{"title":"论特殊线性群上 Cayley 图的直径","authors":"Eitan Porat","doi":"arxiv-2409.06929","DOIUrl":null,"url":null,"abstract":"We prove for the matrix group $G=\\mathrm{SL}_{n}\\left(\\mathbb{F}_{p}\\right)$\nthat there exist absolute constants $c\\in\\left(0,1\\right)$ and $C>0$ such that\nany symmetric generating set $A$, with $\\left|A\\right|\\geq\\left|G\\right|^{1-c}$\nhas covering number $\\leq\nC\\left(\\log\\left(\\frac{\\left|G\\right|}{\\left|A\\right|}\\right)\\right)^{2}.$ This\nresult is sharp up to the value of the constant $C>0$.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"29 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Diameters of Cayley Graphs over Special Linear Groups\",\"authors\":\"Eitan Porat\",\"doi\":\"arxiv-2409.06929\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove for the matrix group $G=\\\\mathrm{SL}_{n}\\\\left(\\\\mathbb{F}_{p}\\\\right)$\\nthat there exist absolute constants $c\\\\in\\\\left(0,1\\\\right)$ and $C>0$ such that\\nany symmetric generating set $A$, with $\\\\left|A\\\\right|\\\\geq\\\\left|G\\\\right|^{1-c}$\\nhas covering number $\\\\leq\\nC\\\\left(\\\\log\\\\left(\\\\frac{\\\\left|G\\\\right|}{\\\\left|A\\\\right|}\\\\right)\\\\right)^{2}.$ This\\nresult is sharp up to the value of the constant $C>0$.\",\"PeriodicalId\":501407,\"journal\":{\"name\":\"arXiv - MATH - Combinatorics\",\"volume\":\"29 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Combinatorics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.06929\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Combinatorics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06929","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On Diameters of Cayley Graphs over Special Linear Groups
We prove for the matrix group $G=\mathrm{SL}_{n}\left(\mathbb{F}_{p}\right)$
that there exist absolute constants $c\in\left(0,1\right)$ and $C>0$ such that
any symmetric generating set $A$, with $\left|A\right|\geq\left|G\right|^{1-c}$
has covering number $\leq
C\left(\log\left(\frac{\left|G\right|}{\left|A\right|}\right)\right)^{2}.$ This
result is sharp up to the value of the constant $C>0$.
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