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

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 text2 or Formula: see text, where Formula: see text is the Jacobson radical of Formula: see text and Formula: see text is a finite ring.