Properties of the Dot Product Graph of a Commutative Ring

TL;DR

This paper studies the properties of the total dot product graph of a commutative ring, exploring its structure, regularity, and parameters like domination, clique, and independence numbers, and classifies when such graphs are planar.

Contribution

It characterizes the structure of total dot product graphs for product rings, determines vertex degrees and regularity conditions, and classifies planar graphs within this class.

Findings

The structure of $TD(R imes S, n)$ relates to $TD(R, n)$ and $TD(S, n)$.

Conditions for regularity of $TD(R, n)$ are established.

Finite rings correspond to finite independence numbers, and planar graphs are classified.

Abstract

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.