{"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}
引用次数: 0
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 $$irr(G)\leq d(d+1)n+\frac{1}{\Delta}\left(\Delta^2-(2d+1)\Delta-d^2-d\right)m.$$