On the essential dot product graph of a commutative ring

Let B be a commutative ring with unity 16= 0, 1 <= m <infinity be an integer and R=BxBx<middle dot><middle dot><middle dot>xB(m times). The total essential dot product graph ET D(R)and the essential zero-divisor dot product graph EZD(R) are undirected graphs with the vertex sets R-& lowast;=R\{(0,0, . . . ,0)}and Z(R)(& lowast;)=Z(R)\{(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(B )(w<middle dot>z) is an essential ideal of B(where w<middle dot>z = w(1)z(1)+w(2)z(2)+<middle dot><middle dot><middle dot>+w(m)z(m)is an element of B).In this paper, we prove some results on connectedness, diameter and girth of ET D(R)and EZD(R). We classify the ring R such that EZD(R) and ET D(R) are planar, outerplanar, and of genus one.