On zero-divisor graphs of skew polynomial rings over non-commutative rings

Abstract

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