The non-zero divisor graph of a ring

Abstract

Let R be a ring, we associate a simple graph Φ(R) to R, with vertices V(R) = R\{0, 1,−1}, where distinct vertices x, y ∈ V(R) are adjacent if and only if either xy ≠ 0 or yx ≠ 0.  In this paper, we prove that if Φ(R) is connected such that R \(\ncong\) Z₂ × Z₄ then the diameter of Φ(R) is almost 2.  Also, we will pay specific attention to investigate the connectivity of certain rings such that, the ring of integers modulo n, Z_n is connected, reduced ring and matrix ring.