on the nilpotent dot product graph of a commutative ring

Let b be a commutative ring with 1 ̸= 0, 1 ≤ m < ∞ be an integer and r = b×b×···×b (m times). in this paper, we introduce two types of (undirected) graphs, total nilpotent dot product graph denoted by tn d(r) and nilpotent dot product graph denoted by zn d(r), in which vertices are from r∗ = r{(0, 0, ..., 0)} and zn (r) ∗ respectively, where zn (r) ∗ = {w ∈ r∗ |wz ∈ n (r), for some z ∈ r∗}. two distinct vertices w = (w1, w2, ..., wm) and z = (z1, z2, ..., zm) are said to be adjacent if and only if w · z ∈ n (b) (where w · z = w1z1 + · · · + wmzm, denotes the normal dot product and n (b) is the set of nilpotent elements of b). we study about connectedness, diameter and girth of the graphs tn d(r) and zn d(r). finally, we establish the relationship between tn d(r), zn d(r), t d(r) and zd(r).