On the Dot Product Graph of a Commutative Ring

Abstract

Let A be a commutative ring with nonzero identity, 1 ≤ n < ∞ be an integer, and R = A × A × … ×A (n times). The total dot product graph of R is the (undirected) graph TD(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 denote 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 TD(R) with vertices Z(R)* = Z(R)∖{(0, 0,…, 0)}. It follows that each edge (path) of the classical zero-divisor graph Γ(R) is an edge (path) of ZD(R). We observe that if n = 1, then TD(R) is a disconnected graph and ZD(R) is identical to the well-known zero-divisor graph of R in the sense of Beck–Anderson–Livingston, and hence it is connected. In this paper, we study both graphs TD(R) and ZD(R). For a commutative ring A and n ≥ 3, we show that TD(R) (ZD(R)) is connected with diameter two (at most three) and with girth three. Among other things, for n ≥ 2, we show that ZD(R) is identical to the zero-divisor graph of R if and only if either n = 2 and A is an integral domain or R is ring-isomorphic to ℤ2 × ℤ2 × ℤ2.