{"title":"关于超立方体的平方图的一些注释","authors":"S. Mirafzal","doi":"10.26493/1855-3974.2621.26f","DOIUrl":null,"url":null,"abstract":"Let $\\Gamma=(V,E)$ be a graph. The square graph $\\Gamma^2$ of the graph $\\Gamma$ is the graph with the vertex set $V(\\Gamma^2)=V$ in which two vertices are adjacent if and only if their distance in $\\Gamma$ is at most two. The square graph of the hypercube $Q_n$ has some interesting properties. For instance, it is highly symmetric and panconnected. In this paper, we investigate some algebraic properties of the graph ${Q^2_n}$. In particular, we show that the graph ${Q^2_n}$ is distance-transitive. We show that the graph ${Q^2_n}$ is an imprimitive distance-transitive graph if and only if $n$ is an odd integer. Also, we determine the spectrum of the graph $Q_n^2$. Finally, we show that when $n>2$ is an even integer, then ${Q^2_n}$ is an automorphic graph, that is, $Q_n^2$ is a distance-transitive primitive graph which is not a complete or a line graph.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2021-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Some remarks on the square graph of the hypercube\",\"authors\":\"S. Mirafzal\",\"doi\":\"10.26493/1855-3974.2621.26f\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $\\\\Gamma=(V,E)$ be a graph. The square graph $\\\\Gamma^2$ of the graph $\\\\Gamma$ is the graph with the vertex set $V(\\\\Gamma^2)=V$ in which two vertices are adjacent if and only if their distance in $\\\\Gamma$ is at most two. The square graph of the hypercube $Q_n$ has some interesting properties. For instance, it is highly symmetric and panconnected. In this paper, we investigate some algebraic properties of the graph ${Q^2_n}$. In particular, we show that the graph ${Q^2_n}$ is distance-transitive. We show that the graph ${Q^2_n}$ is an imprimitive distance-transitive graph if and only if $n$ is an odd integer. Also, we determine the spectrum of the graph $Q_n^2$. Finally, we show that when $n>2$ is an even integer, then ${Q^2_n}$ is an automorphic graph, that is, $Q_n^2$ is a distance-transitive primitive graph which is not a complete or a line graph.\",\"PeriodicalId\":8402,\"journal\":{\"name\":\"Ars Math. Contemp.\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-01-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Ars Math. Contemp.\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.26493/1855-3974.2621.26f\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Ars Math. Contemp.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.26493/1855-3974.2621.26f","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Let $\Gamma=(V,E)$ be a graph. The square graph $\Gamma^2$ of the graph $\Gamma$ is the graph with the vertex set $V(\Gamma^2)=V$ in which two vertices are adjacent if and only if their distance in $\Gamma$ is at most two. The square graph of the hypercube $Q_n$ has some interesting properties. For instance, it is highly symmetric and panconnected. In this paper, we investigate some algebraic properties of the graph ${Q^2_n}$. In particular, we show that the graph ${Q^2_n}$ is distance-transitive. We show that the graph ${Q^2_n}$ is an imprimitive distance-transitive graph if and only if $n$ is an odd integer. Also, we determine the spectrum of the graph $Q_n^2$. Finally, we show that when $n>2$ is an even integer, then ${Q^2_n}$ is an automorphic graph, that is, $Q_n^2$ is a distance-transitive primitive graph which is not a complete or a line graph.