{"title":"Near automorphisms of the complement or the square of a cycle","authors":"Jinxing Zhao","doi":"10.1142/s021949882550286x","DOIUrl":null,"url":null,"abstract":"<p>Let <span><math altimg=\"eq-00001.gif\" display=\"inline\"><mi>G</mi></math></span><span></span> be a graph with vertex set <span><math altimg=\"eq-00002.gif\" display=\"inline\"><mi>V</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, <span><math altimg=\"eq-00003.gif\" display=\"inline\"><mi>f</mi></math></span><span></span> a permutation of <span><math altimg=\"eq-00004.gif\" display=\"inline\"><mi>V</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span>. Define <span><math altimg=\"eq-00005.gif\" display=\"inline\"><msub><mrow><mi>δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo>|</mo><mi>d</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">−</mo><mi>d</mi><mo stretchy=\"false\">(</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>,</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>y</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mo>|</mo></math></span><span></span> and <span><math altimg=\"eq-00006.gif\" display=\"inline\"><msub><mrow><mi>δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo>∑</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, where the sum is taken over all unordered pairs <span><math altimg=\"eq-00007.gif\" display=\"inline\"><mi>x</mi></math></span><span></span>, <span><math altimg=\"eq-00008.gif\" display=\"inline\"><mi>y</mi></math></span><span></span> of distinct vertices of <span><math altimg=\"eq-00009.gif\" display=\"inline\"><mi>G</mi></math></span><span></span>. Let <span><math altimg=\"eq-00010.gif\" display=\"inline\"><mi>π</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span> denote the smallest positive value of <span><math altimg=\"eq-00011.gif\" display=\"inline\"><msub><mrow><mi>δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span> among all permutations <span><math altimg=\"eq-00012.gif\" display=\"inline\"><mi>f</mi></math></span><span></span> of <span><math altimg=\"eq-00013.gif\" display=\"inline\"><mi>V</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span>. A permutation <span><math altimg=\"eq-00014.gif\" display=\"inline\"><mi>f</mi></math></span><span></span> with <span><math altimg=\"eq-00015.gif\" display=\"inline\"><msub><mrow><mi>δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>π</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span> is called a near automorphism of <span><math altimg=\"eq-00016.gif\" display=\"inline\"><mi>G</mi></math></span><span></span>. In this paper, the near automorphisms of the complement or the square of a cycle are characterized. Moreover, <span><math altimg=\"eq-00017.gif\" display=\"inline\"><mi>π</mi><mo stretchy=\"false\">(</mo><mover accent=\"false\"><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mo accent=\"true\">¯</mo></mover><mo stretchy=\"false\">)</mo></math></span><span></span> and <span><math altimg=\"eq-00018.gif\" display=\"inline\"><mi>π</mi><mfenced close=\")\" open=\"(\" separators=\"\"><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfenced></math></span><span></span> are determined.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2024-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebra and Its Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s021949882550286x","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let be a graph with vertex set , a permutation of . Define and , where the sum is taken over all unordered pairs , of distinct vertices of . Let denote the smallest positive value of among all permutations of . A permutation with is called a near automorphism of . In this paper, the near automorphisms of the complement or the square of a cycle are characterized. Moreover, and are determined.
期刊介绍:
The Journal of Algebra and Its Applications will publish papers both on theoretical and on applied aspects of Algebra. There is special interest in papers that point out innovative links between areas of Algebra and fields of application. As the field of Algebra continues to experience tremendous growth and diversification, we intend to provide the mathematical community with a central source for information on both the theoretical and the applied aspects of the discipline. While the journal will be primarily devoted to the publication of original research, extraordinary expository articles that encourage communication between algebraists and experts on areas of application as well as those presenting the state of the art on a given algebraic sub-discipline will be considered.