{"title":"化学树指数增强萨格勒布指数未决问题的完整解决方案","authors":"","doi":"10.1016/j.amc.2024.128983","DOIUrl":null,"url":null,"abstract":"<div><p>One of the crucial problems in combinatorics and graph theory is characterizing extremal structures with respect to graph invariants from the family of chemical trees. Cruz et al. (2020) <span><span>[7]</span></span> presented a unified approach to identify extremal chemical trees for degree-based graph invariants in terms of graph order. The exponential augmented Zagreb index (<em>EAZ</em>) is a well-established graph invariant formulated for a graph <em>G</em> as<span><span><span><math><mi>E</mi><mi>A</mi><mi>Z</mi><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><mi>e</mi></mrow><mrow><msup><mrow><mo>(</mo><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>−</mo><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msup></mrow></msup><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> signifies the degree of vertex <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>, and <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the edge set. Due to some special counting features of <em>EAZ</em>, it was not covered by the aforementioned unified approach. As a result, the exploration of extremal chemical trees for this invariant was posed as an open problem in the same article. The present work focuses on generating a complete solution to this problem. Our findings offer maximal and minimal chemical trees of <em>EAZ</em> in terms of the graph order <em>n</em>.</p></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":null,"pages":null},"PeriodicalIF":3.5000,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Complete solution to open problems on exponential augmented Zagreb index of chemical trees\",\"authors\":\"\",\"doi\":\"10.1016/j.amc.2024.128983\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>One of the crucial problems in combinatorics and graph theory is characterizing extremal structures with respect to graph invariants from the family of chemical trees. Cruz et al. (2020) <span><span>[7]</span></span> presented a unified approach to identify extremal chemical trees for degree-based graph invariants in terms of graph order. The exponential augmented Zagreb index (<em>EAZ</em>) is a well-established graph invariant formulated for a graph <em>G</em> as<span><span><span><math><mi>E</mi><mi>A</mi><mi>Z</mi><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><mi>e</mi></mrow><mrow><msup><mrow><mo>(</mo><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>−</mo><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msup></mrow></msup><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> signifies the degree of vertex <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>, and <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the edge set. Due to some special counting features of <em>EAZ</em>, it was not covered by the aforementioned unified approach. As a result, the exploration of extremal chemical trees for this invariant was posed as an open problem in the same article. The present work focuses on generating a complete solution to this problem. Our findings offer maximal and minimal chemical trees of <em>EAZ</em> in terms of the graph order <em>n</em>.</p></div>\",\"PeriodicalId\":55496,\"journal\":{\"name\":\"Applied Mathematics and Computation\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":3.5000,\"publicationDate\":\"2024-08-01\",\"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/S0096300324004442\",\"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/S0096300324004442","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Complete solution to open problems on exponential augmented Zagreb index of chemical trees
One of the crucial problems in combinatorics and graph theory is characterizing extremal structures with respect to graph invariants from the family of chemical trees. Cruz et al. (2020) [7] presented a unified approach to identify extremal chemical trees for degree-based graph invariants in terms of graph order. The exponential augmented Zagreb index (EAZ) is a well-established graph invariant formulated for a graph G as where signifies the degree of vertex , and is the edge set. Due to some special counting features of EAZ, it was not covered by the aforementioned unified approach. As a result, the exploration of extremal chemical trees for this invariant was posed as an open problem in the same article. The present work focuses on generating a complete solution to this problem. Our findings offer maximal and minimal chemical trees of EAZ in terms of the graph order n.
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.