{"title":"具有交数组{27，20，7；1，4，21}的距离正则图不存在","authors":"K. S. Efimov, A. Makhnev","doi":"10.15826/umj.2020.2.006","DOIUrl":null,"url":null,"abstract":"In the class of distance-regular graphs of diameter 3 there are 5 intersection arrays of graphs with at most 28 vertices and noninteger eigenvalue. These arrays are \\(\\{18,14,5;1,2,14\\}\\), \\(\\{18,15,9;1,1,10\\}\\), \\(\\{21,16,10;1,2,12\\}\\), \\(\\{24,21,3;1,3,18\\}\\), and \\(\\{27,20,7;1,4,21\\}\\). Automorphisms of graphs with intersection arrays \\(\\{18,15,9;1,1,10\\}\\) and \\(\\{24,21,3;1,3,18\\}\\) were found earlier by A.A. Makhnev and D.V. Paduchikh. In this paper, it is proved that a graph with the intersection array \\(\\{27,20,7;1,4,21\\}\\) does not exist.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2020-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"DISTANCE-REGULAR GRAPH WITH INTERSECTION ARRAY {27, 20, 7; 1, 4, 21} DOES NOT EXIST\",\"authors\":\"K. S. Efimov, A. Makhnev\",\"doi\":\"10.15826/umj.2020.2.006\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In the class of distance-regular graphs of diameter 3 there are 5 intersection arrays of graphs with at most 28 vertices and noninteger eigenvalue. These arrays are \\\\(\\\\{18,14,5;1,2,14\\\\}\\\\), \\\\(\\\\{18,15,9;1,1,10\\\\}\\\\), \\\\(\\\\{21,16,10;1,2,12\\\\}\\\\), \\\\(\\\\{24,21,3;1,3,18\\\\}\\\\), and \\\\(\\\\{27,20,7;1,4,21\\\\}\\\\). Automorphisms of graphs with intersection arrays \\\\(\\\\{18,15,9;1,1,10\\\\}\\\\) and \\\\(\\\\{24,21,3;1,3,18\\\\}\\\\) were found earlier by A.A. Makhnev and D.V. Paduchikh. In this paper, it is proved that a graph with the intersection array \\\\(\\\\{27,20,7;1,4,21\\\\}\\\\) does not exist.\",\"PeriodicalId\":36805,\"journal\":{\"name\":\"Ural Mathematical Journal\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-12-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Ural Mathematical Journal\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.15826/umj.2020.2.006\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Ural Mathematical Journal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.15826/umj.2020.2.006","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
DISTANCE-REGULAR GRAPH WITH INTERSECTION ARRAY {27, 20, 7; 1, 4, 21} DOES NOT EXIST
In the class of distance-regular graphs of diameter 3 there are 5 intersection arrays of graphs with at most 28 vertices and noninteger eigenvalue. These arrays are \(\{18,14,5;1,2,14\}\), \(\{18,15,9;1,1,10\}\), \(\{21,16,10;1,2,12\}\), \(\{24,21,3;1,3,18\}\), and \(\{27,20,7;1,4,21\}\). Automorphisms of graphs with intersection arrays \(\{18,15,9;1,1,10\}\) and \(\{24,21,3;1,3,18\}\) were found earlier by A.A. Makhnev and D.V. Paduchikh. In this paper, it is proved that a graph with the intersection array \(\{27,20,7;1,4,21\}\) does not exist.
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