On the essential dot product graph of a commutative ring

Let $\mathcal{B}$ be a commutative ring with unity $1\neq 0$, $1\leq m \infty$ be an integer and $\mathcal{R}=\mathcal{B}\times \mathcal{B} \times\cdots\times \mathcal{B}$ ($m$ times). The total essential dot product graph $ETD(\mathcal{R})$ and the essential zero-divisor dot product graph $EZD(\mathcal{R})$ are undirected graphs with the vertex sets $\mathcal{R}^{*} = \mathcal{R}\setminus \{(0,0,...0)\}$ and $Z(\mathcal{R})^*=Z(\mathcal{R})\setminus \{(0,0,...,0)\}$ respectively. Two distinct vertices $w=(w_1,w_2,...,w_m)$ and $z=(z_1,z_2,...,z_m)$ are adjacent if and only if $ann_\mathcal{B}(w\cdot z)$ is an essential ideal of $\mathcal{B}$ (where $w\cdot z=w_1z_1+w_2z_2+\cdots +w_mz_m\in \mathcal{B}$). In this paper, we prove some results on connectedness, diameter and girth of $ETD(\mathcal{R})$ and $EZD(\mathcal{R})$. We classify the ring $\mathcal{R}$ such that $EZD(\mathcal{R})$ and $ETD(\mathcal{R})$ are planar, outerplanar, and of genus one.