ON THE NILPOTENT DOT PRODUCT GRAPH OF A COMMUTATIVE RING

Abstract

Let $\mathscr{B}$ be a commutative ring with $1\neq 0$, $1\leq m<\infty$ be an integer and $\mathcal{R}=\mathscr{B}\times \mathscr{B}\times \cdot \cdot \cdot \times \mathscr{B}$ ($m$ times). In this paper, we introduce two types of (undirected) graphs, total nilpotent dot product graph denoted by $\mathcal{T_{N}D(\mathcal{R})}$ and nilpotent dot product graph denoted by $\mathcal{Z_ND(\mathcal{R})}$, in which vertices are from $\mathcal{R}^\ast = \mathcal{R}\setminus \{(0,0,...,0)\}$ and $\mathcal{Z_{N}(\mathcal{R})}^*$ respectively, where $\mathcal{Z_{N}(\mathcal{R})}^{*}=\{w\in \mathcal{R}^*| wz\in \mathcal{N(R)}, \mbox{for some }z\in \mathcal{R}^*\} $. Two distinct vertices $w=(w_1,w_2,...,w_m)$ and $z=(z_1,z_2,...,z_m)$ are said to be adjacent if and only if $w\cdot z\in \mathcal{N}(\mathscr{B})$ (where $w\cdot z=w_1z_1+\cdots+w_mz_m$, denotes the normal dot product and $\mathcal{N}(\mathscr{B})$ is the set of nilpotent elements of $\mathscr{B}$). We study about connectedness, diameter and girth of the graphs $\mathcal{T_ND(R)}$ and $\mathcal{Z_ND(R)}$. Finally, we establish the relationship between $\mathcal{T_ND(R)}$, $\mathcal{Z_ND(R)}$, $\mathcal{TD(R)}$ and $\mathcal{ZD(R)}$.