a fixed number of leaves, <em>Appl. Math. Comput.</em> 464 (2024) #128390]</strong> to include both even <span><math><mi>p</mi><mo>≥</mo><mn>6</mn></math></span> and odd <span><math><mi>p</mi><mo>≥</mo><mn>9</mn></math></span>. This paper, in contrast, provides a more comprehensive solution, fully characterizing the problem for all <span><math><mi>p</mi><mo>≥</mo><mn>3</mn></math></span> maximizing the general Sombor index for any <em>α</em>, where <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>0</mn></mrow></msub><mo><</mo><mi>α</mi><mo><</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. In addition, the chemical significance of the <span><math><msub><mrow><mi>SO</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> index over the range <span><math><mo>−</mo><mn>10</mn><mo>≤</mo><mi>α</mi><mo>≤</mo><mn>10</mn></math></span> is explored by using the octane isomers dataset to predict their physicochemical properties. Promising results are obtained when the approximated values of <em>α</em> belong to the set <span><math><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mspace></mspace><mn>1</mn><mo>,</mo><mspace></mspace><mn>8</mn><mo>,</mo><mspace></mspace><mn>10</mn><mo>}</mo></math></span>.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"505 ","pages":"Article 129532"},"PeriodicalIF":3.4000,"publicationDate":"2025-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A complete solution for maximizing the general Sombor index of chemical trees with given number of pendant vertices\",\"authors\":\"Sultan Ahmad , Kinkar Chandra Das\",\"doi\":\"10.1016/j.amc.2025.129532\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>For a graph <em>G</em>, the general Sombor (<span><math><msub><mrow><mi>SO</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>) index is defined as:<span><span><span><math><msub><mrow><mi>SO</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><munder><mo>∑</mo><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></munder><msup><mrow><mo>(</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo></mrow><mrow><mi>α</mi></mrow></msup><mo>,</mo></math></span></span></span> where <span><math><mi>α</mi><mo>≠</mo><mn>0</mn></math></span> is a real number, <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the edge set and <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> denotes the degree of a vertex <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> in <em>G</em>. A chemical tree is a tree in which no vertex has a degree greater than 4, and a pendant vertex is a vertex with degree 1. This paper aims to completely characterize the <em>n</em>− vertex chemical trees with a fixed number of pendant vertices (=<em>p</em>) that maximize the <span><math><msub><mrow><mi>SO</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> index over <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>0</mn></mrow></msub><mo><</mo><mi>α</mi><mo><</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, where <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≈</mo><mn>0.144</mn></math></span> is the unique non-zero root of equation <span><math><mn>4</mn><mo>(</mo><msup><mrow><mn>32</mn></mrow><mrow><mi>α</mi></mrow></msup><mo>−</mo><msup><mrow><mn>25</mn></mrow><mrow><mi>α</mi></mrow></msup><mo>)</mo><mo>+</mo><msup><mrow><mn>8</mn></mrow><mrow><mi>α</mi></mrow></msup><mo>−</mo><msup><mrow><mn>13</mn></mrow><mrow><mi>α</mi></mrow></msup><mo>+</mo><msup><mrow><mn>5</mn></mrow><mrow><mi>α</mi></mrow></msup><mo>−</mo><msup><mrow><mn>10</mn></mrow><mrow><mi>α</mi></mrow></msup><mo>=</mo><mn>0</mn></math></span> and <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≈</mo><mn>3.335</mn></math></span> is the unique non-zero solution of equation <span><math><mn>3</mn><mspace></mspace><mo>(</mo><msup><mrow><mn>17</mn></mrow><mrow><mi>α</mi></mrow></msup><mo>−</mo><msup><mrow><mn>10</mn></mrow><mrow><mi>α</mi></mrow></msup><mo>)</mo><mo>+</mo><mn>3</mn><mspace></mspace><msup><mrow><mo>(</mo><mn>20</mn><mo>)</mo></mrow><mrow><mi>α</mi></mrow></msup><mo>−</mo><msup><mrow><mn>13</mn></mrow><mrow><mi>α</mi></mrow></msup><mo>−</mo><mn>2</mn><mspace></mspace><msup><mrow><mo>(</mo><mn>25</mn><mo>)</mo></mrow><mrow><mi>α</mi></mrow></msup><mo>=</mo><mn>0</mn></math></span>. Since <span><math><msub><mrow><mi>SO</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>SO</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msub></math></span> correspond to the classical forgotten and the Sombor indices of a graph <em>G</em>, respectively, our results apply to both indices. Moreover, Liu et al. <strong>[More on Sombor indices of chemical graphs and their applications to the boiling point of benzenoid hydrocarbons, <em>Int. J. Quantum Chem.