{"title":"Maximum energy bicyclic graphs containing two odd cycles with one common vertex","authors":"Jing Gao, Xueliang Li, Ning Yang, Ruiling Zheng","doi":"10.1016/j.dam.2024.09.014","DOIUrl":null,"url":null,"abstract":"<div><div>The energy of a graph is the sum of the absolute values of all eigenvalues of its adjacency matrix. Let <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>6</mn><mo>,</mo><mn>6</mn></mrow></msubsup></math></span> be the graph obtained from two copies of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>6</mn></mrow></msub></math></span> joined by a path <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>10</mn></mrow></msub></math></span>. In 2001, Gutman and Vidović (2001) conjectured that the bicyclic graph with the maximal energy is <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>6</mn><mo>,</mo><mn>6</mn></mrow></msubsup></math></span>. This conjecture is true for bipartite bicyclic graphs. For non-bipartite bicyclic graphs, Ji and Li (2012) proved the conjecture for bicyclic graphs which have exactly two edge-disjoint cycles such that one of them is even and the other is odd. This paper is to prove the conjecture for bicyclic graphs containing two odd cycles with one common vertex.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166218X24004049","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
The energy of a graph is the sum of the absolute values of all eigenvalues of its adjacency matrix. Let be the graph obtained from two copies of joined by a path . In 2001, Gutman and Vidović (2001) conjectured that the bicyclic graph with the maximal energy is . This conjecture is true for bipartite bicyclic graphs. For non-bipartite bicyclic graphs, Ji and Li (2012) proved the conjecture for bicyclic graphs which have exactly two edge-disjoint cycles such that one of them is even and the other is odd. This paper is to prove the conjecture for bicyclic graphs containing two odd cycles with one common vertex.
期刊介绍：
The aim of Discrete Applied Mathematics is to bring together research papers in different areas of algorithmic and applicable discrete mathematics as well as applications of combinatorial mathematics to informatics and various areas of science and technology. Contributions presented to the journal can be research papers, short notes, surveys, and possibly research problems. The "Communications" section will be devoted to the fastest possible publication of recent research results that are checked and recommended for publication by a member of the Editorial Board. The journal will also publish a limited number of book announcements as well as proceedings of conferences. These proceedings will be fully refereed and adhere to the normal standards of the journal.
Potential authors are advised to view the journal and the open calls-for-papers of special issues before submitting their manuscripts. Only high-quality, original work that is within the scope of the journal or the targeted special issue will be considered.
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