交换环的点积图的性质
Properties of the Dot Product Graph of a Commutative Ring
摘要：
设$R$是有单位元的交换环，$n\geq1$是整数.让$R^{n}=R\times\cdots\times R~(n~times)$。\textit{total dot product}图，用$TD(R,n)$表示，是以$R^{n}-\{(0,0,\ldots,0)\}$的元素为顶点的简单图，并且两个不同的顶点$\mathbf{x}$和$\mathbf{y}$相邻当且仅当$\mathbf{x} \cdot \mathbf{y}=0\in R$，其中$\mathbf{x} \cdot \mathbf{y}$表示$\mathbf{x}$和$\mathbf{y}$的点积。在本文中，我们找到了$TD(R\times S,n)$的结构相对于$TD(R,n)$和$TD(S,n)$的结构。此外，我们还求出了这个图的顶点的度。我们确定它何时是正常的。假设$\mathbb{F}$是一个有限域。证明了若$TD(\mathbb{F},n)\simeq TD(R,m)$，则$n=m$和$R\simeq\mathbb{F}$ .还提出了一些关于控制数的结果。此外，我们给出了$TD(R,n)$的团和独立数的一些结果。证明了环$R$是有限的当且仅当它的独立数是有限的。最后，我们将所有的平面图在这个类。
英文摘要：
Let $R$ be a commutative ring with identity and $n\geq1$ be an integer. Let $R^{n}=R\times\cdots\times R~(n~times)$. The \textit{total dot product} graph, denoted by $TD(R,n)$ is a simple graph with elements of $R^{n}-\{(0,0,\ldots,0)\}$ as vertices, and two distinct vertices $\mathbf{x}$ and $\mathbf{y}$ are adjacent if and only if $\mathbf{x} \cdot \mathbf{y}=0\in R$, where $\mathbf{x} \cdot \mathbf{y}$ denotes the dot product of $\mathbf{x}$ and $\mathbf{y}$. In this paper, we find the structure of $TD(R\times S,n)$ with respect to the structure of $TD(R,n)$ and $TD(S,n)$. In addition, we find the degree of vertices of this graph. We determine when it is regular. Let $\mathbb{F}$ be a finite field. It is shown that if $TD(\mathbb{F},n)\simeq TD(R,m)$, then $n=m$ and $R\simeq\mathbb{F}$. A number of results concerning the domination number are also presented. Furthermore, we give some results on the clique and the independence number of $TD(R,n)$. It is shown that the ring $R$ is finite if and only if its independence number is finite. Finally, we classify all planar graphs within this class.
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