</em> 121 (2021) #26689]</strong> addressed the problem of maximizing the Sombor index for chemical trees with even <span><math><mi>p</mi><mo>≥</mo><mn>6</mn></math></span> only, which was later extended by Du et al. <strong>[On bond incident degree index of chemical trees with a fixed order and a fixed number of leaves, <em>Appl. Math. Comput.</em> 464 (2024) #128390]</strong> to include both even <span><math><mi>p</mi><mo>≥</mo><mn>6</mn></math></span> and odd <span><math><mi>p</mi><mo>≥</mo><mn>9</mn></math></span>. This paper, in contrast, provides a more comprehensive solution, fully characterizing the problem for all <span><math><mi>p</mi><mo>≥</mo><mn>3</mn></math></span> maximizing the general Sombor index for any <em>α</em>, where <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>0</mn></mrow></msub><mo><</mo><mi>α</mi><mo><</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. In addition, the chemical significance of the <span><math><msub><mrow><mi>SO</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> index over the range <span><math><mo>−</mo><mn>10</mn><mo>≤</mo><mi>α</mi><mo>≤</mo><mn>10</mn></math></span> is explored by using the octane isomers dataset to predict their physicochemical properties. Promising results are obtained when the approximated values of <em>α</em> belong to the set <span><math><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mspace></mspace><mn>1</mn><mo>,</mo><mspace></mspace><mn>8</mn><mo>,</mo><mspace></mspace><mn>10</mn><mo>}</mo></math></span>.</div></div>\",\"PeriodicalId\":55496,\"journal\":{\"name\":\"Applied Mathematics and Computation\",\"volume\":\"505 \",\"pages\":\"Article 129532\"},\"PeriodicalIF\":3.4000,\"publicationDate\":\"2025-05-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Applied Mathematics and Computation\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0096300325002589\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics and Computation","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0096300325002589","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
A complete solution for maximizing the general Sombor index of chemical trees with given number of pendant vertices
For a graph G, the general Sombor () index is defined as: where is a real number, is the edge set and denotes the degree of a vertex in G. A chemical tree is a tree in which no vertex has a degree greater than 4, and a pendant vertex is a vertex with degree 1. This paper aims to completely characterize the n− vertex chemical trees with a fixed number of pendant vertices (=p) that maximize the index over , where is the unique non-zero root of equation and is the unique non-zero solution of equation . Since and correspond to the classical forgotten and the Sombor indices of a graph G, respectively, our results apply to both indices. Moreover, Liu et al. [More on Sombor indices of chemical graphs and their applications to the boiling point of benzenoid hydrocarbons, Int. J. Quantum Chem. 121 (2021) #26689] addressed the problem of maximizing the Sombor index for chemical trees with even only, which was later extended by Du et al. [On bond incident degree index of chemical trees with a fixed order and a fixed number of leaves, Appl. Math. Comput. 464 (2024) #128390] to include both even and odd . This paper, in contrast, provides a more comprehensive solution, fully characterizing the problem for all maximizing the general Sombor index for any α, where . In addition, the chemical significance of the index over the range is explored by using the octane isomers dataset to predict their physicochemical properties. Promising results are obtained when the approximated values of α belong to the set .
期刊介绍:
Applied Mathematics and Computation addresses work at the interface between applied mathematics, numerical computation, and applications of systems – oriented ideas to the physical, biological, social, and behavioral sciences, and emphasizes papers of a computational nature focusing on new algorithms, their analysis and numerical results.
In addition to presenting research papers, Applied Mathematics and Computation publishes review articles and single–topics issues.