The Wiener index of the zero-divisor graph of a finite commutative ring with unity

Abstract

Let R be an arbitrary finite commutative ring with unity. The zero-divisor graph of R, denoted by Γ(R), is a graph with vertex set non-zero zero-divisors of R and two of them are connected by an edge if their product is zero. In this paper, we derive a formula for the Wiener index of the graph Γ(R). In the literature, the Wiener index of the graph Γ(R) is known only for R=Zn, the ring of integers modulo n. As applications of our formula, the Wiener index of Γ(R) is explicitly calculated when (i) R is a reduced ring, (ii) R is the ring of integers modulo n, and (iii) more generally R is the product of ring of integers modulo n. The Wiener index of the zero-divisor graph of the ring of Gaussian integers over Zn is also discussed.