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

IF 3.4 2区 数学 Q1 MATHEMATICS, APPLIED
Sultan Ahmad , Kinkar Chandra Das
{"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 ,&nbsp;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>&lt;</mo><mi>α</mi><mo>&lt;</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>&lt;</mo><mi>α</mi><mo>&lt;</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 ,&nbsp;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>&lt;</mo><mi>α</mi><mo>&lt;</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>&lt;</mo><mi>α</mi><mo>&lt;</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}
引用次数: 0

摘要

对于图G,一般Sombor (SOα)指标定义为:SOα(G)=∑vivj∈E(G)(di2+dj2)α,其中α≠0为实数,E(G)为边集,di表示顶点vi在G中的度数。化学树是其中没有顶点的度数大于4的树,垂顶点是度数为1的顶点。本文旨在完整刻画α0<;α<;α1上具有固定数目(=p)的最大SOα指数的n-顶点化学树,其中α0≈0.144是方程4(32α−25α)+8α−13α+5α−10α=0的唯一非零根,α1≈3.335是方程3(17α−10α)+3(20)α−13α−2(25)α=0的唯一非零解。由于SO1和SO12分别对应于图G的经典遗忘指标和Sombor指标,因此我们的结果适用于这两个指标。此外,刘等。[更多关于化学图Sombor指数及其在苯类烃沸点上的应用,[j]。J.量子化学,121(2021)#26689]解决了仅偶p≥6的化学树的Sombor指数最大化问题,该问题后来由Du等人扩展。[关于固定顺序和固定数量的化学树的键关联度指数,Appl。]数学。计算。464(2024)#128390]包括偶数p≥6和奇数p≥9。相比之下,本文给出了一个更全面的解,充分刻画了所有p≥3对任意α的一般Sombor指数最大化的问题,其中α0<;α<α1。此外,利用辛烷同分异构体数据预测其理化性质,探讨了- 10≤α≤10范围内SOα指数的化学意义。当α的近似值属于{−1,1,8,10}集时,得到了令人满意的结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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 (SOα) index is defined as:SOα(G)=vivjE(G)(di2+dj2)α, where α0 is a real number, E(G) is the edge set and di denotes the degree of a vertex vi 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 SOα index over α0<α<α1, where α00.144 is the unique non-zero root of equation 4(32α25α)+8α13α+5α10α=0 and α13.335 is the unique non-zero solution of equation 3(17α10α)+3(20)α13α2(25)α=0. Since SO1 and SO12 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 p6 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 p6 and odd p9. This paper, in contrast, provides a more comprehensive solution, fully characterizing the problem for all p3 maximizing the general Sombor index for any α, where α0<α<α1. In addition, the chemical significance of the SOα index over the range 10α10 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 {1,1,8,10}.
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来源期刊
CiteScore
7.90
自引率
10.00%
发文量
755
审稿时长
36 days
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.
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