ON THE NILPOTENT DOT PRODUCT GRAPH OF A COMMUTATIVE RING

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)}.

ON THE NILPOTENT DOT PRODUCT GRAPH OF A COMMUTATIVE RING Keywords:

Dot product graph , Nilpotent graph , Reduced ring