When unit graphs are isomorphic to unitary Cayley graphs of rings?

Abstract

Let [Formula: see text] be a ring with identity. The unit (respectively, unitary Cayley) graph of [Formula: see text] is a simple graph [Formula: see text] (respectively, [Formula: see text]) with vertex set [Formula: see text], where two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] (respectively, [Formula: see text]) is a unit of [Formula: see text]. In this paper, we explore when [Formula: see text] and [Formula: see text] are isomorphic for a finite ring [Formula: see text]. Among other results, we prove that [Formula: see text] is isomorphic to [Formula: see text] for a finite ring [Formula: see text] if and only if char([Formula: see text])[Formula: see text]=[Formula: see text]2 or [Formula: see text], where [Formula: see text] is the Jacobson radical of [Formula: see text] and [Formula: see text] is a finite ring.