论文标题
奇怪的统治场景的均等
Parity of an odd dominating set
论文作者
论文摘要
对于简单的图形$ g $，带有顶点set $ v（g）= \ {v_1，...，v_n \} $，我们将顶点$ u $的封闭邻域集定义为$ n [u] = \ {v \ {v \ in v（g）\; | \; v \; \ text {与} \相邻; u \; \ text {或} \; V = u \} $和封闭的邻里矩阵$ n（g）$作为矩阵通过设置为$ 1 $ $ g $的邻接矩阵的所有对角线条目获得的矩阵。我们说，如果$ n [u] \ cap s $对于v（g）$中的所有$ u \都是奇怪的，则$ s $是奇怪的。我们证明，$ g $的奇数统治集的奇偶校验等于$ g $等级的奇偶校验，其中$ g $的等级定义为$ n（g）$的列空间的尺寸。使用此结果，我们证明了其中一种推论，我们获得了图形连接无效的一般公式。
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 obtained by setting to $1$ all the diagonal entries of the adjacency matrix of $G$. 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 an odd dominating set of $G$ is equal to the parity of the rank of $G$, where the 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.
