{"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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s021949882550286x","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","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.