Near automorphisms of the complement or the square of a cycle

Pub Date : 2024-05-25 DOI:10.1142/s021949882550286x
Jinxing Zhao
{"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}
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Abstract

Let G be a graph with vertex set V(G), f a permutation of V(G). Define δf(x,y)=|d(x,y)d(f(x),f(y))| and δf(G)=δf(x,y), where the sum is taken over all unordered pairs x, y of distinct vertices of G. Let π(G) denote the smallest positive value of δf(G) among all permutations f of V(G). A permutation f with δf(G)=π(G) is called a near automorphism of G. In this paper, the near automorphisms of the complement or the square of a cycle are characterized. Moreover, π(Cn¯) and πCn2 are determined.

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循环的补集或平方的近自动形态
设 G 是一个有顶点集 V(G) 的图,f 是 V(G) 的置换。定义 δf(x,y)=|d(x,y)-d(f(x),f(y))| 和 δf(G)=∑δf(x,y),其中总和取自 G 中所有无序的不同顶点对 x、y。具有 δf(G)=π(G)的置换 f 称为 G 的近自动形。此外,本文还确定了 π(Cn¯) 和 πCn2 。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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