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

Let $R$ be a ring with identity. The unit (respectively, unitary Cayley) graph of $R$ is a simple graph $G(R)$ (respectively, $\mathrm{\Gamma }(R)$) with vertex set $R$, where two distinct vertices $x$ and $y$ are adjacent if and only if $x+y$ (respectively, $x-y$) is a unit of $R$. In this paper, we explore when $G(R)$ and $\mathrm{\Gamma }(R)$ are isomorphic for a finite ring $R$. Among other results, we prove that $G(R)$ is isomorphic to $\mathrm{\Gamma }(R)$ for a finite ring $R$ if and only if char($R/J(R)$)=2 or $R/J(R)\cong {\mathbb{Z}}_2\times S$, where $J(R)$ is the Jacobson radical of $R$ and $S$ is a finite ring.