{"title":"奇支配集的奇偶性","authors":"A. Batal","doi":"10.31801/cfsuasmas.1051208","DOIUrl":null,"url":null,"abstract":"For a simple graph $G$ with vertex set $V(G)=\\{v_1,...,v_n\\}$, we define the closed neighborhood set of a vertex $u$ as \\\\$N[u]=\\{v \\in V(G) \\; | \\; v \\; \\text{is adjacent to} \\; u \\; \\text{or} \\; v=u \\}$ and the closed neighborhood matrix $N(G)$ as the matrix whose $i$th column is the characteristic vector of $N[v_i]$. We say a set $S$ is odd dominating if $N[u]\\cap S$ is odd for all $u\\in V(G)$. We prove that the parity of the cardinality of an odd dominating set of $G$ is equal to the parity of the rank of $G$, where rank of $G$ is defined as the dimension of the column space of $N(G)$. Using this result we prove several corollaries in one of which we obtain a general formula for the nullity of the join of graphs.","PeriodicalId":44692,"journal":{"name":"Communications Faculty of Sciences University of Ankara-Series A1 Mathematics and Statistics","volume":" ","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Parity of an odd dominating set\",\"authors\":\"A. Batal\",\"doi\":\"10.31801/cfsuasmas.1051208\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For a simple graph $G$ with vertex set $V(G)=\\\\{v_1,...,v_n\\\\}$, we define the closed neighborhood set of a vertex $u$ as \\\\\\\\$N[u]=\\\\{v \\\\in V(G) \\\\; | \\\\; v \\\\; \\\\text{is adjacent to} \\\\; u \\\\; \\\\text{or} \\\\; v=u \\\\}$ and the closed neighborhood matrix $N(G)$ as the matrix whose $i$th column is the characteristic vector of $N[v_i]$. We say a set $S$ is odd dominating if $N[u]\\\\cap S$ is odd for all $u\\\\in V(G)$. We prove that the parity of the cardinality of an odd dominating set of $G$ is equal to the parity of the rank of $G$, where rank of $G$ is defined as the dimension of the column space of $N(G)$. Using this result we prove several corollaries in one of which we obtain a general formula for the nullity of the join of graphs.\",\"PeriodicalId\":44692,\"journal\":{\"name\":\"Communications Faculty of Sciences University of Ankara-Series A1 Mathematics and Statistics\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2022-12-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications Faculty of Sciences University of Ankara-Series A1 Mathematics and Statistics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.31801/cfsuasmas.1051208\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications Faculty of Sciences University of Ankara-Series A1 Mathematics and Statistics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.31801/cfsuasmas.1051208","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

对于顶点集为$V(G)=\{V_1,…,V_n\}$的简单图$G$,我们将顶点$u$的闭邻域集定义为\ \$n[u]=\{V\in V(G;u\;\text{or}\;v=u\}$和闭邻域矩阵$N(G)$作为其第$i$列是$N[v_i]$的特征向量的矩阵。如果$N[u]\cap S$对V(G)$中的所有$u\都是奇数,则我们说集合$S$是奇数支配。我们证明了$G$的奇支配集的基数的奇偶性等于$G$秩的奇偶性,其中$G$阶被定义为$N(G)$的列空间的维数。利用这一结果,我们证明了几个推论,其中一个推论得到了图连接零度的一般公式。
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Parity of an odd dominating set
For a simple graph $G$ with vertex set $V(G)=\{v_1,...,v_n\}$, we define the closed neighborhood set of a vertex $u$ as \\$N[u]=\{v \in V(G) \; | \; v \; \text{is adjacent to} \; u \; \text{or} \; v=u \}$ and the closed neighborhood matrix $N(G)$ as the matrix whose $i$th column is the characteristic vector of $N[v_i]$. We say a set $S$ is odd dominating if $N[u]\cap S$ is odd for all $u\in V(G)$. We prove that the parity of the cardinality of an odd dominating set of $G$ is equal to the parity of the rank of $G$, where rank of $G$ is defined as the dimension of the column space of $N(G)$. Using this result we prove several corollaries in one of which we obtain a general formula for the nullity of the join of graphs.
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