Properties of the Dot Product Graph of a Commutative Ring

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.