LAUSR

In this paper, we continue to study zero-divisor properties of skew polynomial rings R[x; α,δ], where R is an associative ring equipped with an endomorphism α and an α-derivation δ. For an associative ring R, the undirected zero-divisor graph of R is the graph Γ(R) such that the vertices of Γ(R) are all the nonzero zero-divisors of R and two distinct vertices x and y are connected by an edge if and only if xy = 0 or yx = 0. As an application of reversible rings, we investigate the interplay between the ring-theoretical properties of a skew polynomial ring R[x; α,δ] and the graph-theoretical properties of its zero-divisor graph Γ(R[x; α,δ]). Our goal in this paper is to give a characterization of the possible diameters of (Γ(R[x; α,δ])) in terms of the diameter of Γ(R), when the base ring R is reversible and also have the (α,δ)-compatible property. We also completely describe the associative rings all of whose zero-divisor graphs of skew polynomials are complete.