{"title":"环拼接二部图的零性","authors":"Sarula Chang, Jianxi Li, Yirong Zheng","doi":"10.13001/ela.2023.7377","DOIUrl":null,"url":null,"abstract":"For a simple graph $G$, let $\\eta(G)$ and $c(G)$ be the nullity and the cyclomatic number of $G$, respectively. A cycle-spliced bipartite graph is a connected graph in which every block is an even cycle. It was shown by Wong et al. (2022) that for every cycle-spliced bipartite graph $G$, $0\\leq\\eta(G)\\leq c(G)+1$. Moreover, the extremal graphs with $\\eta(G) = c(G)+1$ and $\\eta(G) =0$, respectively, have been characterized. In this paper, we prove that there is no cycle-spliced bipartite graphs $G$ of any order with nullity $\\eta(G)=c(G)$. Furthermore, we also provide a structural characterization on cycle-spliced bipartite graphs $G$ with nullity $\\eta(G)=c(G)-1$.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2023-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Nullities of cycle-spliced bipartite graphs\",\"authors\":\"Sarula Chang, Jianxi Li, Yirong Zheng\",\"doi\":\"10.13001/ela.2023.7377\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For a simple graph $G$, let $\\\\eta(G)$ and $c(G)$ be the nullity and the cyclomatic number of $G$, respectively. A cycle-spliced bipartite graph is a connected graph in which every block is an even cycle. It was shown by Wong et al. (2022) that for every cycle-spliced bipartite graph $G$, $0\\\\leq\\\\eta(G)\\\\leq c(G)+1$. Moreover, the extremal graphs with $\\\\eta(G) = c(G)+1$ and $\\\\eta(G) =0$, respectively, have been characterized. In this paper, we prove that there is no cycle-spliced bipartite graphs $G$ of any order with nullity $\\\\eta(G)=c(G)$. Furthermore, we also provide a structural characterization on cycle-spliced bipartite graphs $G$ with nullity $\\\\eta(G)=c(G)-1$.\",\"PeriodicalId\":50540,\"journal\":{\"name\":\"Electronic Journal of Linear Algebra\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2023-06-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Electronic Journal of Linear Algebra\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.13001/ela.2023.7377\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Electronic Journal of Linear Algebra","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.13001/ela.2023.7377","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Mathematics","Score":null,"Total":0}
For a simple graph $G$, let $\eta(G)$ and $c(G)$ be the nullity and the cyclomatic number of $G$, respectively. A cycle-spliced bipartite graph is a connected graph in which every block is an even cycle. It was shown by Wong et al. (2022) that for every cycle-spliced bipartite graph $G$, $0\leq\eta(G)\leq c(G)+1$. Moreover, the extremal graphs with $\eta(G) = c(G)+1$ and $\eta(G) =0$, respectively, have been characterized. In this paper, we prove that there is no cycle-spliced bipartite graphs $G$ of any order with nullity $\eta(G)=c(G)$. Furthermore, we also provide a structural characterization on cycle-spliced bipartite graphs $G$ with nullity $\eta(G)=c(G)-1$.
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