{"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}
引用次数: 0
Abstract
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.