On the dot product graph of a commutative ring II

Abstract

In 2015, the second-named author introduced the dot product graph associated to a commutative ring A. Let A be a commutative ring with nonzero identity, 1 ≤ n < ∞ be an integer, and R = A × A × · · · × A (n times). We recall that the total dot product graph of R is the (undirected) graph T D(R) with vertices R∗ = R \ {(0, 0, . . . , 0)}, and two distinct vertices x and y are adjacent if and only if x·y = 0 ∈ A (where x·y denotes the normal dot product of x and y). Let Z(R) denote the set of all zero-divisors of R. Then the zero-divisor dot product graph of R is the induced subgraph ZD(R) of T D(R) with vertices Z(R)∗ = Z(R) \ {(0, 0, . . . , 0)}. Let U(R) denote the set of all units of R. Then the unit dot product graph of R is the induced subgraph UD(R) of T D(R) with vertices U(R). In this paper, we study the structure of T D(R), UD(R), and ZD(R) when A = Zn or A = GF(pn), the finite field with pn elements, where n ≥ 2 and p is a prime positive integer.