{"title":"关于度界的图的不规则性","authors":"Dieter Rautenbach, Florian Werner","doi":"10.37236/11948","DOIUrl":null,"url":null,"abstract":"Albertson defined the irregularity of a graph $G$ as $$irr(G)=\\sum\\limits_{uv\\in E(G)}|d_G(u)-d_G(v)|.$$ For a graph $G$ with $n$ vertices, $m$ edges, maximum degree $\\Delta$, and $d=\\left\\lfloor \\frac{\\Delta m}{\\Delta n-m}\\right\\rfloor$, we show $$irr(G)\\leq d(d+1)n+\\frac{1}{\\Delta}\\left(\\Delta^2-(2d+1)\\Delta-d^2-d\\right)m.$$","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Irregularity of Graphs Respecting Degree Bounds\",\"authors\":\"Dieter Rautenbach, Florian Werner\",\"doi\":\"10.37236/11948\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Albertson defined the irregularity of a graph $G$ as $$irr(G)=\\\\sum\\\\limits_{uv\\\\in E(G)}|d_G(u)-d_G(v)|.$$ For a graph $G$ with $n$ vertices, $m$ edges, maximum degree $\\\\Delta$, and $d=\\\\left\\\\lfloor \\\\frac{\\\\Delta m}{\\\\Delta n-m}\\\\right\\\\rfloor$, we show $$irr(G)\\\\leq d(d+1)n+\\\\frac{1}{\\\\Delta}\\\\left(\\\\Delta^2-(2d+1)\\\\Delta-d^2-d\\\\right)m.$$\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-11-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.37236/11948\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.37236/11948","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Albertson defined the irregularity of a graph $G$ as $$irr(G)=\sum\limits_{uv\in E(G)}|d_G(u)-d_G(v)|.$$ For a graph $G$ with $n$ vertices, $m$ edges, maximum degree $\Delta$, and $d=\left\lfloor \frac{\Delta m}{\Delta n-m}\right\rfloor$, we show $$irr(G)\leq d(d+1)n+\frac{1}{\Delta}\left(\Delta^2-(2d+1)\Delta-d^2-d\right)m.$$
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